{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R637",
  "content_hash": null,
  "slug": "r2s6-claim-m12-exact-above-half",
  "type": "claim",
  "title": "The exact satisfiability probability at twelve clauses exceeds one half",
  "summary": "At \\(m=12\\), the exact satisfiability probability is \\(805717285720/\\binom{60}{12}>1/2\\); a seeded simulation places \\(m=13\\) below \\(1/2\\), while the exact thirteen-clause coefficient remains unextracted, so whether 13 is the first below-half value remains open.",
  "relevance": "For Median satisfiability threshold for a six-variable clause set, record r2s6-claim-m12-exact-above-half (“The exact satisfiability probability at twelve clauses exceeds one half”) records a bound, answer, status fact, or structural consequence. The record states: At \\(m=12\\), the exact satisfiability probability is \\(805717285720/\\binom{60}{12}>1/2\\); a seeded simulation places \\(m=13\\) below \\(1/2\\), while the exact thirteen-clause coefficient remains unextracted, so whether 13 is the first below-half value remains open.",
  "relevance_source": "recorded",
  "body": "Dovgal, de Panafieu, and Ravelomanana count satisfiable 2-CNF formulas by labeled variables and distinct clauses. Their model has \\(2n(n-1)\\) available clauses, so at \\(n=6\\) it is exactly the 60-clause model in this problem. Table 5.3 of the published paper, Table 2 in the arXiv version, gives\n\\[\na_{6,12}=805{,}717{,}285{,}720.\n\\]\nThe denominator is\n\\[\n\\binom{60}{12}=1{,}399{,}358{,}844{,}975.\n\\]\nTherefore\n\\[\nP_{12}=\\frac{805{,}717{,}285{,}720}{1{,}399{,}358{,}844{,}975}\n=\\frac{5{,}556{,}670{,}936}{9{,}650{,}750{,}655}\n\\approx0.5757760338695643.\n\\]\nThe exact comparison is \\(2a_{6,12}-\\binom{60}{12}=212{,}075{,}726{,}465>0\\). This settles the lower side of the candidate's proposed crossing.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "all 12-element subsets of the 60 non-tautological two-variable clauses on six labeled Boolean variables",
    "bounds": {
      "variables": {
        "min": 6,
        "max": 6
      },
      "clauses": {
        "min": 12,
        "max": 12
      },
      "possible_clauses": {
        "min": 60,
        "max": 60
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.5070/C63261985",
      "locator": "Sergey Dovgal, Elie de Panafieu, and Vlady Ravelomanana, Exact enumeration of satisfiable 2-SAT formulae, Combinatorial Theory 3(2), 2023, Theorem 4.7 and Table 5.3; arXiv:2108.08067v2, Table 2"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.5070/C63261985",
    "locator": "Sergey Dovgal, Elie de Panafieu, and Vlady Ravelomanana, Exact enumeration of satisfiable 2-SAT formulae, Combinatorial Theory 3(2), 2023, Theorem 4.7 and Table 5.3; arXiv:2108.08067v2, Table 2"
  },
  "relations": [
    {
      "slug": "R636",
      "title": "The exact thirteen-clause coefficient remains to be extracted",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "random-two-sat-six-median",
      "title": "random two sat six median",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}