{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R636",
  "content_hash": null,
  "slug": "r2s6-attempt-exact-m13-audit",
  "type": "attempt",
  "title": "The exact thirteen-clause coefficient remains to be extracted",
  "summary": "The published formula covers this coefficient, while its printed table stops at twelve clauses.",
  "relevance": "For Median satisfiability threshold for a six-variable clause set, record r2s6-attempt-exact-m13-audit (“The exact thirteen-clause coefficient remains to be extracted”) documents a concrete method, search boundary, or failed route. The record states: The published formula covers this coefficient, while its printed table stops at twelve clauses.",
  "relevance_source": "recorded",
  "body": "Theorem 4.7 of Dovgal, de Panafieu, and Ravelomanana gives an exact bivariate coefficient formula for the number \\(a_{n,m}\\) of satisfiable 2-CNFs with \\(n\\) variables and \\(m\\) distinct clauses. Their printed small-value table reaches \\(m=12\\), one column short of the coefficient needed here. The companion repository supplies Python code for bivariate formal power series and 2-SAT generating functions.\n\nThe missing denominator is\n\\[\n\\binom{60}{13}=5{,}166{,}863{,}427{,}600.\n\\]\nTo complete the problem, evaluate the published recurrence at \\((n,m)=(6,13)\\), record the integer \\(a_{6,13}\\), and verify\n\\[\n2a_{6,13}<5{,}166{,}863{,}427{,}600.\n\\]\nA useful independent check is an orbit-weighted enumeration under signed variable permutations, testing each representative by strongly connected components of its implication graph. The exact \\(m=12\\) table value should be reproduced before accepting the \\(m=13\\) output.\n\nThe literature search and the local simulation support the proposed answer 13. The exact inequality at 13 has not been reproduced in this fixture, so the headline question stays open.",
  "status": "inconclusive",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the exact number of satisfiable 13-element subsets of the 60 non-tautological two-variable clauses on six labeled variables",
    "bounds": {
      "variables": {
        "min": 6,
        "max": 6
      },
      "clauses": {
        "min": 13,
        "max": 13
      },
      "all_clause_sets": {
        "min": 5166863427600,
        "max": 5166863427600
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.5070/C63261985",
      "locator": "Dovgal, de Panafieu, and Ravelomanana, Theorem 4.7 and Table 5.3; companion code at GitLab project enumeration-2sat-aux, commit 346079fa"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.5070/C63261985",
    "locator": "Dovgal, de Panafieu, and Ravelomanana, Theorem 4.7 and Table 5.3; companion code at GitLab project enumeration-2sat-aux, commit 346079fa"
  },
  "relations": [
    {
      "slug": "R637",
      "title": "The exact satisfiability probability at twelve clauses exceeds one half",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R638",
      "title": "Satisfiability probability decreases with the number of clauses",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R635",
      "title": "Seeded simulation independently places the crossing between twelve and thirteen",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "random-two-sat-six-median",
      "title": "random two sat six median",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}