{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R705",
  "content_hash": null,
  "slug": "sbmrlc-attempt-formula-convention-audit",
  "type": "attempt",
  "title": "The literature convention matches unrestricted binary diagonals",
  "summary": "MacWilliams's formula, its later transcription, row totals, and direct enumeration all identify the intended matrix family.",
  "relevance": "For Rank log-concavity for symmetric binary matrices through order fifty, record sbmrlc-attempt-formula-convention-audit (“The literature convention matches unrestricted binary diagonals”) documents a concrete method, search boundary, or failed route. The record states: MacWilliams's formula, its later transcription, row totals, and direct enumeration all identify the intended matrix family.",
  "relevance_source": "recorded",
  "body": "MacWilliams's Theorem 2 is the source of the fixed-rank count. The 2011 exposition reproduces the formula as Equation (4.5) and defines \\(\\operatorname{sym}(n,r)\\) as the unrestricted symmetric count. Its separate notation \\(\\operatorname{sym}_0(n,r)\\) imposes a zero diagonal, so the two families cannot be confused in the displayed formula.\n\nAt \\(q=2\\), exact evaluation gives the six rank vectors stated in the candidate record. Their totals are \\(2^1,2^3,2^6,2^{10},2^{15},2^{21}\\), as required when the upper triangle, including every diagonal entry, is free. The independent enumeration in the artifact checks every one of the 2,131,018 matrices in these six orders. This resolves the characteristic-two convention before the formula is used through order 50.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the rank-count formula and diagonal convention needed for symmetric binary matrices of orders 1 through 50",
    "bounds": {
      "matrix_order": {
        "min": 1,
        "max": 50
      },
      "publication_year": {
        "min": 1969,
        "max": 2011
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://www.intlpress.com/site/pub/files/_fulltext/journals/joc/2011/0002/0003/JOC-2011-0002-0003-a002.pdf",
      "locator": "Joel Brewster Lewis, Ricky Ini Liu, Alejandro H. Morales, Greta Panova, Steven V. Sam, and Yan X. Zhang, Matrices with Restricted Entries and q-Analogues of Permutations, Journal of Combinatorics 2(3) (2011), Equation (4.5), Remark 4.1, and the definitions preceding Proposition 4.12; the formula is attributed there to MacWilliams, Theorem 2"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.intlpress.com/site/pub/files/_fulltext/journals/joc/2011/0002/0003/JOC-2011-0002-0003-a002.pdf",
    "locator": "Joel Brewster Lewis, Ricky Ini Liu, Alejandro H. Morales, Greta Panova, Steven V. Sam, and Yan X. Zhang, Matrices with Restricted Entries and q-Analogues of Permutations, Journal of Combinatorics 2(3) (2011), Equation (4.5), Remark 4.1, and the definitions preceding Proposition 4.12; the formula is attributed there to MacWilliams, Theorem 2"
  },
  "relations": [
    {
      "slug": "R706",
      "title": "MacWilliams's product formula gives every rank count",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "symmetric-binary-matrix-rank-log-concavity",
      "title": "symmetric binary matrix rank log concavity",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}