{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R707",
  "content_hash": null,
  "slug": "sbmrlc-claim-strict-log-concavity",
  "type": "claim",
  "title": "The symmetric binary rank distribution is strictly log-concave",
  "summary": "Adjacent quotients from the exact rank formula decrease strictly, proving every requested inequality and the same result in all orders.",
  "relevance": "For Rank log-concavity for symmetric binary matrices through order fifty, record sbmrlc-claim-strict-log-concavity (“The symmetric binary rank distribution is strictly log-concave”) records a bound, answer, status fact, or structural consequence. The record states: Adjacent quotients from the exact rank formula decrease strictly, proving every requested inequality and the same result in all orders.",
  "relevance_source": "recorded",
  "body": "Let \\(R_{n,r}\\) count symmetric \\(n\\times n\\) matrices over \\(\\mathbb F_2\\) of rank \\(r\\), with unrestricted diagonal. Put \\(Q_{n,r}=R_{n,r+1}/R_{n,r}\\) for \\(0\\leq r<n\\). Substitution in the MacWilliams formula gives\n\\[\nQ_{n,2s}=2^{n-2s}-1\n\\]\nand\n\\[\nQ_{n,2s+1}=\\frac{2^{2s+2}}{2^{2s+2}-1}\\left(2^{n-2s-1}-1\\right).\n\\]\nThese quotients decrease strictly. At an even internal rank \\(2s\\), the preceding quotient has both a larger power-of-two factor and a multiplier greater than one:\n\\[\nQ_{n,2s-1}=\\frac{2^{2s}}{2^{2s}-1}\\left(2^{n-2s+1}-1\\right)>2^{n-2s}-1=Q_{n,2s}.\n\\]\nAt an odd internal rank \\(2s+1\\), write \\(a=n-2s-1\\geq1\\). Since \\(2^{2s+2}/(2^{2s+2}-1)\\leq4/3\\),\n\\[\nQ_{n,2s+1}\\leq\\frac43(2^a-1)<2^{a+1}-1=Q_{n,2s}.\n\\]\nThus \\(Q_{n,r-1}>Q_{n,r}\\), which is equivalent to\n\\[\nR_{n,r}^2>R_{n,r-1}R_{n,r+1}\n\\]\nfor every \\(n\\geq2\\) and \\(1\\leq r<n\\). In particular, all 1,225 inequalities requested for \\(1\\leq n\\leq50\\) hold strictly.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "universal",
    "statement": "every positive matrix order n over F_2, with all diagonal entries unrestricted"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1080/00029890.1969.12000160",
      "locator": "Exact adjacent-quotient argument from the audited MacWilliams formula, replayed in sbmrlc-artifact-exact-sweep"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1080/00029890.1969.12000160",
    "locator": "Exact adjacent-quotient argument from the audited MacWilliams formula, replayed in sbmrlc-artifact-exact-sweep"
  },
  "relations": [
    {
      "slug": "R706",
      "title": "MacWilliams's product formula gives every rank count",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R704",
      "title": "Replayable exact rank and log-concavity sweep",
      "object_type": "artifact",
      "relation": "reproduces",
      "direction": "incoming"
    },
    {
      "slug": "symmetric-binary-matrix-rank-log-concavity",
      "title": "symmetric binary matrix rank log concavity",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}