{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R84",
  "content_hash": null,
  "slug": "bst20-claim-exact-maximum",
  "type": "claim",
  "title": "The order-20 maximum is 23,003,136",
  "summary": "Exhaustive exact enumeration gives D_20 = 23,003,136, attained by one 19-bit first row.",
  "relevance": "For Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20, record bst20-claim-exact-maximum (“The order-20 maximum is 23,003,136”) records a bound, answer, status fact, or structural consequence. The record states: Exhaustive exact enumeration gives D_20 = 23,003,136, attained by one 19-bit first row.",
  "relevance_source": "recorded",
  "body": "Let \\(T(t)\\) have entries \\(T_{ij}=t_{|i-j|}\\), where \\(t_0=0\\) and \\(t_1,\\ldots,t_{19}\\in\\{0,1\\}\\). Exhaustive enumeration of all \\(2^{19}=524288\\) bit strings gives\n\\[\nD_{20}=23003136=2^{16}\\,3^3\\,13.\n\\]\nThe unique maximizing string, written as \\(t_1t_2\\cdots t_{19}\\), is\n\\[\n1011001111111001101.\n\\]\nIts matrix determinant is \\(-23003136\\). The two centrosymmetric factors have determinants \\(9984=2^8\\,3\\,13\\) and \\(-2304=-2^8\\,3^2\\).\n\nThe maximizing string is palindromic, so reversal gives the same row. Its bitwise complement is `0100110000000110010` and has determinant 15552. Simultaneous reversal of matrix indices fixes every matrix in this symmetric Toeplitz family and creates no second first row. The exhaustive count of maximizers is one.",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all 524288 zero-diagonal binary symmetric Toeplitz matrices of order 20",
    "bounds": {
      "n": {
        "min": 20,
        "max": 20
      },
      "mask": {
        "min": 0,
        "max": 524287
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/0024-3795(76)90101-4",
      "locator": "Exhaustive certificate in bst20-artifact-exhaustive-certificate, executed independently on 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/0024-3795(76)90101-4",
    "locator": "Exhaustive certificate in bst20-artifact-exhaustive-certificate, executed independently on 2026-07-24"
  },
  "relations": [
    {
      "slug": "R82",
      "title": "Exact exhaustive certificate for D_20",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R83",
      "title": "Each determinant splits into two order-10 factors",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "binary-symmetric-toeplitz-maxdet-20",
      "title": "binary symmetric toeplitz maxdet 20",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}