{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R83",
  "content_hash": null,
  "slug": "bst20-claim-centrosymmetric-factorization",
  "type": "claim",
  "title": "Each determinant splits into two order-10 factors",
  "summary": "Centrosymmetry reduces every order-20 determinant to the product of two exact order-10 determinants.",
  "relevance": "For Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20, record bst20-claim-centrosymmetric-factorization (“Each determinant splits into two order-10 factors”) records a bound, answer, status fact, or structural consequence. The record states: Centrosymmetry reduces every order-20 determinant to the product of two exact order-10 determinants.",
  "relevance_source": "recorded",
  "body": "Index rows and columns from 0. Let \\(B_{ij}=t_{|i-j|}\\) and \\(H_{ij}=t_{19-i-j}\\) for \\(0\\leq i,j<10\\). Reverse the final ten basis vectors. This changes \\(T\\) by a permutation similarity and gives\n\\[\n\\begin{pmatrix}B&H\\\\H&B\\end{pmatrix}.\n\\]\nThe symmetric and skew-symmetric subspaces for the exchange matrix then give the block diagonal form\n\\[\n(B+H)\\oplus(B-H).\n\\]\nConsequently\n\\[\n\\det T=\\det(B+H)\\det(B-H).\n\\]\nThis is the even-order specialization of the standard decomposition for symmetric centrosymmetric matrices. The displayed change of basis also proves the identity directly.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "every zero-diagonal binary symmetric Toeplitz matrix of order 20",
    "bounds": {
      "n": {
        "min": 20,
        "max": 20
      }
    },
    "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": "A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra and its Applications 13 (1976), 275-288, Theorem 2 and the even-order symmetric/skew-symmetric decomposition"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/0024-3795(76)90101-4",
    "locator": "A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra and its Applications 13 (1976), 275-288, Theorem 2 and the even-order symmetric/skew-symmetric decomposition"
  },
  "relations": [
    {
      "slug": "R82",
      "title": "Exact exhaustive certificate for D_20",
      "object_type": "artifact",
      "relation": "enables",
      "direction": "outgoing"
    },
    {
      "slug": "R84",
      "title": "The order-20 maximum is 23,003,136",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "binary-symmetric-toeplitz-maxdet-20",
      "title": "binary symmetric toeplitz maxdet 20",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}