{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R594",
  "content_hash": null,
  "slug": "pp12-claim-bruck-ryser-silent",
  "type": "claim",
  "title": "Bruck-Ryser gives no obstruction at order 12",
  "summary": "The theorem's sum-of-two-squares condition applies to orders congruent to 1 or 2 modulo 4, while 12 is congruent to 0.",
  "relevance": "For A projective plane of order 12, record pp12-claim-bruck-ryser-silent (“Bruck-Ryser gives no obstruction at order 12”) records a bound, answer, status fact, or structural consequence. The record states: The theorem's sum-of-two-squares condition applies to orders congruent to 1 or 2 modulo 4, while 12 is congruent to 0.",
  "relevance_source": "recorded",
  "body": "Bruck and Ryser prove that if a projective plane of order \\(n\\) exists and \\(n\\equiv1\\) or \\(2\\pmod4\\), then \\(n\\) must be a sum of two integer squares. Since\n\\[\n12\\equiv0\\pmod4,\n\\]\nthe theorem is silent here.\n\nThe elementary determinant condition is also consistent. If \\(B\\) is a \\(157\\times157\\) incidence matrix, then\n\\[\nBB^{\\mathsf T}=12I_{157}+J_{157}.\n\\]\nThe eigenvalues on the right are \\(169=13^2\\) once and 12 with multiplicity 156. Hence\n\\[\n\\det(B)^2=13^2\\,12^{156}\n\\quad\\text{and}\\quad\n|\\det(B)|=13\\,12^{78}.\n\\]\nThis is an integer square identity, so it supplies no contradiction. Any order-12 exclusion needs information beyond this arithmetic test.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the projective-plane form of the Bruck-Ryser theorem and the incidence determinant identity at order 12",
    "bounds": {
      "order": {
        "min": 12,
        "max": 12
      },
      "matrix_dimension": {
        "min": 157,
        "max": 157
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.4153/CJM-1949-009-2",
      "locator": "R. H. Bruck and H. J. Ryser, The Nonexistence of Certain Finite Projective Planes, Canadian Journal of Mathematics 1 (1949), 88-93, Theorem 1; order-12 substitution and determinant calculation replayed in pp12-artifact-model-audit"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4153/CJM-1949-009-2",
    "locator": "R. H. Bruck and H. J. Ryser, The Nonexistence of Certain Finite Projective Planes, Canadian Journal of Mathematics 1 (1949), 88-93, Theorem 1; order-12 substitution and determinant calculation replayed in pp12-artifact-model-audit"
  },
  "models": [],
  "relations": [
    {
      "slug": "R597",
      "title": "Existence at order 12 remains open",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R592",
      "title": "Exact OA and 11-MOLS computational formulation",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "projective-plane-order-12",
      "title": "projective plane order 12",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}