{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R380",
  "content_hash": null,
  "slug": "ho668-claim-mod64-near-miss",
  "type": "claim",
  "title": "The 2025 modular construction fails exact orthogonality",
  "summary": "Eliahou's matrix has Gram matrix congruent to 668I modulo 64, while every row has 26 nonzero off-diagonal dot products over the integers.",
  "relevance": "For A Hadamard matrix of order 668, record ho668-claim-mod64-near-miss (“The 2025 modular construction fails exact orthogonality”) records a bound, answer, status fact, or structural consequence. The record states: Eliahou's matrix has Gram matrix congruent to 668I modulo 64, while every row has 26 nonzero off-diagonal dot products over the integers.",
  "relevance_source": "recorded",
  "body": "Eliahou gives two run-length encoded sequences \\(q,s\\in\\{\\pm1\\}^{167}\\), forms the quadruple\n\\[\n(A,B,C,D)=(s,s',(sq),(sq)'),\n\\]\nand inserts its four circulant matrices into the Goethals-Seidel array. Here the prime operation keeps the first 84 signs and reverses the signs of the last 83.\n\nThe independent replay reconstructs all 446,224 signs of the resulting \\(668\\)-square matrix \\(H\\). Every off-diagonal entry of \\(HH^{\\mathsf T}\\) is divisible by 64, so\n\\[\nHH^{\\mathsf T}\\equiv668I_{668}\\pmod {64}.\n\\]\nExact integer products expose the gap. Each row is orthogonal to 641 of the other rows and has 26 nonzero products. Per row, their values and multiplicities are\n\\[\n-512:2,\\ -320:2,\\ -256:2,\\ -192:4,\\ -64:4,\\ 128:6,\\ 256:4,\\ 384:2.\n\\]\nIn particular, \\(HH^{\\mathsf T}\\ne668I_{668}\\). The construction is a verified 64-modular Hadamard matrix and supplies no real Hadamard matrix of order 668.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "Eliahou's specified 64-modular Goethals-Seidel matrix of order 668",
    "bounds": {
      "order": {
        "min": 668,
        "max": 668
      },
      "modulus": {
        "min": 64,
        "max": 64
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://ajc.maths.uq.edu.au/pdf/93/ajc_v93_p422.pdf",
      "locator": "Shalom Eliahou, A 64-modular Hadamard matrix of order 668, Australasian Journal of Combinatorics 93(2) (2025), 422-427, Theorem 2.3 and Fact 3.1 on pages 424-426; independently reconstructed in ho668-artifact-replay-mod64"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://ajc.maths.uq.edu.au/pdf/93/ajc_v93_p422.pdf",
    "locator": "Shalom Eliahou, A 64-modular Hadamard matrix of order 668, Australasian Journal of Combinatorics 93(2) (2025), 422-427, Theorem 2.3 and Fact 3.1 on pages 424-426; independently reconstructed in ho668-artifact-replay-mod64"
  },
  "relations": [
    {
      "slug": "R377",
      "title": "Exact replay of the 64-modular order-668 matrix",
      "object_type": "artifact",
      "relation": "validates",
      "direction": "incoming"
    },
    {
      "slug": "R379",
      "title": "Order 668 remains open",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "hadamard-order-668",
      "title": "hadamard order 668",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}