{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R379",
  "content_hash": null,
  "slug": "ho668-claim-currently-open",
  "type": "claim",
  "title": "Order 668 remains open",
  "summary": "No real Hadamard matrix of order 668 is known, and no nonexistence proof is known.",
  "relevance": "For A Hadamard matrix of order 668, record ho668-claim-currently-open (“Order 668 remains open”) records a bound, answer, status fact, or structural consequence. The record states: No real Hadamard matrix of order 668 is known, and no nonexistence proof is known.",
  "relevance_source": "recorded",
  "body": "The current answer to the candidate is open. Cati and Pasechnik's construction database, revised on August 30, 2025, states that all known Hadamard orders through 1208 are implemented in SageMath and lists 668, 716, 892, and 1132 as the unknown orders in that range. Their Section 3 gives the direct query\n```\nhadamard_matrix(668, existence=True)\nUnknown\n```\nThe result `Unknown` means that SageMath has no construction and makes no nonexistence assertion.\n\nThe appendix gives a second check. Table 2 lists a construction for each known order \\(4n\\) with odd \\(n<300\\); the entry at \\(n=167\\) is blank. Table 4 records \\(167(3)\\), meaning that the least known exponent \\(m\\) for an order \\(2^m167\\) construction is \\(m=3\\). Thus order \\(1336=8\\cdot167\\) is known, while the desired order \\(668=4\\cdot167\\) is the missing exponent-two case.\n\nEliahou's 2025 primary paper calls 668 the smallest open case and constructs the 64-modular near miss checked in the companion artifact. Suksmono's 2025 search paper independently lists 668, 716, and 892 as unresolved below 1000. Epoch AI still identifies 668 as the smallest unknown order on the access date. These sources report the state of knowledge. They cannot prove that a matrix does not exist.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the published construction status of real Hadamard matrices at order 668 as checked on 2026-07-24",
    "bounds": {
      "order": {
        "min": 668,
        "max": 668
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2411.18897v2",
      "locator": "Matteo Cati and Dmitrii V. Pasechnik, A database of constructions of Hadamard matrices, arXiv:2411.18897v2: page 1, lines listing unknown orders; Section 3, page 5, the order-668 SageMath query; Appendix Tables 2 and 4, pages 15 and 17"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2411.18897v2",
    "locator": "Matteo Cati and Dmitrii V. Pasechnik, A database of constructions of Hadamard matrices, arXiv:2411.18897v2: page 1, lines listing unknown orders; Section 3, page 5, the order-668 SageMath query; Appendix Tables 2 and 4, pages 15 and 17"
  },
  "relations": [
    {
      "slug": "R378",
      "title": "Construction and citation audit",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R380",
      "title": "The 2025 modular construction fails exact orthogonality",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "hadamard-order-668",
      "title": "hadamard order 668",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}