{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R378",
  "content_hash": null,
  "slug": "ho668-attempt-construction-literature-audit",
  "type": "attempt",
  "title": "Construction and citation audit",
  "summary": "Current tables and recent primary papers agree that 668 is unresolved; the strongest explicit order-668 object found is modular.",
  "relevance": "For A Hadamard matrix of order 668, record ho668-attempt-construction-literature-audit (“Construction and citation audit”) documents a concrete method, search boundary, or failed route. The record states: Current tables and recent primary papers agree that 668 is unresolved; the strongest explicit order-668 object found is modular.",
  "relevance_source": "recorded",
  "body": "The audit followed the candidate's Epoch citation into the current construction tables and primary papers. Kharaghani and Tayfeh-Rezaie's `A Hadamard matrix of order 428`, Journal of Combinatorial Designs 13 (2005), 435-440, DOI 10.1002/jcd.20043, constructs order 428. It thereby moved the smallest unknown order to 668 and contains no order-668 construction.\n\nCati and Pasechnik checked the published construction record by implementing every known order through 1208. Their revised table leaves \\(4\\cdot167\\) empty and reports `Unknown` for SageMath's order-668 existence query. Eliahou's later paper, received September 24, 2025, provides explicit data at order 668. Its equation is the congruence\n\\[\nHH^{\\mathsf T}\\equiv668I\\pmod {64},\n\\]\nwhich allows nonzero off-diagonal multiples of 64. The exact replay finds those multiples and confirms the paper's own description of the true order-668 matrix as elusive.\n\nNo supplied citation claims a verified real matrix of order 668. The search also found no downloadable real witness to test. A future resolution should include either a complete \\(668\\)-square sign matrix with an exact Gram check or a construction theorem whose hypotheses are verified at the stated parameters.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "published real and modular Hadamard constructions bearing on order 668 through 2026-07-24",
    "bounds": {
      "order": {
        "min": 668,
        "max": 668
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/2411.18897",
      "locator": "Cati and Pasechnik, arXiv:2411.18897v2, Sections 1, 3, and 6 plus Appendix Tables 2 and 4; Eliahou 2025, Fact 3.1 and Conclusion; Kharaghani and Tayfeh-Rezaie 2005, DOI 10.1002/jcd.20043"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2411.18897",
    "locator": "Cati and Pasechnik, arXiv:2411.18897v2, Sections 1, 3, and 6 plus Appendix Tables 2 and 4; Eliahou 2025, Fact 3.1 and Conclusion; Kharaghani and Tayfeh-Rezaie 2005, DOI 10.1002/jcd.20043"
  },
  "relations": [
    {
      "slug": "R379",
      "title": "Order 668 remains open",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R377",
      "title": "Exact replay of the 64-modular order-668 matrix",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "hadamard-order-668",
      "title": "hadamard order 668",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}