{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R98",
  "content_hash": null,
  "slug": "ca32-claim-currently-open",
  "type": "claim",
  "title": "Existence at order 32 remains open",
  "summary": "No verified order-32 Costas permutation or exhaustive nonexistence certificate was found in the audited primary literature.",
  "relevance": "For Existence of a Costas array of order 32, record ca32-claim-currently-open (“Existence at order 32 remains open”) records a bound, answer, status fact, or structural consequence. The record states: No verified order-32 Costas permutation or exhaustive nonexistence certificate was found in the audited primary literature.",
  "relevance_source": "recorded",
  "body": "The candidate remains unresolved. Vulakh and Finkel describe order 32 as unsolved in their 2022 peer-reviewed paper. Their search produced near solutions rather than a Costas permutation. The 2022 paper `The Density of Costas Arrays Decays Exponentially` likewise states that existence at order 32 remains unknown.\n\nA fresh 2026 paper by Gulec and Abolghasemi reports that exhaustive enumeration reaches order 29 and bases its higher-order experiments on Beard's public collection of known arrays. Beard's database page says that its generated collection reaches order 1030, while exhaustive-search additions stop at orders 28 and 29. These records provide a current cross-check against a published witness. Database absence cannot establish nonexistence.\n\nThe acceptance condition therefore remains unchanged. A positive resolution needs a 32-entry permutation whose 496 displacement vectors pass an exact duplicate check. A negative resolution needs an exhaustive search certificate that covers every symmetry class.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the published existence status of Costas arrays of order 32, checked on 2026-07-25",
    "bounds": {
      "order": {
        "min": 32,
        "max": 32
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/s00500-022-06969-1",
      "locator": "David Vulakh and Raphael Finkel, Parallel m-dimensional relative ant colony optimization (mDRACO) for the Costas-array problem, Soft Computing 26 (2022), 5765-5772, Conclusion and Appendix"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/s00500-022-06969-1",
    "locator": "David Vulakh and Raphael Finkel, Parallel m-dimensional relative ant colony optimization (mDRACO) for the Costas-array problem, Soft Computing 26 (2022), 5765-5772, Conclusion and Appendix"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R99",
      "title": "The published heuristic search reached 17 violations",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R97",
      "title": "Standard construction and record audit",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R96",
      "title": "Exact Costas-permutation verifier",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "costas-array-order-32",
      "title": "costas array order 32",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}