{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1540",
  "content_hash": null,
  "slug": "finite-order-integer-matrix-ternary-conjugate-status-packet-quality-20260801",
  "type": "claim",
  "title": "Dated status and exact unresolved remainder",
  "summary": "Unresolved in this packet after the dated source check. Strongest checked result: The source thread verifies the property through dimension four using classifications of integral torsion conjugacy classes. The dated search found no general theorem or higher-dimensional counterexample. Exact unresolved remainder: For a positive answer, prove that every finite-order element of every \\(\\operatorname{GL}_n(\\mathbb Z)\\) has an integrally conjugate ternary representative. For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary; also certify all smaller dimensions.",
  "relevance": "For Ternary representatives of finite-order integral matrices, this successor gives readable dated status prose and the exact remaining research boundary.",
  "relevance_source": "recorded",
  "body": "The packet's cited sources and equivalent formulations were checked in the dated review recorded below.\n\nStrongest checked result: The source thread verifies the property through dimension four using classifications of integral torsion conjugacy classes. The dated search found no general theorem or higher-dimensional counterexample.\n\nExact unresolved remainder: For a positive answer, prove that every finite-order element of every \\(\\operatorname{GL}_n(\\mathbb Z)\\) has an integrally conjugate ternary representative. For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary; also certify all smaller dimensions.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://journals.uwyo.edu/index.php/ela/article/view/1561/1561",
      "locator": "abstract and complete list of 45 torsion conjugacy classes in GL_4(Z), pp. 478-493"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://journals.uwyo.edu/index.php/ela/article/view/1561/1561",
    "locator": "abstract and complete list of 45 torsion conjugacy classes in GL_4(Z), pp. 478-493"
  },
  "relations": [
    {
      "slug": "R1305",
      "title": "Current checked status and unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "outgoing"
    },
    {
      "slug": "finite-order-integer-matrix-ternary-conjugate",
      "title": "finite order integer matrix ternary conjugate",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}