{
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  "ref": "R1305",
  "content_hash": null,
  "slug": "finite-order-integer-matrix-ternary-conjugate-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. 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.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. On 2026-07-27 both MathOverflow answers and all their comments were checked. They reduce dimensions at most four to published lists and report ternary representatives in every listed class. Tahara classifies the relevant low-dimensional finite subgroups, and Yang's 2015 Electronic Journal of Linear Algebra paper lists 45 torsion conjugacy classes in \\(\\operatorname{GL}_4(\\mathbb Z)\\). These checks set the first possible counterexample dimension at five. Rational canonical form is insufficient because rational conjugacy can split into several integral conjugacy classes. A search must enumerate integral lattices or integral conjugacy classes. A matrix outside the ternary alphabet is not itself a counterexample. One must certify that its entire \\(\\operatorname{GL}_n(\\mathbb Z)\\)-conjugacy class contains no ternary matrix. Trap: allowing conjugation by \\(\\operatorname{GL}_n(\\mathbb Q)\\), changing the lattice, or using a larger bounded entry set answers a weaker question.\n\nA complete resolution must satisfy: 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",
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    "citation": {
      "url": "https://mathoverflow.net/questions/346842/is-every-finite-order-unimodular-matrix-conjugate-to-a-0-1-1-matrix",
      "locator": "Dataset references and independent 2026-08-01 status search."
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  "source": {
    "url": "https://mathoverflow.net/questions/346842/is-every-finite-order-unimodular-matrix-conjugate-to-a-0-1-1-matrix",
    "locator": "Dataset references and independent 2026-08-01 status search."
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  "relations": [
    {
      "slug": "R1304",
      "title": "Complete the stated acceptance conditions",
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      "slug": "R1540",
      "title": "Dated status and exact unresolved remainder",
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    {
      "slug": "finite-order-integer-matrix-ternary-conjugate",
      "title": "finite order integer matrix ternary conjugate",
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}