{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R918",
  "content_hash": null,
  "slug": "fib-claim-determinant-range-formally-proved",
  "type": "claim",
  "title": "The determinant is always minus one, zero, or one",
  "summary": "For every integer n >= 1, the determinant of the Fibonacci-sum indicator matrix M_n belongs to {-1,0,1}.",
  "relevance": "For fib problem determinant range; fib problem nonzero support, record fib-claim-determinant-range-formally-proved (“The determinant is always minus one, zero, or one”) records a bound, answer, status fact, or structural consequence. The record states: For every integer n >= 1, the determinant of the Fibonacci-sum indicator matrix M_n belongs to {-1,0,1}.",
  "relevance_source": "recorded",
  "body": "The Lean declaration TheoremDB.Fibonacci.fibSumMatrix_det_range has type DeterminantRangeStatement, the exact formal counterpart of the canonical problem. Its proof applies the total-unimodularity theorem to the full matrix M_n, obtains the determinant as a signed value, and eliminates the three possible signs to conclude that the determinant is -1, 0, or 1. The TheoremDB Lean verifier accepted this declaration in the pinned Lean and mathlib world, which establishes the stated determinant range for every n.",
  "status": "established",
  "evidence_grade": "mathematical_argument",
  "scope": {
    "kind": "universal",
    "statement": "every matrix size n >= 1"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/513340/is-the-determinant-of-this-fibonacci-sum-indicator-matrix-always-1-0-or/513372",
      "locator": "formal/lean/TheoremDB/Fibonacci/DeterminantRange.lean"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/513340/is-the-determinant-of-this-fibonacci-sum-indicator-matrix-always-1-0-or/513372",
    "locator": "formal/lean/TheoremDB/Fibonacci/DeterminantRange.lean"
  },
  "models": [],
  "relations": [
    {
      "slug": "fib-problem-determinant-range",
      "title": "Fibonacci-sum indicator determinant conjecture",
      "object_type": "problem",
      "relation": "resolves",
      "direction": "outgoing",
      "metadata": {
        "verification_record": "fib-formalization-determinant-range-lean-v1-lean-5e5b725441ec-verified-e4e988240074"
      }
    },
    {
      "slug": "R309",
      "title": "Every nonsingular Fibonacci-sum matrix is unimodular",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "outgoing",
      "metadata": {
        "reason": "The exact determinant-range statement has since been verified in Lean. The stronger total-unimodularity claim keeps its separate review status."
      }
    },
    {
      "slug": "R920",
      "title": "The determinant is always minus one, zero, or one",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "incoming",
      "metadata": {
        "reason": "Pairs the verified determinant-range result with its complete human-readable proof."
      }
    },
    {
      "slug": "R919",
      "title": "Every Fibonacci-sum matrix is totally unimodular",
      "object_type": "claim",
      "relation": "strengthens",
      "direction": "incoming"
    }
  ]
}