{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R863",
  "content_hash": null,
  "slug": "fib-formalization-support-card-divisibility",
  "type": "formalization",
  "title": "Fibonacci odd square cover",
  "summary": "Lean proves that every Eulerian selected support is covered with odd multiplicity by finitely many contained support squares.",
  "relevance": "This supplies the graph-theoretic core of Fibonacci support divisibility.",
  "relevance_source": "recorded",
  "body": "The proof works with arbitrary even sub-supports. A maximal incident vertex exposes a Fibonacci support square containing at least three current edges. Toggling that square preserves every row and column parity and strictly reduces the edge count, so well-founded induction produces the odd square cover. The two-square incidence bound then turns the cover into a four-edge partition.",
  "status": "draft",
  "evidence_grade": "unverified_formalization",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "formalization",
    "runtime": "lean-4.33.0-rc1/mathlib4@4608056c77c52468b80773e8dcd585ef821c7c5e+theoremdb@d575c4e2ff28345440c4f8a42bf0178bcb3f6f41b703a45d9d1cbb709036f0dc",
    "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/OddSquareCover.lean"
    },
    "missing": [
      "source",
      "command",
      "expected_output"
    ]
  },
  "formal_statement": "theorem fibSubmatrixSupport_has_odd_square_cover (n k : ℕ) (f : Fin k → Fin n) (g : Fin k → Fin n) (hf : f.Injective) (hg : g.Injective) (hrows : HasEvenRowSums ((fibSumMatrix n).submatrix f g)) (hcols : HasEvenColumnSums ((fibSumMatrix n).submatrix f g)) : ∃ faces : Finset (SelectedSupportSquare n k f g), ∀ edge ∈ fibSubmatrixSupport n k f g, Odd (faces.filter fun S => edge ∈ S.edges).card := by\n  -- Kernel-checked square-toggle induction in OddSquareCover.lean",
  "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/OddSquareCover.lean"
  },
  "relations": [
    {
      "slug": "R861",
      "title": "Fibonacci support divisibility condition",
      "object_type": "formalization",
      "relation": "depends_on",
      "direction": "incoming"
    },
    {
      "slug": "R865",
      "title": "Fibonacci support four-cycle classification",
      "object_type": "formalization",
      "relation": "depends_on",
      "direction": "outgoing"
    }
  ]
}