{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R864",
  "content_hash": null,
  "slug": "fib-formalization-camion-minimal-obstruction",
  "type": "formalization",
  "title": "Almost-TU determinant and nonvanishing cofactors",
  "summary": "Lean proves that a least-order bad sign minor has determinant plus or minus two and none of its cofactors vanish.",
  "relevance": "For fib problem determinant range; fib problem nonzero support, record fib-formalization-camion-minimal-obstruction (“Almost-TU determinant and nonvanishing cofactors”) states a machine-checkable theorem or proof obligation. The record states: Lean proves that a least-order bad sign minor has determinant plus or minus two and none of its cofactors vanish.",
  "relevance_source": "recorded",
  "body": "The proof uses the adjugate identities and minimality. Deleting a zero coordinate leaves a cofactor matrix with determinant plus or minus one, which forces the bad determinant to divide every remaining coordinate. An adjugate column rules out zero cofactors. A signed difference of two adjugate columns forces the determinant to divide two.",
  "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/Matrix/Camion.lean"
    },
    "missing": [
      "source",
      "command",
      "expected_output"
    ]
  },
  "formal_statement": "theorem minimal_bad_square_minor_det_and_adjugate_nonzero {m n : Type*} [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] (A : Matrix m n ℤ) (hentries : ∀ i j, A i j ∈ Set.range SignType.cast) (k : ℕ) (f : Fin k → m) (g : Fin k → n) (hf : f.Injective) (hg : g.Injective) (hbad : (A.submatrix f g).det ∉ Set.range SignType.cast) (hminimal : ∀ (l : ℕ), l < k → ∀ (f' : Fin l → m) (g' : Fin l → n), f'.Injective → g'.Injective → (A.submatrix f' g').det ∈ Set.range SignType.cast) (hk : 3 ≤ k) : ((A.submatrix f g).det = 2 ∨ (A.submatrix f g).det = -2) ∧ ∀ i j, (A.submatrix f g).adjugate i j ≠ 0 := by\n  exact ⟨minimal_bad_square_minor_det_eq_two_or_neg_two A k f g hf hg hbad hminimal hk, minimal_bad_square_minor_adjugate_nonzero A k f g hf hg hbad hminimal (by omega)⟩",
  "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/Matrix/Camion.lean"
  },
  "models": [],
  "relations": [
    {
      "slug": "R860",
      "title": "Camion's total-unimodularity criterion",
      "object_type": "formalization",
      "relation": "depends_on",
      "direction": "incoming"
    }
  ]
}