{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R195",
  "content_hash": null,
  "slug": "dbm-claim-pascal-minors",
  "type": "claim",
  "title": "The required Pascal blocks are unimodular",
  "summary": "Lucas's theorem identifies binary disjointness with the symmetric Pascal matrix modulo 2, whose leading and adjacent border minors have determinant \\(\\pm1\\).",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Lucas's theorem gives\n\\[\n\\mathbf1_{i\\mathbin{\\&}j=0}=\\binom{i+j}{i}\\pmod2.\n\\]\nWrite \\(P_r=(\\mathbf1_{i\\mathbin{\\&}j=0})_{0\\leq i,j<r}\\). For a power of two, \\(P_r\\) is a Kronecker power of \\(\\left(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\right)\\), hence unimodular. If \\(r=q+s\\) with \\(q\\) the largest power of two at most \\(r\\), its binary block decomposition gives \\(|\\det P_r|=|\\det P_q\\det P_s|=1\\). Sorting the rows and columns of the top-border block \\(C_r\\) by parity reduces it inductively to blocks \\(P_{\\lceil r/2\\rceil}\\) and \\(C_{\\lfloor r/2\\rfloor}\\), so \\(|\\det C_r|=1\\).\n\nMereb proves the border-minor result in a broader Pascal-matrix setting. Bacher and Chapman give the leading-principal determinant theorem.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every finite leading principal block and top-border block of the symmetric Pascal matrix modulo 2"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/pdf/math/0212144",
      "locator": "Bacher and Chapman, Symmetric Pascal matrices modulo p, Theorem 1.1; Mereb, arXiv:2210.12913, Corollary 1"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/pdf/math/0212144",
    "locator": "Bacher and Chapman, Symmetric Pascal matrices modulo p, Theorem 1.1; Mereb, arXiv:2210.12913, Corollary 1"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R194",
      "title": "Exact determinant of the disjoint-binary-support matrix",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "disjoint-binary-support-determinant",
      "title": "disjoint binary support determinant",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}