{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R248",
  "content_hash": null,
  "slug": "fcptp-claim-topology-reduction",
  "type": "claim",
  "title": "First homology reduces to two connectivity counts and Euler characteristic",
  "summary": "The identity \\(\\beta_1=c_{26}+h_6-\\chi\\) and the exact \\(3\\times3\\times3\\) probability \\(4355/16384\\) are established, while the capped \\(4\\times4\\times4\\) transfer stopped after 27 of 64 voxels; the requested four-cube numerator remains undetermined.",
  "relevance": "For Exact tunnel probability for site percolation on a four by four by four cubical box, record fcptp-claim-topology-reduction (“First homology reduces to two connectivity counts and Euler characteristic”) records a bound, answer, status fact, or structural consequence. The record states: The identity \\(\\beta_1=c_{26}+h_6-\\chi\\) and the exact \\(3\\times3\\times3\\) probability \\(4355/16384\\) are established, while the capped \\(4\\times4\\times4\\) transfer stopped after 27 of 64 voxels; the requested four-cube numerator remains undetermined.",
  "relevance_source": "recorded",
  "body": "Let \\(c_{26}(X)\\) be the number of components of the chosen voxels under vertex-touching adjacency. Let \\(h_6(X)\\) be the number of face-connected components of unchosen voxels that miss the boundary of the box. Then\n\\[\n\\beta _1(X;\\mathbb F_2)=c_{26}(X)+h_6(X)-\\chi(X).\n\\]\n\nThe closed cubes that meet at a face, edge, or vertex belong to the same connected component, which gives \\(\\beta _0(X)=c_{26}(X)\\). The bounded components of \\(\\mathbb R^3\\setminus X\\) correspond to face-connected unchosen-voxel components that do not meet the box boundary. Alexander duality gives \\(\\beta _2(X)=h_6(X)\\), and \\(\\beta _3(X)=0\\) for a bounded subset of \\(\\mathbb R^3\\). Substitution in\n\\[\n\\chi(X)=\\beta _0(X)-\\beta _1(X)+\\beta _2(X)\n\\]\nproves the formula. The transfer computes \\(\\chi\\) directly as the number of present vertices minus present edges plus present faces minus selected cubes.",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "universal",
    "statement": "every finite closed union X of unit cubes contained in a rectangular three-dimensional cubical box"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://archive.intlpress.com/site/pub/files/_fulltext/journals/hha/2003/0005/0002/HHA-2003-0005-0002-a008.pdf",
      "locator": "Kaczynski, Mischaikow, and Mrozek, Computing Homology, Homology Homotopy and Applications 5(2), 233-256 (2003), for cubical chain complexes; the displayed reduction also uses Euler-Poincare and Alexander duality"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://archive.intlpress.com/site/pub/files/_fulltext/journals/hha/2003/0005/0002/HHA-2003-0005-0002-a008.pdf",
    "locator": "Kaczynski, Mischaikow, and Mrozek, Computing Homology, Homology Homotopy and Applications 5(2), 233-256 (2003), for cubical chain complexes; the displayed reduction also uses Euler-Poincare and Alexander duality"
  },
  "models": [],
  "relations": [
    {
      "slug": "R245",
      "title": "Exact two-connectivity frontier transfer for the three-cube box",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "four-cube-site-percolation-tunnel-probability",
      "title": "four cube site percolation tunnel probability",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}