{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R446",
  "content_hash": null,
  "slug": "lee7-claim-integral-homology",
  "type": "claim",
  "title": "The integral homology is Z, Z squared, Z, 0, 0",
  "summary": "Exact boundary ranks and Smith forms give H_0 = Z, H_1 = Z^2, H_2 = Z, and zero homology in degrees three and above, with no torsion.",
  "relevance": "For Homotopy type of a Lee-metric Rips complex on the 7 by 7 torus, record lee7-claim-integral-homology (“The integral homology is Z, Z squared, Z, 0, 0”) records a bound, answer, status fact, or structural consequence. The record states: Exact boundary ranks and Smith forms give H_0 = Z, H_1 = Z^2, H_2 = Z, and zero homology in degrees three and above, with no torsion.",
  "relevance_source": "recorded",
  "body": "Exact clique enumeration gives chain-group ranks\n\\[\n(\\operatorname{rank}C_0,\\ldots,\\operatorname{rank}C_4)\n=(49,294,490,294,49).\n\\]\nThe boundary maps \\(\\partial_1,\\ldots,\\partial_4\\) have ranks\n\\[\n(48,244,245,49).\n\\]\nIt follows that the Betti numbers are\n\\[\n(1,2,1,0,0).\n\\]\n\nThe Smith normal form of each boundary matrix has only \\(1\\) among its nonzero invariant factors, with multiplicities \\(48,244,245,49\\), respectively. Thus every boundary image is primitive in its target chain group. Each homology group is a subgroup of a free quotient \\(C_i/\\operatorname{im}\\partial_{i+1}\\), so it is torsion-free. Hence\n\\[\nH_i(Y;\\mathbb Z)\\cong\n\\begin{cases}\n\\mathbb Z,&i=0,2,\\\\\n\\mathbb Z^2,&i=1,\\\\\n0,&i\\geq3.\n\\end{cases}\n\\]\nThis agrees with the published homotopy equivalence.",
  "status": "reproduced",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "complete integral simplicial homology of the 49-vertex Lee-threshold-two complex",
    "bounds": {
      "side_length": {
        "min": 7,
        "max": 7
      },
      "lee_threshold": {
        "min": 2,
        "max": 2
      },
      "homological_degree": {
        "min": 0,
        "max": 4
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2502.07134v2",
      "locator": "Integral computation in lee7-artifact-exact-cliques-and-smith; comparison with Adams et al., Theorem 5.8"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2502.07134v2",
    "locator": "Integral computation in lee7-artifact-exact-cliques-and-smith; comparison with Adams et al., Theorem 5.8"
  },
  "relations": [
    {
      "slug": "R445",
      "title": "The complex is homotopy equivalent to the 2-torus",
      "object_type": "claim",
      "relation": "implies",
      "direction": "incoming"
    },
    {
      "slug": "R443",
      "title": "Exact clique enumeration and Smith-normal-form certificate",
      "object_type": "artifact",
      "relation": "reproduces",
      "direction": "incoming"
    },
    {
      "slug": "lee-rips-torus-7-homotopy",
      "title": "lee rips torus 7 homotopy",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}