{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R163",
  "content_hash": null,
  "slug": "cset11-claim-certified-interval",
  "type": "claim",
  "title": "The certified interval is 74 through 387,420,489",
  "summary": "An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18.",
  "relevance": "For Largest first-homology torsion from forty-five triangles on eleven vertices, record cset11-claim-certified-interval (“The certified interval is 74 through 387,420,489”) records a bound, answer, status fact, or structural consequence. The record states: An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18.",
  "relevance_source": "recorded",
  "body": "Let \\(M(K)\\) be the maximum in the question. The exact replay in this record constructs a 45-face complex with\n\\[\nH_1(K;\\mathbb Z)\\cong\\mathbb Z/74\\mathbb Z,\n\\]\nso \\(M(K)\\ge74\\).\n\nKalai's torsion theorem states that the torsion part of \\(H_{d-1}\\) for a \\(d\\)-complex on \\(n\\) vertices has order at most\n\\[\n(\\sqrt{d+1})^{\\binom{n-2}{d}}.\n\\]\nAt \\((n,d)=(11,2)\\), this is\n\\[\n(\\sqrt3)^{\\binom92}=3^{18}=387{,}420{,}489.\n\\]\nConsequently\n\\[\n\\boxed{74\\le M(K)\\le387{,}420{,}489}.\n\\]\nThe construction improves the candidate record's lower bound of 51. The search used determinant-preserving basis exchanges followed by exact integer checks. It did not enumerate all \\(\\binom{165}{45}\\) face sets, so the exact maximum remains open.",
  "status": "reported",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all 45-face subcomplexes of the complete two-dimensional complex on eleven labeled vertices whose integral first homology is finite",
    "bounds": {
      "vertices": {
        "min": 11,
        "max": 11
      },
      "triangular_faces": {
        "min": 45,
        "max": 45
      },
      "available_triangles": {
        "min": 165,
        "max": 165
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1707.09271",
      "locator": "Andrew Newman, Small simplicial complexes with prescribed torsion in homology, Theorem 4 part 1 as quoted from Kalai; lower construction and Smith certificate in this record"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1707.09271",
    "locator": "Andrew Newman, Small simplicial complexes with prescribed torsion in homology, Theorem 4 part 1 as quoted from Kalai; lower construction and Smith certificate in this record"
  },
  "relations": [
    {
      "slug": "R164",
      "title": "A 45-face complex has H1 isomorphic to Z/74Z",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R162",
      "title": "The literature supplies the framework and universal bound, with no exact n=11 table located",
      "object_type": "attempt",
      "relation": "contextualizes",
      "direction": "incoming"
    },
    {
      "slug": "complete-skeleton-eleven-torsion",
      "title": "complete skeleton eleven torsion",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}