{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R164",
  "content_hash": null,
  "slug": "cset11-claim-order-74-construction",
  "type": "claim",
  "title": "A 45-face complex has H1 isomorphic to Z/74Z",
  "summary": "The reduced boundary matrix has determinant 74 and Smith diagonal 1 repeated 44 times followed by 74.",
  "relevance": "For Largest first-homology torsion from forty-five triangles on eleven vertices, record cset11-claim-order-74-construction (“A 45-face complex has H1 isomorphic to Z/74Z”) records a bound, answer, status fact, or structural consequence. The record states: The reduced boundary matrix has determinant 74 and Smith diagonal 1 repeated 44 times followed by 74.",
  "relevance_source": "recorded",
  "body": "Write `A` for vertex 10. Take the following faces, in the displayed order:\n\n`359 038 235 037 68A 57A 248 016 569 058 13A 568 129 127 024 025 236 09A 45A 046 159 147 679 28A 168 46A 189 479 148 238 36A 27A 49A 067 37A 345 789 15A 02A 134 578 267 249 039 01A`.\n\nDelete the ten star edges \\(0i\\) from the edge chain group. The remaining edges \\(ij\\), with \\(1\\le i<j\\le10\\), give a 45-element basis for the cycle group after the usual spanning-tree reduction. For an oriented face \\(a<b<c\\), its reduced boundary column is\n\\[\n[bc]-[ac]+[ab],\n\\]\nwith every term containing vertex 0 omitted. This gives the integer presentation matrix \\(B\\) used by the replay.\n\nFraction-free elimination gives \\(\\det B=74\\). Deleting row 0, indexed by edge 12, and column 1, indexed by face 038, gives a 44 by 44 minor of determinant \\(-1\\). The gcd of all 44-row minors is therefore 1. The determinantal-divisor characterization of Smith normal form now gives\n\\[\n\\operatorname{SNF}(B)=\\operatorname{diag}(1^{44},74).\n\\]\nThus the matrix has full rational rank, \\(H_2(K;\\mathbb Z)=0\\), and\n\\[\nH_1(K;\\mathbb Z)\\cong\\operatorname{coker}B\\cong\\mathbb Z/74\\mathbb Z.\n\\]\nThe face list and boundary matrix have SHA-256 digests `cdd0175fca03a927e3ae5b11da108dc2ca61d045f15a4c3442ecfa9bbc928f85` and `a3b4e556bd16ddd806f8199a4aad171acb521102d12e3a27e3f65beb8b991c15`.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the displayed 45 triangular faces on vertices 0 through 10 and their complete one-skeleton",
    "bounds": {
      "vertices": {
        "min": 11,
        "max": 11
      },
      "edges": {
        "min": 55,
        "max": 55
      },
      "triangular_faces": {
        "min": 45,
        "max": 45
      },
      "boundary_rank": {
        "min": 45,
        "max": 45
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1707.09271",
      "locator": "Exact integer certificate in cset11-artifact-boundary-smith-replay, produced 2026-07-25"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1707.09271",
    "locator": "Exact integer certificate in cset11-artifact-boundary-smith-replay, produced 2026-07-25"
  },
  "relations": [
    {
      "slug": "R163",
      "title": "The certified interval is 74 through 387,420,489",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R161",
      "title": "Exact reduced boundary and Smith-form replay",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "complete-skeleton-eleven-torsion",
      "title": "complete skeleton eleven torsion",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}