{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R391",
  "content_hash": null,
  "slug": "hr4-claim-n7-exact-fvector",
  "type": "claim",
  "title": "The n=7 complex has 209,570,782,049 faces and reduced Euler characteristic -3937",
  "summary": "Exact deletion-contraction gives every face count of VR(Q_7;4), maximum face cardinality 29, and reduced Euler characteristic -3937. These invariants constrain any future homology table and carry no torsion conclusion by themselves.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-claim-n7-exact-fvector (“The n=7 complex has 209,570,782,049 faces and reduced Euler characteristic -3937”) records a bound, answer, status fact, or structural consequence. The record states: Exact deletion-contraction gives every face count of VR(Q_7;4), maximum face cardinality 29, and reduced Euler characteristic -3937.",
  "relevance_source": "recorded",
  "body": "The forbidden graph on \\(Q_7\\) joins words at Hamming distance 5, 6, or 7. It is regular of degree\n\\[\n\\binom75+\\binom76+\\binom77=29\n\\]\nand has 1,856 edges. The exact independence polynomial in `hr4-artifact-exact-fvectors-six-seven` has degree 29 and coefficient sum 209,570,782,049. Its alternating evaluation gives ordinary Euler characteristic \\(-3936\\), hence reduced Euler characteristic \\(-3937\\).\n\nThe face count is coefficient-independent and supplies a checksum for any later boundary computation. Euler characteristic records only an alternating sum of free ranks. It neither detects torsion nor identifies individual Betti numbers. Combined with `hr4-claim-published-n7-rank-lower-bounds`, any complete rational table must have \\(b_7\\ge3107\\), \\(b_{15}\\ge110\\), and reduced Euler characteristic \\(-3937\\).",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the complete f-vector and Euler characteristic of VR(Q_7;4)",
    "bounds": {
      "cube_dimension": {
        "min": 7,
        "max": 7
      },
      "scale": {
        "min": 4,
        "max": 4
      },
      "face_cardinality": {
        "min": 0,
        "max": 29
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Exact deletion-contraction computation in hr4-artifact-exact-fvectors-six-seven"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Exact deletion-contraction computation in hr4-artifact-exact-fvectors-six-seven"
  },
  "models": [],
  "relations": [
    {
      "slug": "R381",
      "title": "Exact deletion-contraction f-vectors for n=6 and n=7",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R393",
      "title": "Propagation gives rational rank bounds 3107 and 110 at n=7",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R388",
      "title": "Torsion-free through n=5 and no 2-primary torsion at n=6",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "hypercube-rips-scale-four-torsion-free",
      "title": "hypercube rips scale four torsion free",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}