{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R383",
  "content_hash": null,
  "slug": "hr4-attempt-coefficient-connectivity-audit",
  "type": "attempt",
  "title": "A mod-two vanishing table does not prove integral 6-connectivity",
  "summary": "The 2026 cohomology paper infers 6-connectivity from mod-two homology vanishing through degree 6. That coefficient calculation leaves odd-primary integral torsion possible, so the inference needs another argument.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-attempt-coefficient-connectivity-audit (“A mod-two vanishing table does not prove integral 6-connectivity”) documents a concrete method, search boundary, or failed route. The record states: The 2026 cohomology paper infers 6-connectivity from mod-two homology vanishing through degree 6.",
  "relevance_source": "recorded",
  "body": "The source audit checked arXiv:2605.00705v2, revised 2026-07-17. Its introduction says that Ripser calculated the first 15 homology groups of \\(\\operatorname{VR}(Q_6;4)\\) with \\(\\mathbb Z/2\\) coefficients, found the first nonzero group in degree 7, and thereby showed 6-connectivity. Page 5 repeats exact 6-connectivity and states \\(H_7(-;\\mathbb Z)\\ne0\\).\n\nVanishing over \\(\\mathbb F_2\\) in degrees 1 through 6 excludes free summands and 2-primary torsion in the corresponding universal-coefficient range. It permits odd-primary torsion. Homological vanishing over one field also supplies no direct vanishing of the homotopy groups needed for connectivity. The paper's rigorous total-domination bound gives only 3-connectivity, while the exact invariant in this packet improves that route to 4-connectivity.\n\nThe audit located no additional proof in the paper that closes the coefficient gap. This is a source-scope advisory about the connectivity sentence. It does not affect the canonical torsion question's definition.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the coefficient logic used for the claimed 6-connectivity of VR(Q_6;4) in arXiv:2605.00705v2",
    "bounds": {
      "cube_dimension": {
        "min": 6,
        "max": 6
      },
      "scale": {
        "min": 4,
        "max": 4
      },
      "vanishing_degree": {
        "min": 1,
        "max": 6
      },
      "coefficient_characteristic": {
        "min": 2,
        "max": 2
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/2605.00705",
      "locator": "Introduction page 2, Theorem 2.3, Theorem 2.6, and discussion on page 5"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2605.00705",
    "locator": "Introduction page 2, Theorem 2.3, Theorem 2.6, and discussion on page 5"
  },
  "relations": [
    {
      "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"
    }
  ]
}