{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R388",
  "content_hash": null,
  "slug": "hr4-claim-current-torsion-boundary",
  "type": "claim",
  "title": "Torsion-free through n=5 and no 2-primary torsion at n=6",
  "summary": "The complexes VR(Q_n;4) are torsion-free for n at most 5. For n=6, the rational reduced Betti numbers are 239 in degree 7 and 14 in degree 15, and every integral homology group has trivial 2-primary torsion. Odd-primary torsion at n=6 and the full torsion question for every n at least 7 remain unresolved.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-claim-current-torsion-boundary (“Torsion-free through n=5 and no 2-primary torsion at n=6”) records a bound, answer, status fact, or structural consequence. The record states: The complexes VR(Q_n;4) are torsion-free for n at most 5.",
  "relevance_source": "recorded",
  "body": "For \\(1\\leq n\\leq4\\), every pair of cube vertices has Hamming distance at most four. The Rips complex is a simplex, so its integral homology is \\(H_0\\cong\\mathbb Z\\) with all higher groups zero. For \\(n=5\\), the only forbidden pairs are the sixteen complementary pairs. The complex is the join of sixteen copies of \\(S^0\\), hence is \\(S^{15}\\). Thus \\(H_0\\cong H_{15}\\cong\\mathbb Z\\), with all other groups zero.\n\nAt \\(n=6\\), Galetto, Montaño, and Wellner report\n\\[\n\\widetilde H_j(\\operatorname{VR}(Q_6;4);\\mathbb Q)\\cong\n\\begin{cases}\n\\mathbb Q^{239},&j=7,\\\\\n\\mathbb Q^{14},&j=15,\\\\\n0,&j\\ne7,15,\n\\end{cases}\n\\]\nand state that the Betti numbers over \\(\\mathbb F_2\\) are the same. The universal coefficient theorem gives\n\\[\n\\dim_{\\mathbb F_2}H_j(K;\\mathbb F_2)\n=b_j(K;\\mathbb Q)+t_j+t_{j-1},\n\\]\nwhere \\(t_i\\) is the number of cyclic 2-primary summands of \\(H_i(K;\\mathbb Z)\\). Equality of the rational and mod-two Betti numbers in every degree forces every \\(t_i\\) to vanish.\n\nThe exact deletion-contraction replay supplies a separate Euler check. Its complete face count gives \\(\\widetilde\\chi=-253\\), agreeing with \\(-239-14\\). It also finds 2,932,100,733 faces including the empty face and maximum face cardinality 22.\n\nThis settles the prime 2 part at \\(n=6\\). The checked sources and computations supply no Smith certificate or odd-characteristic rank table that excludes odd-primary torsion. The canonical universal question therefore remains open.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all homological degrees of VR(Q_n;4) for 1 <= n <= 6, with the n=6 torsion conclusion restricted to the prime 2",
    "bounds": {
      "cube_dimension": {
        "min": 1,
        "max": 6
      },
      "scale": {
        "min": 4,
        "max": 4
      },
      "torsion_prime_at_n6": {
        "min": 2,
        "max": 2
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2606.20784v1",
      "locator": "Galetto, Montaño, and Wellner, Appendix C.3; exact Euler replay in hr4-artifact-exact-fvectors-six-seven; universal-coefficient inference written here"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2606.20784v1",
    "locator": "Galetto, Montaño, and Wellner, Appendix C.3; exact Euler replay in hr4-artifact-exact-fvectors-six-seven; universal-coefficient inference written here"
  },
  "relations": [
    {
      "slug": "R392",
      "title": "Published field computations for VR(Q_6;4)",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R381",
      "title": "Exact deletion-contraction f-vectors for n=6 and n=7",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R391",
      "title": "The n=7 complex has 209,570,782,049 faces and reduced Euler characteristic -3937",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R389",
      "title": "The minimum domination witness spans an explicit 5-cycle",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R394",
      "title": "The n=6 forbidden graph has total domination number 12",
      "object_type": "claim",
      "relation": "strengthens",
      "direction": "incoming"
    },
    {
      "slug": "R386",
      "title": "Total domination cannot certify 6-connectivity at n=6",
      "object_type": "attempt",
      "relation": "attempts",
      "direction": "incoming"
    },
    {
      "slug": "R383",
      "title": "A mod-two vanishing table does not prove integral 6-connectivity",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R387",
      "title": "Coordinate inclusion gives split injections in homology",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R384",
      "title": "Reduce the n=6 complex before exact odd-prime and Smith computations",
      "object_type": "attempt",
      "relation": "attempts",
      "direction": "incoming"
    },
    {
      "slug": "hypercube-rips-scale-four-torsion-free",
      "title": "hypercube rips scale four torsion free",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}