{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R392",
  "content_hash": null,
  "slug": "hr4-claim-published-field-homology-n6",
  "type": "claim",
  "title": "Published field computations for VR(Q_6;4)",
  "summary": "Adams and Virk report mod-two homology through degree 15, with dimensions 239 in degree 7 and 14 in degree 15. Galetto, Montaño, and Wellner report the complete rational table with the same two dimensions and explicitly state that the rational and mod-two Betti numbers agree in every degree.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-claim-published-field-homology-n6 (“Published field computations for VR(Q_6;4)”) records a bound, answer, status fact, or structural consequence. The record states: Adams and Virk report mod-two homology through degree 15, with dimensions 239 in degree 7 and 14 in degree 15.",
  "relevance_source": "recorded",
  "body": "Section 6.4.4 of Adams and Virk reports a Ripser computation using about 180 GB of memory:\n\\[\nH_q(\\operatorname{VR}(Q_6;4);\\mathbb F_2)\\cong\n\\begin{cases}\n0,&1\\leq q\\leq6,\\\\\n\\mathbb F_2^{239},&q=7,\\\\\n0,&8\\leq q\\leq14,\\\\\n\\mathbb F_2^{14},&q=15.\n\\end{cases}\n\\]\nTheir displayed table ends at degree 15.\n\nAppendix C.3 of Galetto, Montaño, and Wellner reports a Polymake computation over \\(\\mathbb Q\\) taking about 18 hours and at least 800 GB of memory. It gives reduced rational homology of ranks 239 and 14 in degrees 7 and 15, respectively, and zero in every other degree. The authors then state that \\(X^{6,4}\\) has the same Betti numbers over \\(\\mathbb Z/2\\) and \\(\\mathbb Q\\) in every degree. That all-degree statement, rather than the earlier table truncated at degree 15, supports the universal-coefficient conclusion in the status record.\n\nThe associated repository was inspected at commit `15eebf49305d28a50cadc591aba6320f42e99bc3`. Its 454-byte `Polymake/n6r4_homology.poly` script constructs the full Rips complex and asks Polymake for rational homology. The repository supplies no stored output certificate and declares no repository-wide source license. This packet cites the paper's reported table and does not copy repository code.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "published rational and mod-two homology reports for VR(Q_6;4), preserving each source's stated coefficient and degree scope",
    "bounds": {
      "cube_dimension": {
        "min": 6,
        "max": 6
      },
      "scale": {
        "min": 4,
        "max": 4
      },
      "displayed_mod_two_degree": {
        "min": 1,
        "max": 15
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2606.20784",
      "locator": "Appendix C.3 and https://github.com/galettof/VietorisRipsHypercube/tree/15eebf49305d28a50cadc591aba6320f42e99bc3/Polymake; Adams and Virk arXiv:2309.06222v1, Section 6.4.4"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2606.20784",
    "locator": "Appendix C.3 and https://github.com/galettof/VietorisRipsHypercube/tree/15eebf49305d28a50cadc591aba6320f42e99bc3/Polymake; Adams and Virk arXiv:2309.06222v1, Section 6.4.4"
  },
  "relations": [
    {
      "slug": "R388",
      "title": "Torsion-free through n=5 and no 2-primary torsion at n=6",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R393",
      "title": "Propagation gives rational rank bounds 3107 and 110 at n=7",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R389",
      "title": "The minimum domination witness spans an explicit 5-cycle",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R385",
      "title": "The current sources leave odd-primary torsion open",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "hypercube-rips-scale-four-torsion-free",
      "title": "hypercube rips scale four torsion free",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}