{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R385",
  "content_hash": null,
  "slug": "hr4-attempt-primary-source-audit",
  "type": "attempt",
  "title": "The current sources leave odd-primary torsion open",
  "summary": "A 2026-07-28 audit found field-valued homology, connectivity bounds, propagation results, and representation decompositions in proved ranges. It found no integral Smith computation or theorem deciding odd-primary torsion at scale four.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-attempt-primary-source-audit (“The current sources leave odd-primary torsion open”) documents a concrete method, search boundary, or failed route. The record states: A 2026-07-28 audit found field-valued homology, connectivity bounds, propagation results, and representation decompositions in proved ranges.",
  "relevance_source": "recorded",
  "body": "The audit searched the exact target and nearby formulations using `VR(Q_6;4) homology`, `hypercube Vietoris-Rips torsion`, `scale four integral homology`, folded-cube independence-complex terminology, and facet terminology. It checked the complete current versions of five primary sources.\n\nAdamaszek and Adams prove the scale-two wedge-of-spheres theorem and list larger scales as open. Adams and Virk give homology lower bounds at every scale, publish the mod-two \\(n=6,r=4\\) table through degree 15, and retain the scale-four homotopy and collapse questions. Briggs, Feng, and Wells repeat that mod-two table, construct further nonzero field-valued classes, and leave the dimensions of nontrivial homology open. Bendersky, Elia, and Grbić give general connectivity and coconnectivity bounds, with the coefficient-scope concern recorded separately. Galetto, Montaño, and Wellner determine representation decompositions for \\(r\\leq3\\) and \\(r=n-1\\); Appendix C.3 reports the rational \\(n=6,r=4\\) computation.\n\nNo checked source computes integral Smith form at \\((n,r)=(6,4)\\), discusses torsion in this complex, or proves torsion-freeness for every \\(n\\). Search silence is recorded as a dated audit result rather than evidence of nonexistence.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "dated prior-art search for integral torsion in scale-four Vietoris-Rips complexes of binary hypercubes",
    "family": "VR(Q_n;4) over integral coefficients"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/2309.06222",
      "locator": "Adamaszek-Adams arXiv:2103.01040v3; Adams-Virk arXiv:2309.06222v1; Briggs-Feng-Wells arXiv:2408.01288v2; Bendersky-Elia-Grbić arXiv:2605.00705v2; Galetto-Montaño-Wellner arXiv:2606.20784v1"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2309.06222",
    "locator": "Adamaszek-Adams arXiv:2103.01040v3; Adams-Virk arXiv:2309.06222v1; Briggs-Feng-Wells arXiv:2408.01288v2; Bendersky-Elia-Grbić arXiv:2605.00705v2; Galetto-Montaño-Wellner arXiv:2606.20784v1"
  },
  "relations": [
    {
      "slug": "R392",
      "title": "Published field computations for VR(Q_6;4)",
      "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"
    }
  ]
}