{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R384",
  "content_hash": null,
  "slug": "hr4-attempt-odd-prime-homology",
  "type": "attempt",
  "title": "Reduce the n=6 complex before exact odd-prime and Smith computations",
  "summary": "The next computation should construct a certified Morse or symmetry reduction of the n=6 independence complex, then compute exact boundary ranks over F_3 and integral Smith data. A finite prime screen must retain its prime-by-prime scope.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-attempt-odd-prime-homology (“Reduce the n=6 complex before exact odd-prime and Smith computations”) documents a concrete method, search boundary, or failed route. The record states: The next computation should construct a certified Morse or symmetry reduction of the n=6 independence complex, then compute exact boundary ranks over F_3 and integral Smith data.",
  "relevance_source": "recorded",
  "body": "The unreduced complex has 2,932,100,733 faces including the empty face, so direct boundary assembly reproduces the published high-memory barrier. The graph description and exact f-vector provide fixed inputs and checksums for a smaller chain model. The explicit 64-term cycle in `hr4-claim-explicit-cross-polytopal-five-cycle` supplies a focused first integral filling test.\n\nA useful next run has three stages. First, build an acyclic discrete Morse matching or an equivariant orbit-chain reduction for the independence complex of \\(Q_6\\) with antipodal matching, and emit the critical cells together with a checkable matching certificate. Second, map the explicit 5-cycle into the reduced complex, test whether it has an integral filling or finite annihilator, and compute ranks over \\(\\mathbb F_3\\) in degrees 5 through 21. The 4-connectivity result handles lower degrees, while the exact f-vector shows that the complex has dimension 21. Third, compute Smith data or determinantal-divisor certificates for every reduced boundary matrix.\n\nUse a first-pass budget of 48 wall hours, 256 GiB of memory, and 1 TiB of scratch storage. Stop and report an inconclusive reduction if the matching fails acyclicity validation, reaches a resource cap, or leaves a chain model too large to materialize every required boundary. Matching rational ranks over \\(\\mathbb F_3\\) would exclude 3-primary torsion only. Repeating for selected odd primes remains a finite screen. A proof of complete torsion-freeness needs integral Smith form or a separate bound that limits possible torsion primes.",
  "status": "next_experiment",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "bounded",
    "statement": "a proposed characteristic-three and integral reduced-chain computation for VR(Q_6;4)",
    "bounds": {
      "cube_dimension": {
        "min": 6,
        "max": 6
      },
      "scale": {
        "min": 4,
        "max": 4
      },
      "coefficient_characteristic": {
        "min": 3,
        "max": 3
      },
      "homological_degree": {
        "min": 5,
        "max": 21
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "locator": "Plan based on the exact graph, f-vector, connectivity, and coefficient boundary recorded in this packet"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Plan based on the exact graph, f-vector, connectivity, and coefficient boundary recorded in this packet"
  },
  "models": [],
  "relations": [
    {
      "slug": "R390",
      "title": "The n=6 complex is an independence complex of Q_6 with antipodal edges",
      "object_type": "claim",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "R389",
      "title": "The minimum domination witness spans an explicit 5-cycle",
      "object_type": "claim",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "R388",
      "title": "Torsion-free through n=5 and no 2-primary torsion at n=6",
      "object_type": "claim",
      "relation": "attempts",
      "direction": "outgoing"
    },
    {
      "slug": "hypercube-rips-scale-four-torsion-free",
      "title": "hypercube rips scale four torsion free",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}