{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R389",
  "content_hash": null,
  "slug": "hr4-claim-explicit-cross-polytopal-five-cycle",
  "type": "claim",
  "title": "The minimum domination witness spans an explicit 5-cycle",
  "summary": "The twelve-vertex total-domination witness induces six disjoint forbidden edges, so its full Rips subcomplex is a 5-sphere. The resulting 64-term integral cycle dies over Q and F_2; its integral class is a concrete odd-torsion test.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-claim-explicit-cross-polytopal-five-cycle (“The minimum domination witness spans an explicit 5-cycle”) records a bound, answer, status fact, or structural consequence. The record states: The twelve-vertex total-domination witness induces six disjoint forbidden edges, so its full Rips subcomplex is a 5-sphere.",
  "relevance_source": "recorded",
  "body": "In the transformed coordinates of `hr4-claim-n6-forbidden-graph-reduction`, the twelve-vertex witness is\n`0, 15, 44, 28, 49, 50, 4, 40, 24, 11, 53, 54`.\nIts induced forbidden graph consists of exactly the six edges\n\\[\n\\{0,4\\},\\ \\{11,15\\},\\ \\{24,28\\},\\ \\{40,44\\},\\ \\{49,53\\},\\ \\{50,54\\}.\n\\]\nThe full Rips subcomplex on these vertices is therefore the independence complex of six disjoint edges, which is the join of six copies of \\(S^0\\) and hence a copy of \\(S^5\\). An explicit integral fundamental cycle is the simplicial-join product\n\\[\nz=([0]-[4])*([11]-[15])*([24]-[28])*([40]-[44])*([49]-[53])*([50]-[54]),\n\\]\nexpanded into 64 oriented 5-simplices.\n\nThe published complete rational and mod-two tables both have \\(H_5=0\\), so the image of \\([z]\\) vanishes over \\(\\mathbb Q\\) and \\(\\mathbb F_2\\). The packet does not determine whether \\([z]\\) is zero integrally or has finite odd order. Testing an integral filling or annihilator for this small explicit cycle is a focused first case for the recommended Smith computation.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the full subcomplex and coefficient images of one explicit twelve-vertex cycle in VR(Q_6;4)",
    "bounds": {
      "cube_dimension": {
        "min": 6,
        "max": 6
      },
      "scale": {
        "min": 4,
        "max": 4
      },
      "homological_degree": {
        "min": 5,
        "max": 5
      },
      "cycle_vertices": {
        "min": 12,
        "max": 12
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Witness and adjacency rule in hr4-artifact-total-domination-twelve and hr4-claim-n6-forbidden-graph-reduction; coefficient vanishing in hr4-claim-published-field-homology-n6; 2-primary conclusion in hr4-claim-current-torsion-boundary"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Witness and adjacency rule in hr4-artifact-total-domination-twelve and hr4-claim-n6-forbidden-graph-reduction; coefficient vanishing in hr4-claim-published-field-homology-n6; 2-primary conclusion in hr4-claim-current-torsion-boundary"
  },
  "relations": [
    {
      "slug": "R382",
      "title": "Exact total-domination search for the n=6 forbidden graph",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R392",
      "title": "Published field computations for VR(Q_6;4)",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R388",
      "title": "Torsion-free through n=5 and no 2-primary torsion at n=6",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R384",
      "title": "Reduce the n=6 complex before exact odd-prime and Smith computations",
      "object_type": "attempt",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "hypercube-rips-scale-four-torsion-free",
      "title": "hypercube rips scale four torsion free",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}