{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R390",
  "content_hash": null,
  "slug": "hr4-claim-n6-forbidden-graph-reduction",
  "type": "claim",
  "title": "The n=6 complex is an independence complex of Q_6 with antipodal edges",
  "summary": "After an explicit linear change of coordinates over F_2, VR(Q_6;4) is the independence complex of the 64-vertex graph obtained by adding the antipodal perfect matching to the ordinary 6-cube.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-claim-n6-forbidden-graph-reduction (“The n=6 complex is an independence complex of Q_6 with antipodal edges”) records a bound, answer, status fact, or structural consequence. The record states: After an explicit linear change of coordinates over F_2, VR(Q_6;4) is the independence complex of the 64-vertex graph obtained by adding the antipodal perfect matching to the ordinary 6-cube.",
  "relevance_source": "recorded",
  "body": "The nonedges of the Rips graph on \\(Q_6\\) join pairs whose difference has Hamming weight 5 or 6. Write \\(\\mathbf1\\) for the all-one vector and \\(s_i=\\mathbf1+e_i\\) for \\(1\\leq i\\leq6\\). The six weight-five vectors \\(s_i\\) form a basis of \\(\\mathbb F_2^6\\), and\n\\[\n\\sum_{i=1}^6s_i=\\mathbf1.\n\\]\nIn coefficient coordinates for this basis, the six weight-five generators become \\(e_1,\\ldots,e_6\\), while the weight-six generator becomes \\(\\mathbf1\\). Thus the forbidden-pair graph is isomorphic to the graph on \\(\\{0,1\\}^6\\) with the ordinary cube edges and the antipodal perfect matching. A Rips face is exactly an independent set in this graph.\n\nSome authors use `folded cube` for this 64-vertex graph, while others reserve that name for an antipodal quotient. The description above fixes the convention used in this packet. The executable f-vector artifact checks the coordinate map against all 2,016 unordered vertex pairs.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the complete forbidden-pair graph for VR(Q_6;4), including all 2,016 unordered vertex pairs",
    "bounds": {
      "cube_dimension": {
        "min": 6,
        "max": 6
      },
      "scale": {
        "min": 4,
        "max": 4
      },
      "vertices": {
        "min": 64,
        "max": 64
      },
      "unordered_pairs_checked": {
        "min": 2016,
        "max": 2016
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Elementary basis calculation and exhaustive pair check in hr4-artifact-exact-fvectors-six-seven"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Elementary basis calculation and exhaustive pair check in hr4-artifact-exact-fvectors-six-seven"
  },
  "models": [],
  "relations": [
    {
      "slug": "R381",
      "title": "Exact deletion-contraction f-vectors for n=6 and n=7",
      "object_type": "artifact",
      "relation": "evidences",
      "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"
    }
  ]
}