{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R394",
  "content_hash": null,
  "slug": "hr4-claim-total-domination-twelve",
  "type": "claim",
  "title": "The n=6 forbidden graph has total domination number 12",
  "summary": "The exact value gamma_t=12 improves the published degree-only connectivity certificate: the cited Chudnovsky-Meshulam total-domination theorem proves that VR(Q_6;4) is 4-connected.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-claim-total-domination-twelve (“The n=6 forbidden graph has total domination number 12”) records a bound, answer, status fact, or structural consequence. The record states: The exact value gamma_t=12 improves the published degree-only connectivity certificate: the cited Chudnovsky-Meshulam total-domination theorem proves that VR(Q_6;4) is 4-connected.",
  "relevance_source": "recorded",
  "body": "Under the graph reduction in this packet, \\(\\operatorname{VR}(Q_6;4)=I(G)\\), where \\(G\\) is the 64-vertex graph \\(Q_6\\) with antipodal matching. The exact set-cover artifact proves\n\\[\n\\gamma_t(G)=12.\n\\]\nTheorem 2.3 quoted by Bendersky, Elia, and Grbić states that if \\(\\gamma_t(G)>2k\\), then \\(I(G)\\) is \\((k-1)\\)-connected. Taking \\(k=5\\) gives 4-connectivity because \\(12>10\\).\n\nThis result proves integral homology vanishes through degree 4. It falls short of the degree-6 vanishing asserted elsewhere from a mod-two computation.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "total domination and the resulting integral connectivity lower bound for VR(Q_6;4)",
    "bounds": {
      "cube_dimension": {
        "min": 6,
        "max": 6
      },
      "scale": {
        "min": 4,
        "max": 4
      },
      "total_domination_number": {
        "min": 12,
        "max": 12
      },
      "certified_connectivity": {
        "min": 4,
        "max": 4
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2605.00705",
      "locator": "Bendersky, Elia, and Grbić, Theorem 2.3; exact gamma_t computation in hr4-artifact-total-domination-twelve"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2605.00705",
    "locator": "Bendersky, Elia, and Grbić, Theorem 2.3; exact gamma_t computation in hr4-artifact-total-domination-twelve"
  },
  "relations": [
    {
      "slug": "R382",
      "title": "Exact total-domination search for the n=6 forbidden graph",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R388",
      "title": "Torsion-free through n=5 and no 2-primary torsion at n=6",
      "object_type": "claim",
      "relation": "strengthens",
      "direction": "outgoing"
    },
    {
      "slug": "R386",
      "title": "Total domination cannot certify 6-connectivity at n=6",
      "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"
    }
  ]
}