{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R386",
  "content_hash": null,
  "slug": "hr4-attempt-total-domination-six-connectivity",
  "type": "attempt",
  "title": "Total domination cannot certify 6-connectivity at n=6",
  "summary": "The attractive total-domination route stops at 4-connectivity because the exact invariant is 12. The cited theorem would require gamma_t greater than 14 to certify 6-connectivity.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-attempt-total-domination-six-connectivity (“Total domination cannot certify 6-connectivity at n=6”) documents a concrete method, search boundary, or failed route. The record states: The attractive total-domination route stops at 4-connectivity because the exact invariant is 12.",
  "relevance_source": "recorded",
  "body": "The action was to compute \\(\\gamma_t\\) exactly and feed it into the total-domination connectivity theorem. The search examined every candidate size allowed by the elementary lower bound until it found the optimum 12.\n\nTo obtain 6-connectivity from Theorem 2.3, set \\(k=7\\). Its strict hypothesis becomes \\(\\gamma_t(G)>14\\). The exact value 12 violates that hypothesis. The same theorem certifies 4-connectivity and cannot certify 5-connectivity, whose hypothesis would be \\(\\gamma_t(G)>12\\).\n\nThis failure boundary is exact. Improving the total-domination search or finding a different minimum witness cannot extend this theorem's result for the same graph. Another topological argument is required.",
  "status": "failed",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the total-domination method applied to 6-connectivity of VR(Q_6;4)",
    "bounds": {
      "cube_dimension": {
        "min": 6,
        "max": 6
      },
      "scale": {
        "min": 4,
        "max": 4
      },
      "target_connectivity": {
        "min": 6,
        "max": 6
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/2605.00705",
      "locator": "Theorem 2.3 and exact total-domination artifact in this packet"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2605.00705",
    "locator": "Theorem 2.3 and exact total-domination artifact in this packet"
  },
  "relations": [
    {
      "slug": "R382",
      "title": "Exact total-domination search for the n=6 forbidden graph",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R394",
      "title": "The n=6 forbidden graph has total domination number 12",
      "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"
    }
  ]
}