{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination",
  "title": "The γ–θ conjecture in eternal domination",
  "canonical_statement": "The $\\gamma$–$\\theta$ conjecture asserts that for every finite graph $G$,\n$$\n\\gamma(G)=\\gamma^\\infty(G)\n\\quad\\Longrightarrow\\quad\n\\gamma(G)=\\theta(G),\n$$\nwhere $\\gamma$ is the domination number, $\\gamma^\\infty$ is the eternal domination number in the one-guard-moves model, and $\\theta$ is the vertex clique-cover number.\n\nThe conjecture is false. The complement $G$ of the 243-vertex ternary Golay graph satisfies\n$$\n\\gamma(G)=\\gamma^\\infty(G)=3<\\theta(G).\n$$",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination:event",
      "problem_id": "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination",
      "type": "disproved",
      "status": "candidate",
      "occurred_at": "2026-09-05T00:00:00.000Z",
      "title": "Github",
      "summary": "Let $H$ be the Berlekamp–van Lint–Seidel graph on $243$ vertices, the Cayley graph of $\\mathbb Z_3^5$ with strongly regular parameters\n$$\n(243,22,1,2),\n$$\nand let $G=\\overline H$.\n\nThe proof establishes\n$$\n\\gamma(G)=3\n$$\nbecause every pair has a common neighbor in $H$, while an $H$-triangle gives a dominating triple in $G$.\n\nIt then proves\n$$\n\\gamma^\\infty(G)=3\n$$\nby showing that the family of all dominating triples is closed under a legal response to every attack: after moving one guard to the attacked vertex, another dominating triple can always be obtained.\n\nFinally, a double-counting argument shows that $H$ is not 3-colorable, hence\n$$\n\\theta(G)=\\chi(H)>3.\n$$\nTherefore\n$$\n\\gamma(G)=\\gamma^\\infty(G)=3<\\theta(G).\n$$",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination:attempt"
      ],
      "method_family_ids": [
        "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination:method"
      ],
      "source_assertion_ids": [
        "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination:attempt",
      "problem_id": "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination",
      "model": "GPT-6 Astra (pre-release)",
      "provider": "OpenAI",
      "attempted_at": "2026-09-05T00:00:00.000Z",
      "human_collaborators": [
        "Tom Adamczewski"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "GPT-6 Astra found the counterexample and wrote the Lean proof. It recognized that the complement of the classical 243-vertex Berlekamp–van Lint–Seidel ternary Golay graph provides the needed structure, proved $\\gamma=3$, constructed an indefinitely repeatable three-guard defense proving $\\gamma^\\infty=3$, and proved $\\theta>3$ by a coloring/double-counting argument. The underlying Golay graph was known since 1973; the new contribution is identifying its complement as a counterexample and proving the eternal domination strategy.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Challenge.lean: the compared statement and definitions",
          "url": "https://github.com/tadamcz/gamma-theta/blob/main/Challenge.lean",
          "kind": "lean-statement",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/gamma-theta",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The γ–θ conjecture in eternal domination",
          "url": "https://vibemathed.com/problem/counterexample-to-the-gamma-theta-conjecture-in-eternal-domination",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination:verification",
      "problem_id": "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination",
      "solution_event_id": "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Lean-checked, statement unaudited. Checked here on 6 September 2026 from a clone of tadamcz/gamma-theta at d64cce5: 1,987 lines; the only sorry outside Challenge.lean is the unused '.disproof' stub in submission/Spec.lean, which the README explains (the compared theorem asserts the existence of the counterexample); zero axiom declarations, no native_decide; the audit folder's docker logs print the theorem's axioms as the standard three. The statement was AI-autoformalized in Epoch's wikipedia run; the definitions of the eternal dominating family (one-guard model), domination number and clique cover number were read here and look right, but that is one reading, not an audit. The repository's Python check of 5,889,840 attacks was not replayed here.",
      "sources": [
        {
          "label": "VibeMathed: The γ–θ conjecture in eternal domination",
          "url": "https://vibemathed.com/problem/counterexample-to-the-gamma-theta-conjecture-in-eternal-domination",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/gamma-theta",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Github",
      "url": "https://github.com/tadamcz/gamma-theta",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The γ–θ conjecture in eternal domination",
      "url": "https://vibemathed.com/problem/counterexample-to-the-gamma-theta-conjecture-in-eternal-domination",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination:signal:verify_now",
      "problem_id": "vibemathed:counterexample-to-the-gamma-theta-conjecture-in-eternal-domination",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}