VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
The $\gamma$–$\theta$ conjecture asserts that for every finite graph $G$,
$$
\gamma(G)=\gamma^\infty(G)
\quad\Longrightarrow\quad
\gamma(G)=\theta(G),
$$
where $\gamma$ 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.
The conjecture is false. The complement $G$ of the 243-vertex ternary Golay graph satisfies
$$
\gamma(G)=\gamma^\infty(G)=3<\theta(G).
$$
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-6 Astra (pre-release)
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.
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.
Let $H$ be the Berlekamp–van Lint–Seidel graph on $243$ vertices, the Cayley graph of $\mathbb Z_3^5$ with strongly regular parameters
$$
(243,22,1,2),
$$
and let $G=\overline H$.
The proof establishes
$$
\gamma(G)=3
$$
because every pair has a common neighbor in $H$, while an $H$-triangle gives a dominating triple in $G$.
It then proves
$$
\gamma^\infty(G)=3
$$
by 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.
Finally, a double-counting argument shows that $H$ is not 3-colorable, hence
$$
\theta(G)=\chi(H)>3.
$$
Therefore
$$
\gamma(G)=\gamma^\infty(G)=3<\theta(G).
$$
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
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.