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
Determine the exact proper hat-guessing number of the complete graph on five vertices with one edge removed. The existing bounds left $\mathrm{HG}_P(K_5-e)$ in $\{7,8\}$.
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-5.6 Pro
OpenAI GPT-5.6 Pro contributed substantively to literature search, problem selection, construction search, proof development, code generation, computational verification, adversarial critique, and manuscript preparation. The central $\mathbb F_2^3$ construction and Hall-completion proof were developed in a model-assisted process. Matthew Protti selected and framed the target, directed and evaluated the work, required exact checks, set the scope, approved disclosure, and accepts responsibility.
No independent mathematical review yet. The public disclosure contains a self-contained proof, a seven-entry finite certificate, and a dependency-free Python verifier. The verifier has been run successfully from a clean clone and from the v0.1 release archive; it reconstructs the residual incidence graph, a saturating matching, and a complete 6,720-entry strategy, then checks all 8,400 proper colorings. This remains author-controlled verification.
The proper hat-guessing number of K5-e - public research disclosure v0.1
The exact value is $\mathrm{HG}_P(K_5-e)=8$. The lower bound uses explicit legal twin-player rules over $\mathbb F_2^3$. After those rules cover 3,024 of the 8,400 proper colorings, the residual-coloring/local-view incidence graph has left degree three and right degree at most three; Hall's theorem supplies consistent guesses for the three clique players. The release also proves a general sufficient twin-completion lemma for $K_n-e$. It does not solve the full $K_n-e$ family or determine $\mathrm{HG}_P(C_5)$.
Known method families
construction (source-reported)
Source-reported tools: construction.
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.