The proper hat-guessing number of $K_6-e$

Precise statement

Determine the exact proper hat-guessing number of the complete graph on six vertices with one edge removed. The known general bounds leave $\mathrm{HG}_P(K_6-e)\in\{9,10\}$.

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, target selection, construction search, proof development, code generation, exact verification, adversarial critique, and manuscript preparation. Matthew Protti selected and framed the target, directed and evaluated the work, required exact checks, determined the claim scope, approved disclosure, and accepts responsibility.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

A separate adversarial technical review independently regenerated the principal finite core and prompted the scope and proof-presentation corrections incorporated into this public version. The dependency-free verifier regenerates the group, repairs, coordinate-line checks, label classification, Witt-design check, and residual right-degree census. Conventional journal peer review and Lean verification are not claimed.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-09-03

   Reviewed public research disclosure v0.1 (GitHub)

   We prove $\mathrm{HG}_P(K_6-e)=10$. The lower bound uses two order-sensitive twin-player rules obtained by deleting and repairing one point of an explicit sharply four-transitive eleven-point permutation group. On every coordinate line the repaired rules are derangement permutations, are pointwise unequal, and have fixed-point-free composition. Hall's theorem completes the strategy on the four clique vertices. The release also classifies all repairable orbit labels and proves an even-$n$ obstruction for set-symmetric line-permutation twin rules. It does not solve the general $K_n-e$ family.

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.