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

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 3, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-co-developed. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The proper hat-guessing number of $K_6-e$ · $\mathrm{HG}_P(K_6-e)=10$

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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VibeMathed record: The proper hat-guessing number of $K_6-e$

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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.

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