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The proper hat-guessing number of K7−eK_7-e

We prove HGP(K7−e)=12\mathrm{HG}_P(K_7-e)=12. One lower-bound proof uses two explicit block-disjoint S(5,6,12)S(5,6,12) Witt designs. A second uses orbit maps from an explicitly regenerated sharply five-transitive twelve-point permutation group, combining one set-symmetric and one order-sensitive rule. Both satisfy a general coordinate-line twin-completion criterion, and Hall's theorem completes the clique strategy. The release also proves a disjoint completion-design theorem, an even-nn obstruction scoped to set-symmetric line-permutation twins in this sufficient framework, and a prime-admissibility theorem for the design parameters. It does not solve the general Kn−eK_n-e problem or K8−eK_8-e.

Exact FrontierDelta

proved→proved

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Occurred: Oct 29, 2026

Delta type: REGISTRY REVISION

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 $K_7-e$ · $\mathrm{HG}_P(K_7-e)=12$

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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GPT-5.6 Pro
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Corrects The proper hat-guessing number of K7−eK_7-e →

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We prove HGP(K7−e)=12\mathrm{HG}_P(K_7-e)=12. One lower-bound proof uses two explicit block-disjoint S(5,6,12)S(5,6,12) Witt designs. A second uses orbit maps from an explicitly regenerated sharply five-transitive twelve-point permutation group, combining one set-symmetric and one order-sensitive rule. Both satisfy a general coordinate-line twin-completion criterion, and Hall's theorem completes the clique strategy. The release also proves a disjoint completion-design theorem, an even-nn obstruction scoped to set-symmetric line-permutation twins in this sufficient framework, and a prime-admissibility theorem for the design parameters. It does not solve the general Kn−eK_n-e problem or K8−eK_8-e. parent of this event

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The proper hat-guessing number of K7−eK_7-e parent of this event

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The proper hat-guessing number of K7−eK_7-e evidence for this event

Four-page K7-e preprint source evidence for this event

Hat guessing with proper colorings (Adriaensen et al.) evidence for this event

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The proper hat-guessing number of $K_7-e$ — Mathematical Frontier Network