What was already known
Adriaensen and coauthors gave a general upper bound but left this Kₙ−e endpoint case unresolved. Exact values for n=5, 6 and 7 are separate earlier results.
AI involvement
The public submission credits GPT-5.6 Pro with construction search, certificate design, proof development and exact verification. Protti directed and repaired the work and set the claim scope.
The model contributed the retained equivariant construction, residual matching and certificate design under Protti’s direction.
Importance
The exact strategy settles the eight-vertex case of the proper hat-guessing problem.
Limitations
The source reports exact certificate regeneration and exhaustive checking, but no named independent domain-expert review. A separate shared-completion companion remains review-pending.
Technical verification
The source reports exact regeneration of the PGL action and matching, an independent orbit checker and a C++ sweep of all 138,378,240 proper colourings with zero uncovered cases. This certifies the finite strategy on a source-reported basis.
Formal checking
Exact group normalization, residual matching certificate and exhaustive game check
- Checker
- Data-only orbit/matching verifier and independent C++ full-colouring checker
- Scope
- Winning fourteen-colour strategy for K8−e over 138,378,240 proper colourings
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The source credits Protti with direction and scope repair, but does not document his own explanation of the central matching construction. Few is the evidence default.
Correction and publication history
The exact strategy settles the eight-vertex case of the proper hat-guessing problem.
Read this published revision