What was already known
Adriaensen and coauthors proved the general bound HG_P(G)≤|V(G)|+χ(G)−1. The earlier K5−e construction does not settle K6−e.
AI involvement
GPT-5.6 Pro developed the repaired twin-rule construction and proof under Matthew Protti’s direction. Protti chose the target, evaluated the result and accepted responsibility for the release.
The model contributed the retained deletion-repair construction and proof under Protti’s direction; stronger autonomy is not documented.
Importance
Determines a previously open member of the K_n−e family.
Limitations
This finite construction does not determine the general K_n−e family. The supplied adversarial reviewer is not identified as a human expert.
Technical verification
A dependency-free verifier reportedly regenerated the 7,920-element action, repairs and 2,880 lines, checked all 181,440 proper colourings and the residual degree bound. The reported adversarial replay supports the finite core.
Formal checking
Exact permutation-group regeneration, exhaustive line/colouring and Hall-degree checks
- Checker
- K6−e dependency-free verifier and supplied adversarial finite-core replay
- Scope
- HG_P(K6−e)=10 construction, line criterion and 181,440 colourings
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Protti’s evaluation and review are reported, but no separate human explanation of the order-sensitive repair mechanism is documented. Few is the evidence default.
Correction and publication history
Determines a previously open member of the K_n−e family.
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