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Weak Order on the MacNeille Completion of Bruhat Order

Colin Defant

Defant’s manuscript displays an interval in the weak order on six-by-six alternating sign matrices whose Möbius value is 1, although it has none of the specified parabolic forms. Its open-interval order complex is therefore not contractible, contrary to the Hamaker–Reiner dichotomy.

Sources for this record

What was already known

Hamaker and Reiner conjectured that open intervals outside specified parabolic forms are contractible, while the specified intervals have sphere homotopy type.

AI involvement AI-autonomous

Defant asked ChatGPT 5.4 Pro to prove the conjecture; it returned the counterexample. Defant incorporated and explained it. His later correction of a model error concerns a different lemma.

The paper's abstract and §7 explicitly say ChatGPT autonomously found the disproof, returning a counterexample when prompted to prove the conjecture.

Importance Notable

The finite interval disproves the published topology conjecture.

Limitations

The paper does not document a separate completed check or executable certificate for the displayed interval. Defant’s later maximal-parabolic identity and shelling results are separate contributions.

Technical verification Reported

The v2 paper displays a finite ASM_6 interval and states its Möbius value, but it does not document a named complete human check or an exact computational certificate for that interval. A displayed example is evidence for the claim, not a documented checking method.

Human understanding Full

Defant identifies the precise failure: the interval is outside the specified parabolic form but has Möbius value 1, so its open order complex cannot be contractible.

Correction and publication history