What was already known
Mason’s flat-count log-concavity conjecture was open. Results for characteristic-polynomial coefficients and independent-set counts do not prove it.
AI involvement
ChatGPT found a variant of the graph after Matt Larson directed a search toward large F5-representable matroids and high-index failures. Larson supplied the counting proof and paper.
The disclosed model search found the counterexample variant under Larson’s directions; the proof and exposition are human work.
Importance
The explicit graphic counterexample resolves the flat-count log-concavity question while leaving the weaker unimodality question separate.
Limitations
This example does not itself refute unimodality. A later human-led paper develops nonunimodal examples using additional constructions.
Technical verification
Larson supplies and claims authorship of a complete human combinatorial count for the AI-found construction, ending in an explicit strict inequality. This is a source-reported completed proof check of the construction, not a reproduced independent calculation.
Independent checking
Human dual-matroid cyclic-set enumeration and exact integer inequality in Theorem 2.3
- Checker
- Matt Larson, author
- Scope
- Example 2.2 violates Mason’s flat-count log-concavity at index 74
- Basis
- Reported by the linked checker
Human understanding
Larson explains the generalized-theta graph, its dual parallel classes, the cyclic-set counts and why unequal class sizes create failure.
Correction and publication history
The explicit graphic counterexample resolves the flat-count log-concavity question while leaving the weaker unimodality question separate.
Read this published revision