What was already known
White proposed progressively weaker basis-exchange statements. Special matroid classes and Farber’s version through rank four were known; a separate result had already disproved the stronger Gröbner-basis version.
AI involvement
ChatGPT 5.5 Pro found a matroid isomorphic to Larson’s example during his directed search. Larson established the structural invariant and wrote the proof.
The paper specifically credits ChatGPT with finding the rank-nine counterexample, within Larson’s directed search; Larson supplied the proof.
Importance
The counterexample settles the symmetric-exchange generation claim in the negative.
Limitations
It does not refute White’s weaker quadratic-generation conjecture. Larson labels model SAT observations about ranks five and six unverified.
Technical verification
Larson supplies a complete human structural proof for the AI-found example in Theorem 1.5. His unverified ChatGPT SAT claims about ranks five and six are outside the checked contribution.
Formal checking
Human proof using the invariant s(B) on complementary basis pairs
- Checker
- Matt Larson, author
- Scope
- Example 1.4 refutes Farber’s Conjecture 1.2 and White’s Conjecture 1.1(2)
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Larson explains why the block and split-pair invariant separates symmetric-exchange components in the rank-nine matroid.
Correction and publication history
The counterexample settles the symmetric-exchange generation claim in the negative.
Read this published revision