White's Conjecture on Matroids
Statement
White conjectured that the symmetric exchange binomials generate the toric ideal of a matroid. This is now known to be false; a rank binary matroid constitutes a counterexample.
Context
A named conjecture from 1980 on the toric ideal of a matroid, with a substantial literature of proved special cases behind it. Placed just above Rota's flat unimodality conjecture at 36: better known in the algebraic-combinatorics community, and forty-five years old when it fell.
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Attempts
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The author prompted ChatGPT-5.5 Pro to check White's conjecture and it claimed to prove the result for ranks and . After further prompting to check for random matroids of ranks , , , and , it found a matroid isomorphic to the given counterexample.
ChatGPT-5.5 Pro and Claude Opus 4.8 were used for proofreading and generating the figures.
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