Anthropic mathematician Levent Alpöge says he used Anthropic’s Claude Fable 5 model to find a counterexample that disproves the Jacobian conjecture, an algebraic-geometry problem that had stood unresolved since 1939.
Alpöge posted the result on X on July 20. The Claude Fable 5-assisted construction is a polynomial map from three-dimensional space to itself whose Jacobian determinant is a constant −2 — exactly the condition the conjecture says should force the map to be reversible. Instead, three distinct points, (0, 0, −1/4), (1, −3/2, 13/2) and (−1, 3/2, 13/2), all map to the same output, (−1/4, 0, 0), meaning the map has no inverse. That single example is enough to make the conjecture false for every dimension of three or higher; the two-dimensional case remains open.
Independently checked within a day
Mathematicians moved quickly to check the claim. Fields medalist Terence Tao wrote on his blog on July 21 that he had personally verified the determinant and the three-point collision, and described the cancellations behind the construction as “a massive miracle” — the polynomial satisfies 1,329 equations against only 360 degrees of freedom, a coincidence he said would be “highly unlikely” to stumble on by brute-force search. Fields medalist Timothy Gowers called it the first time he had seen an LLM solve a problem outside his own field that was nevertheless “big enough that I had very definitely heard of it,” while cautioning it was still “a counterexample, so not in ’end of mathematics’ territory.” Other mathematicians published a verification preprint checking the arithmetic independently, and the construction has since been generalized to every dimension above two.
Alpöge credited Akhil Mathew, an algebraic geometer at the University of Chicago, with posing the question, and said Fable 5 helped him search for the example over a few hours. Mathematician John D. Cook, among others, cautioned against overstating the AI’s role: Claude functioned as “an inanimate tool in the hand of a mathematician,” directed by Alpöge rather than working independently. Even so, several researchers said the episode — systematically searching for an object with specific algebraic properties, rather than writing a proof outright — points to a role for AI models in mathematical discovery distinct from proof-checking.
Not yet peer-reviewed
The result has not gone through formal peer review, and the mathematicians who checked it stress that arithmetic verification is not the same as a vetted publication. It follows OpenAI’s contested claim that GPT-5.6 Sol Ultra had proved a different long-standing conjecture, adding to a short but growing list of open problems shaped by AI-assisted discovery.