Levent Alpöge, a mathematician working at AI company Anthropic, announced on X that he had found a counterexample to the Jacobian conjecture — one of the oldest open problems in algebraic geometry — using the company's large language model Claude Fable 5, released to the public only a few weeks earlier.

The Jacobian conjecture concerns polynomial functions, which map points in space to other points. Mathematicians test how "nicely" such a function behaves by computing its Jacobian determinant; if that determinant is a constant, non-zero number, the function never folds or crushes space at any point. The conjecture, stated in two dimensions by Czech mathematician Ludwig Kraus in 1884 and generalized by German mathematician Ott-Heinrich Keller in 1939, holds that any such function must be reversible by another polynomial function.

The problem is so compelling that Fields Medalist Stephen Smale put it on his 1998 list of Mathematical Problems for the Next Century. Over the decades it survived many attempted proofs — including by famed mathematicians Beniamino Segre and Wolfgang Gröbner — all of which collapsed under scrutiny. Computational checks confirmed the two-dimensional case up to degree 100, but the general case resisted every effort.

Alpöge's counterexample is deceptively simple: a polynomial mapping in three dimensions with a constant Jacobian determinant of -2 that moves multiple input points to the same output point, making it impossible to reverse. The formula is short enough to fit in a single X post, which made it easy for other mathematicians to verify. It shows the conjecture is false in every dimension above two, while the original two-dimensional version remains open.

The result is the latest in a string of AI-assisted mathematical breakthroughs, including OpenAI's disproof of the unit distance conjecture and amateur mathematician Liam Price's proof of Erdős problem 1196. But Alpöge's discovery has a different flavor: the difficulty lay not in an intricate construction or a lengthy proof, but in searching an enormous space of possible polynomial mappings to find one with the right properties. It suggests large language models may be as valuable for discovering unexpected mathematical objects as for constructing proofs. Details of how Alpöge prompted the model have not yet been made public.

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