The result that ended an 87-year-old mathematical conjecture is short enough to read in a few seconds, and any competent mathematician can check it by hand. That combination — brutally hard to find, trivial to verify — is why the mathematical world has spent the past week arguing about what a machine did on a Sunday evening.
While the FIFA World Cup final played out, Levent Alpöge, a mathematician at Harvard and the AI company Anthropic, posted on X that the Jacobian conjecture is false. He credited Akhil Mathew of the University of Chicago with suggesting the problem, and Anthropic's Fable 5 model, released to the public only weeks earlier, with doing the work during the match. The counterexample ran to 216 characters.
The conjecture concerns polynomial maps — rules that take a set of numbers standing for a point in space and move it somewhere else. One test of how gently a map treats space is a quantity called the Jacobian determinant. If it is a non-zero constant everywhere, the map never crushes or folds space at any point, and the conjecture holds that such a map must always be reversible by another polynomial map that returns every point to where it started. Alpöge and the model produced a map from three-dimensional space to itself that has the required constant determinant and nonetheless cannot be undone. One valid counterexample is enough.
A problem with a history of wrong answers
Ludwig Kraus posed the two-dimensional version in 1884; Ott-Heinrich Keller generalised it to any number of dimensions in 1939. Stephen Smale thought it important enough to include in his 1998 list of problems for the coming century. It also collected famous failures: claimed proofs by Beniamino Segre and Wolfgang Gröbner both turned out to contain subtle errors, and a young Yitang Zhang was told he had failed at the problem, years before his celebrated work on gaps between primes. Because the counterexample lives in three dimensions, the two-dimensional case Kraus first asked about is untouched.
Independent mathematicians have since checked the calculation. "This is a pretty big deal," Abhishek Saha of Queen Mary University of London told New Scientist, calling it probably the biggest conjecture in which AI has played a significant role. The Fields medallist Timothy Gowers said it was the first time an AI had settled a problem outside his own field that was large enough for him to have heard of it. Bartósz Naskręcki cautioned that this was no one-line prompt — searching for such an object takes genuine insight — and an OpenAI researcher, Aaron Lou, said an internal version of the company's Codex model had found essentially the same counterexample independently.
Not everyone is impressed, and the dissent is instructive. Andrew Blumberg of Columbia University, who works on a project testing AI systems on research mathematics, said the episode did not shift his expectations. A counterexample ends an argument; a proof explains why something is true. Smale wanted the conjecture settled for what its solution might reveal. That part of the problem is still open.