When Kevin Buzzard woke up in London on Monday morning, the calculation had already been checked. Overnight, specialists had worked through what a casual post on X had claimed – and it held up. By midday, the pure mathematics department at Imperial College London was talking about nothing else. "It is a big day," Buzzard told Fortune magazine, adding that it was a great time to be alive.
The trigger was Levent Alpöge, a mathematician who works at the AI company Anthropic. While the football World Cup final was being played in New York, he set the language model Claude Fable 5, released to the public only weeks earlier, on a question that had stood open since 1939: the Jacobian conjecture. His terse announcement that the conjecture was false drew more than 20 million views, according to Fortune. It was Akhil Mathew of the University of Chicago who had pointed him to the problem.
A counterexample instead of a proof
The conjecture comes from algebraic geometry and can be pictured through a map: a polynomial function shifts the points of a space, and the Jacobian determinant measures how strongly it squeezes or folds that space. If the determinant is a non-zero constant everywhere, the assumption went, then the function must be fully reversible by a second polynomial function. Ludwig Kraus stated the two-dimensional case as early as 1884; Ott-Heinrich Keller generalised it to any number of dimensions in 1939.
That is exactly where the newly found example breaks down. Its determinant is a constant −2 at every point, yet three different starting points land on one and the same destination. Anyone who sees only the result can no longer tell which of the three points was meant – so the mapping cannot be reversed.
Nobody stumbled on it for so long, and not for lack of ambition. Stephen Smale put the question on his 1998 list of mathematical problems for the coming century. Attempted proofs by prominent figures such as Beniamino Segre and Wolfgang Gröbner failed on subtle errors; computer calculations at least confirmed the two-dimensional case up to polynomial degree 100.
For many mathematicians, the episode above all shows where the models currently have their strength. "Artificial intelligence is excellent at finding counterexamples," said Fields medallist Jacob Tsimerman of the University of Toronto at the International Congress of Mathematicians in July. He himself, he added, would never have the patience to work through countless candidates.
A residue of unease remains. One can verify that the result is correct, Mathew told Fortune – but it would be nice to be able to tell a story along with it. The pattern is not new in any case: in May, an OpenAI model disproved an 80-year-old Erdős conjecture. In June, 16 researchers from 15 universities used the Leiden Declaration to call for binding rules on transparency, attribution and peer review before AI changes what mathematical knowledge even means.