While more than 60 million people in the United States watched the FIFA World Cup final on July 19, an artificial intelligence model was hard at work solving—or technically, disproving—a decades-old math problem.

Mathematician Levent Alpöge, of A.I. company Anthropic and Harvard University, used Anthropic’s Claude Fable 5 to approach a mathematical conjecture: an idea that mathematicians believe to be true but has not yet been proven or disproven. He then posted an A.I.-assisted disproof on social media. It has since been verified by several independent mathematicians.

“hello there the jacobian conjecture is false,” Alpöge wrote on X. He thanked Akhil Mathew, a mathematician at the University of Chicago, for suggesting the problem and his “other close friend fable” for working during the match.

“This is a pretty big deal,” Abhishek Saha, a mathematician at Queen Mary University of London, tells New Scientist’s Matthew Sparkes. “Probably, this is the biggest conjecture that A.I. has played a significant role [in proving or disproving] so far in mathematics.”

hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final

— levent (@alpoge) July 20, 2026

((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3,…

In 1884, Czech mathematician Ludwig Kraus proposed a version of the Jacobian conjecture, which was expanded into its current form by German mathematician Ott-Heinrich Keller in 1939.

The conjecture concerns polynomial functions, which behave like mathematical machines that take in one set of numbers and output a new set, writes Melissa Lee, a mathematician at Monash University in Australia, for the Conversation. Put in a set of numbers representing points in space—like coordinates on a map—and a function will move those numbers to new points according to the rules of the function.

“You can test how ‘nicely’ a function moves everything around in space by calculating something called the Jacobian determinant. If the Jacobian determinant is always a constant number that is not zero, then the function never folds or crushes space around a particular point,” Lee writes.

According to the conjecture, if the determinant is a nonzero constant, there should always be another function you can use to reverse all points to their original positions. Theoretically, it should be simple to disprove by finding a counterexample function that makes all the points merge and has a nonzero constant as the Jacobian determinant—but it’s proven far more challenging to do in real life. Mathematician Stephen Smale, a winner of the Fields Medal, often referred to as the Nobel Prize of mathematics, even included it on a 1998 list of 18 major mathematical problems for the 21st century.

Alpöge’s X post contains such a function. Using Fable, he found a three-dimensional function that has a constant Jacobian determinant and merges multiple input points, which makes it irreversible. This shows that the conjecture is false for spaces that are three or more dimensions, though it still could be true for lower-dimensional spaces.

Many outside researchers describe the result as a milestone for A.I.-assisted mathematics. Still, finding a single counterexample isn’t nearly as big a deal as proving the Jacobian conjecture, which hasn’t been done, Columbia University mathematician Andrew Blumberg tells Mashable’s Timothy Beck Werth. It mostly shows that there are “a lot of polynomials,” which are hard for humans alone to check.

A.I. could help prove conjectures, but at this point, the models aren’t able to provide reliable, step-by-step instructions for solving them, reports Fortune’s Eva Roytburg. Humans can fill more than 100 pages documenting just one proof.

Blumberg describes A.I.’s role with an analogy: “Suppose that Moses came down from the mountain with tablets, and on the tablet was written, ‘Cancer can be cured.’ Would you care? You don’t just want the answer to the question. You want to learn something from the answer,” he tells Mashable. “The reason Smale thought this problem was important is because he thought that if we solved it, we would understand more things about the about the way nature is structured.”

Nevertheless, the advance is widely seen as outpacing expectations for how soon A.I. would become useful in advanced mathematical research. In 2025, forefront models were able to solve five out of six problems at the International Mathematical Olympiad. And in May, OpenAI announced that a model from the company helped researchers solve an 80-year-old problem posed by Hungarian mathematician Paul Erdős. Following the announcement, OpenAI researcher Sébastien Bubeck told the Wall Street Journal’s Ben Cohen that the prospect of that breakthrough would have seemed laughable even “a month ago.”

Quick fact: Mathematicians urge caution about A.I.

After that “Erdős problem” was solved with the help of A.I., 16 mathematicians published the Leiden Declaration on Artificial Intelligence and Mathematics, which is endorsed by the International Mathematical Union. It calls on researchers to disclose their use of A.I., properly credit earlier work and publish their results in peer-reviewed outlets. It also warns against announcing claims before the mathematical community can evaluate them through formal channels.

Although the Jacobian counterexample is exciting, the shift toward A.I. in pure mathematics is “very rapid and very unsettling … especially for junior mathematicians,” says Mathew, who Alpöge mentioned in his X post, to Fortune.

It gets at the “how” without explaining the “why,” he adds. “One can check out that it’s correct, but it would be nice to be able to tell a story.”