AI counterexample settles 1939 Jacobian conjecture

An AI model has resolved the Jacobian conjecture, a problem in pure mathematics that dates to 1939, Fortune reported. The result, announced by Anthropic employee Levant Alpöge, was checked in the Lean proof system and has sharpened concern among mathematicians about how fast AI is entering their field.

Kevin Buzzard, a mathematician at Imperial College London, told Fortune that the result had been verified by the time he woke up in London the morning after Alpöge posted it. Alpöge’s post on X had drawn more than 20 million views at the time Fortune reported the story.

What the result showed

The Jacobian conjecture is associated with German mathematician Ott-Heinrich Keller and is tied to the Jacobian determinant, a tool named for Carl Gustav Jacob Jacobi. Fortune described the problem as concerning mathematical “maps” and when outputs allow a mathematician to determine the inputs that produced them.

According to Fortune, Alpöge’s construction met the Jacobian determinant condition everywhere in space, with the determinant fixed at -2. It still sent three different starting points to the same endpoint, providing a counterexample to the conjecture.

Buzzard called the result “very exciting” in Fortune’s account and said it points toward the kind of “supermathematician” that Google Deep Learning scientist Christian Szegedy warned about roughly five years ago. The breakthrough follows other AI-assisted results, including models solving five of six International Mathematical Olympiad problems in mid-2025 and an OpenAI model disproving an 80-year-old Erdős conjecture in combinatorial geometry in May, Fortune reported.

Verification and understanding

The result also exposes a gap that mathematicians see in current AI work. Akhil Mathew, a University of Chicago mathematician whom Alpöge credited with suggesting the problem, told Fortune that the answer can be checked, but mathematicians still want an explanation that gives the result a clear mathematical story.

Buzzard told Fortune that advanced mathematics is less about calculation than reasoning. A formal proof is a sequence of logical steps that other mathematicians can examine, and some important proofs can take hundreds of pages and months of expert review.

He said current language models still struggle to produce long, delicate proofs because they can fill gaps with plausible but wrong material. Buzzard’s work on Lean, a language for machine-checking proofs, is central to the shift: Fortune reported that the Jacobian result had already been checked in Lean by the time Buzzard saw it.

A changing profession

Mathew described the moment to Fortune as “a very rapid and very unsettling change,” especially for early-career mathematicians. In June, 16 researchers from 15 universities issued the Leiden Declaration on Artificial Intelligence and Mathematics, calling for standards on transparency, credit and peer review as AI tools become more capable.

The debate is unfolding as the profession faces financial pressure. Fortune reported that federal funding for mathematics research has fallen roughly 72% under Trump administration cuts to the National Science Foundation, while PhD admissions at leading research universities are down 15% this fall for a second year of decline.

Some technologists have welcomed the change. Fortune cited Y Combinator president Garry Tan, who responded on X by invoking a return to the era of the “gentleman scientist.” But Fortune noted that Alpöge is not an amateur outsider: he is a Harvard valedictorian who has spent years using algorithms on this class of problem.

Buzzard said humans may retain a role in deciding which problems deserve attention. He told Fortune that machines remain poor at posing worthwhile mathematical questions, while the field’s best-known problems often bear the names of the people who asked them.

This story draws on original reporting from Fortune.