AI Just Disproved an 87-Year-Old Math Conjecture
While the world watched the World Cup final, a mathematician and an AI model quietly broke a problem that had stood since 1939. The proof fit in a single tweet. Whether it means much is the more interesting question.
July 21, 2026 – 1:04 pm
Image by: Canva
On Sunday evening, as Spain and Argentina played out the World Cup final, mathematician Levent Alpöge posted a short message on X:
The Jacobian conjecture is false.
He thanked friends for their interest in the problem, one of whom, "fable," had been working during the match. That friend was Fable 5, Anthropic’s latest AI model. Together, they had produced a counterexample to a problem that had resisted mathematicians for 87 years, all within 216 characters.
What Was Actually Broken
The Jacobian conjecture, set out by Ott-Heinrich Keller in 1939, states that a certain kind of polynomial map with a non-zero constant Jacobian determinant must be reversible with a neat polynomial inverse. It became one of the field’s most stubborn open problems. Stephen Smale included it on his famous 1998 list of challenges for the 21st century, and New Scientist calls it the hardest math problem an AI has yet cracked.
The Model Behind It
Fable 5 is Anthropic’s newest frontier model, a public version of the system once called Claude Mythos, which Anthropic described as too capable to release. Alpöge, a Harvard fellow, lists an Anthropic affiliation.
The result is easy to check even if it was hard to find. The counterexample is a plain list of polynomials that any mathematician can verify by hand. Finding it was the needle-in-a-haystack part, and that’s where the machine came in.
Mathematicians’ Reaction
The reaction online was loud. Timothy Gowers, a Fields medallist, said it was the first time an AI had solved a problem outside his own area significant enough for him to have heard of it. He stopped short of declaring the end of mathematics, since a counterexample is not a grand theory, but he called it amazing.
He was not alone. Mathematician Daniel Litt posted at 2am that he could not stop laughing. He called it incredible.
Part of the delight is historical. As Stanford number theorist Jared Duker Lichtman noted, the conjecture is notorious for attracting wrong proofs, and it once helped sink the PhD of Yitang Zhang. He was told he had failed at it, then later became famous for work on gaps between prime numbers. An AI settling it in a tweet is, to many, poetic.
It’s also the latest sign of AI turning up in serious science, from biosecurity to pure math. And rivals were watching closely.