Published 22 July 2026 · Updated 17 September 2026
A one-line formula just overturned a mathematics problem that had stood for 87 years, and it was written during the World Cup final. Anthropic researcher Levent Alpoge used Fable 5 to produce a proof that disproves the Jacobian conjecture, a famous algebra rule that has stumped mathematicians since 1939.
The result came packaged as a deliberately casual post on X, where Alpoge thanked his “close friend fable for working during the World Cup final.” That understatement masked a significant moment: the problem had previously seen several claimed solutions unravel, but experts say this proof is short enough to verify directly.
A problem decades in the making
The Jacobian conjecture sits at the intersection of algebra and geometry. It predicts that polynomial maps with a non-zero constant Jacobian determinant must be invertible. Although the statement sounds simple, it resisted formal proof for close to a century, with some of the best mathematical minds in the world failing to close the gap.
Famed mathematician T.T. Moh predicted in 2008 that a solution to the conjecture could take humans “another 100 years.” That estimate now looks overly cautious. Alpoge’s proof arrives in a year that has already seen frontier models solve open Erdős problems and crack the 80-year-old unit distance puzzle.
Why this matters beyond mathematics
The speed at which AI is shortening the timeline for long-standing mathematical problems is hard to ignore. Mathematicians have spent decades building machinery around the Jacobian conjecture, publishing partial results and developing specialised tools. A frontier model solved it as a side project during a major sporting event.
That shift matters for fields far beyond pure mathematics. Medicine, engineering, climate science, and cryptography all rely on hard problems that have taken human lifetimes to advance. If frontier models maintain this pace, the next five to ten years could produce findings that currently seem far outside reach.
The concern is not that AI will replace mathematicians. It is that the scale of what becomes solvable in hours instead of decades will force a rethink of how research teams are built, what problems are worth attacking, and how quickly society can adapt to new knowledge.
For now, the reaction from the mathematical community has been measured but impressed. A proof this short can be checked by hand, and early commentary suggests no obvious flaw. If it holds, the Jacobian conjecture will join a growing list of once-impossible tasks that frontier AI has made look easy.
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How this analysis is researched, sourced and corrected: Editorial standards and corrections. Spotted an error? Contact Philip Hall.
