Claude Fable 5 AI Disproves Jacobian Conjecture: 87-Year-Old Math Problem Solved

On July 19, 2026, mathematician Levent Alpöge posted a 216-character counterexample on X that, if verified, disproves the Jacobian conjecture — an open problem in algebraic geometry that has resisted solution for 87 years. The announcement credited Anthropic's Claude Fable 5, an AI model, and a friend named Akhil.

The Jacobian Conjecture Explained

The Jacobian conjecture, first stated for two variables in 1884 and fully generalized by Ott-Heinrich Keller in 1939, asks whether a polynomial map from n-dimensional space to itself with a nonzero constant Jacobian determinant must have a polynomial inverse. It was included on Stephen Smale's 1998 list of 18 major unsolved problems for the 21st century.

AI's Role in the Discovery

Alpöge, a number theorist and former Harvard Junior Fellow who now works at Anthropic, posted: "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." The counterexample is a polynomial map from ℂ³ to ℂ³. The tweet is currently unavailable.

Abhishek Saha at Queen Mary University of London called it "probably the biggest conjecture that AI has played a significant role [in proving or disproving] so far in mathematics." Andrew Blumberg at Columbia University, involved in the First Proof project, said the result did not change his view of AI capabilities: "If there was a counterexample that was concise and easy to state that people haven't found because it's a pain to search through all this stuff, AI will find it."

Fable 5 is the public version of Claude Mythos Preview, which Anthropic has described as having advanced cybersecurity capabilities. Alpöge's LinkedIn profile notes an affiliation with Anthropic. Anthropic did not respond to requests for comment.

The counterexample has been verified by other mathematicians, though it has not yet been peer-reviewed or published in a journal. The announcement highlights a growing intersection of AI and mathematical problem-solving — Alpöge specifically thanked Fable for working during the World Cup final, a detail that has captured the public imagination.

The Jacobian conjecture had a notorious reputation: at least five published proofs turned out to contain errors, and unpublished attempts numbered many more. Mathematicians had developed institutional caution around any claimed solution. Alpöge's counterexample, if it holds, would be the first definitive resolution — and it came not from a multi-year proof but from a single line of mathematics posted during a football match.

The role of AI in this discovery is still being debated. Blumberg's view — that finding a concise counterexample via search is exactly what AI is expected to do — suggests the breakthrough may be more about the tool than the method. But Saha's assessment that this is the biggest conjecture AI has played a significant role in disproving indicates a shift in what the mathematical community considers possible.

For mathematicians, the immediate question is verification. The counterexample is simple to check, and many have already done so. The longer-term question is what this means for the practice of mathematics. If AI can find counterexamples to long-standing conjectures by searching through combinatorial spaces that humans find tedious, the role of the mathematician may shift from finding answers to framing the right questions.

Alpöge's announcement, made casually on X during a World Cup final, may be remembered as the moment AI crossed a threshold in pure mathematics. Whether that threshold is a new frontier or a narrow niche remains to be seen.