Anthropic's Claude Fable 5 Helps Disprove Jacobian Conjecture

Mathematician Levent Alpöge, affiliated with Anthropic and Harvard University, used Anthropic's Claude Fable 5 model to disprove the Jacobian conjecture, a longstanding mathematical problem dating back to 1884. The conjecture, expanded by Ott-Heinrich Keller in 1939, posits that if a polynomial function's Jacobian determinant is a nonzero constant, the function should be reversible. Alpöge posted a counterexample on X on July 20, 2026, during the FIFA World Cup final, which was verified by independent mathematicians. The function merges multiple input points in three-dimensional space, making it irreversible, thus disproving the conjecture for dimensions three and higher. Abhishek Saha of Queen Mary University of London called it the biggest conjecture AI has significantly impacted. However, Andrew Blumberg of Columbia University noted that finding a single counterexample is less significant than proving the conjecture, which remains unproven. The result is seen as a milestone for AI-assisted mathematics, outpacing expectations. In related developments, AI helped solve an 80-year-old problem by Paul Erdős in May, prompting the Leiden Declaration on AI and Mathematics, which urges disclosure and peer review. Akhil Mathew of the University of Chicago, who suggested the problem, expressed concerns about the rapid shift toward AI in pure mathematics.