Harvard Mathematician and AI Model Disprove Jacobian Conjecture

On July 20, 2026, Harvard mathematician Levent Alpöge announced that he and Anthropic's Fable AI model had disproven the Jacobian Conjecture, an 87-year-old mathematical problem. The conjecture, proposed by Eduard Ott-Heinrich Keller in 1939, deals with the invertibility of polynomial mappings in complex space. Alpöge, a 33-year-old researcher at Harvard University, sought a counterexample to prove the conjecture false. He shared the counterexample, a concise 216-character string, on X (formerly Twitter), crediting both a friend for encouragement and Anthropic's Fable model for its computational assistance during the World Cup final. The Jacobian Conjecture posits that if a system of polynomial equations can be used to generate new coordinates from original ones, then those original coordinates can be recovered using polynomial equations. This process is analogous to reversing a transformation on a grid. The counterexample provided by Alpöge and Fable challenges this fundamental assumption. While the work has not yet completed the formal peer-review process, several mathematicians have reportedly verified the arithmetic and performed independent checks. These checks utilized tools such as SymPy, a Python library for symbolic mathematics, and Lean, a formal proof assistant. Further verification involved other AI systems, including OpenAI's GPT models, which were used for fact-checking, sometimes yielding unexpected or humorous outcomes. The collaboration highlights the growing role of AI in advanced mathematical research and problem-solving.