AI-Assisted Discovery Claims to Disprove 87-Year-Old Jacobian Conjecture

An artificial intelligence model has been credited with helping to find a counterexample to the Jacobian Conjecture, an unsolved problem in algebraic geometry that has stumped mathematicians since 1939. The breakthrough was announced by Levent Alpoge, a mathematician formerly at Harvard and now working at Anthropic, the company behind the AI model Claude Fable 5.

According to Alpoge, while watching the FIFA Club World Cup final on Sunday, he asked Claude Fable 5 to work on the Jacobian Conjecture after a friend brought up the problem. Instead of proving the conjecture true, the AI produced a counterexample that appears to show the conjecture is false. The result, a 216-character mathematical formula, was shared by Alpoge on X (formerly Twitter) and has garnered nearly 30 million views.

"Hello there, the Jacobian conjecture is false," Alpoge wrote. "Thanks to my close friend Akhil for asking about it and my other close friend Fable for working during the World Cup final."

The Jacobian Conjecture, proposed by German mathematician Otto Keller in 1939, states that a certain type of polynomial map with a constant, non-zero Jacobian determinant must always be invertible. It became one of the most well-known unsolved problems in algebraic geometry and was included in mathematician Stephen Smale's list of major open mathematical problems in 1998.

The counterexample generated by Claude Fable 5 is a simple polynomial formula in three variables. It has the required constant Jacobian determinant of -2, but three different inputs produce exactly the same output, meaning the mapping is not invertible—directly contradicting the conjecture.

Leading mathematicians, including Terence Tao and Jared Duker Lichtman, have reviewed the counterexample and confirmed that it successfully disproves the long-standing conjecture, according to Alpoge. However, the two-dimensional version of the problem remains unsolved.

This is not the first instance of AI contributing to a major mathematical breakthrough. In May, OpenAI announced that one of its research models had found a counterexample to the Erdos Unit Distance Conjecture. Researchers note that AI can explore unconventional lines of reasoning that humans might overlook, though all such findings require careful verification by the mathematical community.