Tiny counterexample puts AI at center of mathematics debate

Levent Alpöge, a mathematician at Anthropic, announced on X that he had found a counterexample to the Jacobian conjecture using Claude Fable 5. The long-running problem in algebraic geometry concerns whether certain polynomial functions with a constant non-zero Jacobian determinant must always have a polynomial inverse.

The counterexample is strikingly compact and easy for other mathematicians to check. Alpöge found a function in three dimensions with a constant Jacobian determinant of -2 that sends multiple input points to the same output point, making it non-reversible. The result shows the conjecture is false for every dimension larger than 2, while the original two-dimensional case remains open.

The discovery follows other recent mathematical advances involving large language models, including OpenAI’s disproof of the unit distance conjecture and Liam Price’s proof of Erdős’ problem 1196. Unlike breakthroughs built around lengthy or intricate arguments, this case appears to highlight AI’s ability to navigate enormous search spaces and uncover unexpectedly simple mathematical objects.