AI model finds simple counterexample to 87-year-old Jacobian conjecture

Mathematician Levent Alpöge, working at the AI company Anthropic, announced on X that he had found a counterexample to the Jacobian conjecture using the large language model Claude Fable 5. The conjecture, first stated in two dimensions by Ludwig Kraus in 1884 and generalized to any dimension by Ott-Heinrich Keller in 1939, proposes that a polynomial mapping with a constant non-zero Jacobian determinant must be invertible by another polynomial mapping.

The Jacobian determinant measures whether a function folds or crushes space; a constant non-zero value suggests the mapping is locally well-behaved. For decades, mathematicians attempted to prove the general case or find a counterexample, with computational verification confirming the two-dimensional version for polynomials up to degree 100. No general proof or disproof had been found until now.

Alpöge's counterexample is a polynomial function in three dimensions with a constant Jacobian determinant of -2 that maps multiple distinct input points to the same output point, making it non-invertible. The formula is short enough to fit in a single social media post, allowing other mathematicians to verify it quickly.

Because the counterexample works in three dimensions, it automatically disproves the conjecture for all higher dimensions as well. The original two-dimensional version of the conjecture remains unsolved.

The discovery adds to a recent series of mathematical breakthroughs assisted by large language models, including OpenAI's disproof of the unit distance conjecture and a proof of Erdős's problem 1196 by amateur mathematician Liam Price.

Unlike some prior AI-assisted results that involved lengthy proofs, this counterexample is notable for its simplicity. The difficulty lay in searching an enormous space of possible polynomial mappings to find one with the required properties, a task the AI model helped navigate.

Details of the specific prompts and model outputs have not been made public. Researchers note that the result illustrates how AI may be valuable for discovering unexpected mathematical objects, not only for constructing complex proofs.

The implications for mathematical practice are still unfolding, but the episode demonstrates a new mode of collaboration between human mathematicians and AI systems in exploring long-standing open problems.