Claude Fable 5 helps refute Jacobian conjecture in higher dimensions

Levent Alpöge, a mathematician at Anthropic, used Claude Fable 5 to find a compact counterexample to the Jacobian conjecture, a longstanding problem in algebraic geometry. The example is a polynomial mapping 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 generalized conjecture is false for every dimension larger than 2, while the original two-dimensional case remains unsolved. The conjecture says that when a polynomial mapping has a non-zero constant Jacobian determinant there should always be another polynomial function that reverses it, returning all points to their starting positions.

The discovery stands out because the counterexample is short enough to fit into a single X post and was easy for other mathematicians to verify. Unlike AI-assisted advances built around intricate constructions or long proofs, the challenge appears to have been navigating a vast search space of polynomial mappings, pointing to a role for AI in finding unexpected mathematical objects as well as constructing proofs.