Claude-assisted find challenges the Jacobian conjecture

Levent Alpöge, a mathematician at Anthropic, announced on X that he had found a counterexample to the Jacobian conjecture using Claude Fable 5. The conjecture concerns when polynomial functions that move points through space can be reversed by another polynomial function, based on whether their Jacobian determinant is a non-zero constant.

The two-dimensional version dates to 1884, and the broader formulation to 1939. The problem has attracted failed proof attempts by prominent mathematicians, while computational work had shown it holds in two dimensions for polynomials up to degree 100.

Alpöge found a compact 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 counterexample indicates the conjecture is false for every dimension larger than 2, while the two-dimensional case remains open.

The discovery differs from some recent AI-assisted math results because the final object is simple enough to verify quickly. The challenge appears to have been navigating a huge search space of polynomial mappings, suggesting AI systems may help mathematicians uncover unexpected structures as well as construct proofs.