Claude Fable 5 helped produce a counterexample to the Jacobian conjecture

Levent Alpöge, a mathematician at Anthropic, wrote on X that he has found a counterexample to the Jacobian conjecture, the algebraic geometry problem Ott-Heinrich Keller stated for any number of dimensions in 1939. He got it with Claude Fable 5, which shipped to the public a few weeks ago.

The function lives in three dimensions and its Jacobian is the constant -2, yet it sends several distinct points to the same image, so it isn’t invertible. That makes the conjecture false in every dimension above two. The two-dimensional case, stated by Ludwig Kraus in 1884, is still open.

Beniamino Segre and Wolfgang Gröbner had both attempted proofs, and errors were found in their arguments. In two dimensions the conjecture has been confirmed computationally for polynomials of degree up to 100.