Claude Fable 5 helps overturn major Jacobian conjecture cases

Levent Alpöge, a mathematician at Anthropic, used Claude Fable 5 to uncover a simple counterexample to the Jacobian conjecture, a long-standing problem in algebraic geometry. The counterexample shows the conjecture is false in three dimensions and above, while the original two-dimensional version remains open.

The Jacobian conjecture concerns polynomial functions that move points in space. If the Jacobian determinant is always a non-zero constant, the conjecture predicts that another polynomial function should be able to reverse the mapping and return every point to its starting position. 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 two-dimensional version was stated by Ludwig Kraus in 1884, and Ott-Heinrich Keller generalized it to any number of dimensions in 1939. Earlier claimed proofs failed after subtle errors were found, though restricted cases have been validated. The discovery adds to a run of AI-assisted mathematical advances and suggests large language models may be useful not only for proofs, but for searching vast spaces of possible mathematical objects.