Mathematician uses Claude Fable to disprove Jacobian conjecture

Levent Alpöge, a mathematician at Anthropic, posted a counterexample to the Jacobian conjecture on the social platform X. The conjecture, open since 1939, holds that a polynomial map with a constant, non-zero Jacobian determinant must be invertible.

Alpöge's example is a three-variable polynomial transformation with a Jacobian determinant fixed at -2, which satisfies that premise. Three distinct input points collide onto the same output, contradicting the conjecture's conclusion. The map disproves the conjecture in three dimensions and, by extension, every dimension above two; only the original two-dimensional case remains open.

Alpöge credited Claude Fable 5 with constructing the example. The polynomial and the colliding points were compact enough that other mathematicians verified the result within hours of the post.

A problem open for 87 years being settled by a short social media post rather than a paper says something about how mathematical results now spread.