Anthropic's Claude Fable 5 AI disproves Jacobian conjecture in dimensions three and higher

A mathematician working at Anthropic has announced that the long‑standing Jacobian conjecture, a problem that has challenged researchers for more than a century, has been disproved in dimensions three and higher. The breakthrough came after the researcher employed the AI model Claude Fable 5 to generate and test a remarkably simple counterexample that violates the conjecture’s predictions. The original two‑dimensional version of the conjecture remains open.

The Jacobian conjecture, first proposed in 1939, asserts that any polynomial map from \(\mathbb{R}^n\) to itself with a non‑zero constant Jacobian determinant must be invertible. Over the decades, mathematicians have produced numerous partial results and special‑case proofs, but a general proof or disproof has eluded the field. The counterexample identified by the Anthropic mathematician shows that the conjecture fails when \(n \ge 3\), providing a concrete family of polynomial maps that satisfy the Jacobian condition yet are not bijective. The AI model assisted by exploring vast algebraic structures and identifying patterns that guided the construction of the counterexample.

This development underscores the growing role of artificial intelligence in pure mathematics, offering new tools for hypothesis generation and verification. While the result settles the conjecture in higher dimensions, the two‑dimensional case remains unsolved, and mathematicians will now need to reassess related conjectures and techniques in light of the AI‑enabled counterexample.