AI-discovered Jacobian Conjecture counterexample unsettles mathematics

On July 20, Anthropic employee Levent Alpöge used Fable 5 to find a counterexample to the Jacobian Conjecture, an 87-year-old problem first proposed by Ott-Heinrich Keller in 1939. The result is a three-variable polynomial mapping with a Jacobian determinant identically −2 that is generally three-to-one, not one-to-one, and mathematicians worldwide independently verified it within hours after Columbia University mathematician David Speyer shared the equations online.

The finding follows a burst of AI-driven mathematics results in 2026. On May 20, OpenAI said a general-purpose reasoning model overturned Paul Erdős’s Unit Distance Conjecture from 1946, while DeepMind’s AlphaProof Nexus autonomously solved nine open Erdős problems, including two that had remained unresolved for 56 years. The new wave differs from earlier systems because it targets more prominent problems and shows greater autonomy in proof generation and verification.

The Leiden AI and Mathematics Manifesto, released June 2, 2026 and endorsed by the International Mathematical Union, warns that AI could overwhelm peer review, obscure attribution, and let tech companies steer mathematical priorities toward problems suited to their models. Supporters including Terence Tao and Peter Scholze argue that mathematicians must retain authority over evaluation, citations, research values and the judgment of “what mathematics is worth doing.”