Claude AI Cracks 86-Year-Old Jacobian Conjecture

An AI model has produced a counterexample to the Jacobian Conjecture, a cornerstone open problem in algebraic geometry since 1939. Fields Medalist Terence Tao shared a ChatGPT analysis of the result, and the math community is now verifying the claim.

The Jacobian Conjecture has stood as one of algebraic geometry's most famous unsolved problems for 86 years. First posed by Ott-Heinrich Keller in 1939, it asks whether every polynomial map with a constant non-zero Jacobian determinant must have a polynomial inverse.

Claude Fable — an instance of Anthropic's Claude AI — has now produced a specific counterexample over ℂ³: a three-variable polynomial map with Jacobian determinant −2 that maps three distinct inputs to the same output, making it non-invertible.

The mathematics community took serious notice when Fields Medalist Terence Tao shared a detailed ChatGPT conversation tracing through the logic step by step. His analysis lent immediate credibility to what might otherwise have been dismissed as an AI hallucination.

Commenters on Hacker News noted the historic weight of the moment: a major unsolved conjecture, potentially resolved not by a human mathematician but surfaced by an AI assistant. The counterexample, first posted by Harvard mathematician Levent Alpöge, has been discussed across multiple HN threads.

If verified, an 86-year chapter in pure mathematics closes — and a new one opens on the role of AI in mathematical discovery.

Sources: HN: Claude Fable counterexample, Terence Tao's ChatGPT analysis, HN: Digestion of the counterexample, Wikipedia: Jacobian Conjecture