Anthropic's Fable model produces counterexample to the Jacobian Conjecture

Harvard mathematician Levent Alpöge said he had used Anthropic’s Claude Fable 5 to find a counterexample to the Jacobian Conjecture, an open problem in algebraic geometry posed in 1939. He announced the result on social media, describing a three-variable polynomial map whose Jacobian determinant is a constant, non-zero value everywhere, yet which sends three distinct points to the same output — violating the injectivity the conjecture would require. Because the map can be padded with untouched extra variables, the counterexample also disproves the conjecture in every dimension above three.

The claim was checked quickly rather than slowly: within hours, other mathematicians, including Jared Duker Lichtman and Qiaochu Yuan, said they had independently verified the algebra by direct substitution. No peer-reviewed publication had appeared, and Anthropic had made no official announcement of the result at the time it circulated. The exact model version, prompt history and degree of human steering behind the discovery were not independently auditable — commentators noted the account rested on Alpöge’s own description of the workflow with a collaborator.

The result was narrower than headlines suggested. The original two-dimensional Jacobian Conjecture, the version most mathematicians consider central, was untouched; only the general, higher-dimensional statement fell. Commentators framed the episode as evidence that language models could be useful in domains with an exact, checkable verifier — an algebraic identity that anyone could confirm by substitution — rather than as evidence of general mathematical reasoning.

The episode landed alongside a cluster of other frontier-model results in the same week, including reports that several systems had scored full marks at the 2026 International Mathematical Olympiad, and was cited in subsequent commentary as one data point in a broader argument about how close current models were to research-level mathematical capability.