Anthropic researcher uses Claude Fable 5 to refute 87-year-old Jacobian conjecture

Levent Alpöge, a researcher at Anthropic, shared on X that the company's AI system Claude Fable 5 enabled him to identify a counterexample to the Jacobian conjecture, a problem that has persisted for 87 years. The constructed function operates in three dimensions, maintains a constant Jacobian determinant of -2, but maps distinct inputs to the same output, proving it non-invertible. This finding suggests the conjecture fails for dimensions beyond two, although the original two-dimensional formulation remains open.

Alpöge noted that the counterexample is concise enough to fit within a single social-media post, facilitating verification by other mathematicians. The episode underscores a shift in how AI tools are employed, moving from proof assistance to generating novel mathematical results. Confirmation by the broader mathematical community will be required before the claim is accepted as definitive.