Anthropic researcher credits Claude Fable 5 with a Jacobian Conjecture counterexample

Levent Alpoge, a number theorist at Anthropic, said the company’s Claude Fable 5 model helped him produce an explicit counterexample to the Jacobian Conjecture, an algebraic-geometry problem open since 1939. He laid it out in a post on X early on July 20, 2026.

Alpoge published a specific polynomial map, F=(P,Q,R) from C^3 to C^3 with a constant Jacobian determinant of -2, that sends three distinct points to the same output. If it holds, that collision disproves the general form of the conjecture, first posed by Ott-Heinrich Keller in 1939. Alpoge credited Claude Fable 5, an Anthropic AI model, and a collaborator he named only as “Akhil.”

The appeal of the claim is that it is checkable. Alpoge shared Wolfram Alpha links to verify the determinant calculation, and said a valid counterexample would likely also falsify the related Dixmier and Poisson conjectures. The two-variable case of the Jacobian Conjecture, he said, remains open.

The caveats are large. As of this writing there is no formal paper, no full model transcript, no peer review and no official Anthropic statement; the result rests on a personal social-media post. A counterexample is easier to check than a proof, because confirming that one explicit map behaves as claimed is a finite calculation, but independent mathematicians still need to verify the algebra and that the map genuinely violates the conjecture’s conditions.

Mathematicians were taking the claim seriously precisely because the algebra is explicit and reproducible, rather than a black-box assertion. Whether it survives scrutiny, and how much of the work Claude Fable 5 did versus the humans, will decide whether this counts as a modest machine-assisted result or a landmark for AI in mathematics.