Claude Fable 5 AI finds counterexample to 87-year-old Jacobian

On August 6, 2026, Levent Alpöge, a mathematician at Anthropic, announced a counterexample to the Jacobian conjecture, an 87-year-old problem in algebraic geometry. He found it using the company’s large language model, Claude Fable 5.

The counterexample is a polynomial function in three dimensions with a constant Jacobian determinant of –2. It maps multiple distinct input points to a single output, making it non-reversible. The finding disproves the conjecture for every dimension larger than two. The two-dimensional case, first stated by Ludwig Kraus in 1884, remains unresolved.

Alpöge shared the formula in a single post on X, and its brevity allowed other mathematicians to quickly verify the result.

The Jacobian conjecture, generalized by Ott-Heinrich Keller in 1939, posited that any polynomial mapping with a nonzero constant Jacobian determinant has a polynomial inverse. Fields Medalist Stephen Smale included it in his 1998 list of “Mathematical Problems for the Next Century.” Despite many flawed proofs over the years, the conjecture was proven true under various restrictions and computationally verified in two dimensions for polynomials up to degree 100.

Previous AI-assisted mathematical breakthroughs, such as OpenAI’s disproof of the unit distance conjecture and Liam Price’s proof of Erdős’ problem 1196, relied on combining ideas from different areas. Alpöge’s result highlights a different strength: navigating enormous search spaces to uncover simple, unexpected objects.

Anthropic has not disclosed how Alpöge prompted Claude Fable 5. Still, the discovery suggests AI may be as useful for finding mathematical objects as for constructing proofs.