Claude Helps Disprove 87-Year-Old Jacobian Conjecture

The Jacobian conjecture, posed by German mathematician Ott-Heinrich Keller in 1939, holds that a polynomial map over a characteristic-zero field with a nonzero constant Jacobian determinant must have a polynomial inverse. The deceptively simple claim became a central open problem in algebraic geometry and was included in Stephen Smale’s 1998 list of mathematical challenges for the 21st century. A valid counterexample would overturn nearly nine decades of work and demonstrate how frontier AI can assist high-level mathematical discovery.

On July 20, 2026, Anthropic mathematician Levent Alpöge posted on X a three-variable complex polynomial map he said was found with help from Claude Fable 5. The compact construction has a Jacobian determinant fixed at −2, yet maps three distinct points to the same output, making it non-injective and disproving the conjecture in dimension three; adding identity coordinates extends the result to every dimension n≥3. The calculation is short enough for symbolic verification, though the announcement was not accompanied by a formal paper or conventional peer review.