Anthropic researcher uses AI to disprove 87-year-old math conjecture

On the night of July 19, mathematician Levent Alpöge posted a brief message on X that may represent the most difficult mathematical problem yet solved with the help of artificial intelligence. Using Anthropic's Claude Fable 5, Alpöge produced a counterexample disproving the Jacobian conjecture, one of the longest-standing open problems in algebraic geometry.news.ycombinator+1

The conjecture, first stated for two variables in 1884 and generalized in 1939, posits that if a polynomial function from an n-dimensional space to itself has a nonzero constant Jacobian determinant, then the function must have a polynomial inverse. It is one of mathematician Steve Smale's 18 problems for the 21st century.wikipedia

A Concise Counterexample

What Alpöge posted was not a proof of the conjecture but its negation: a polynomial map in three-dimensional complex space with a constant Jacobian determinant of negative two that nonetheless fails to be invertible. Three distinct input points — (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) — all map to the same output, (-1/4, 0, 0), proving the function is not one-to-one and therefore cannot have an inverse.newscientist+1

The counterexample is just 216 characters long, according to New Scientist. Alpöge included Wolfram Alpha verification checks in his original thread.explainx+1

Implications and Verification

According to the Secret Blogging Seminar, a mathematics blog run by professional mathematicians, the counterexample has drawn immediate attention from the community, with mathematician Andy Jiang providing a near-elegant geometric construction of the map.sbseminar.wordpress

The result disproves the conjecture for dimensions greater than two. For the special case of two variables, the conjecture remains an open problem. As reported by New Scientist, experts have called this the most difficult mathematical problem yet solved by AI.newscientist+1

Status and Skepticism

As of publication, the counterexample awaits formal peer review and posting to arXiv. However, several mathematicians have noted that a counterexample of this nature can be verified computationally — checking that the Jacobian determinant is constant and that three points collide requires only polynomial arithmetic. Alpöge, who holds a PhD from Princeton and previously served as a Junior Fellow at Harvard's Society of Fellows, joined Anthropic in 2024.linkedin+2