Anthropic mathematician uses Claude to find Jacobian conjecture counterexample above two dimensions

SciTechDaily, adapting an article originally published in The Conversation, reported that Levent Alpöge, a mathematician at Anthropic, announced a counterexample to the Jacobian conjecture found with help from Anthropic's large language model Claude Fable 5. The Jacobian conjecture, generalized to any number of dimensions by Ott-Heinrich Keller in 1939 after a two-dimensional statement by Ludwig Kraus in 1884, says that a polynomial mapping whose Jacobian determinant is a non-zero constant should always have a polynomial inverse. Many claimed proofs over decades failed under scrutiny, while restricted cases and two-dimensional computational checks up to high degree supported the statement. Alpöge found a short three-dimensional polynomial function with constant Jacobian determinant of -2 that sends multiple input points to the same output, so it is not reversible. SciTechDaily said the example is short enough to fit in a single post on X and was easy for other mathematicians to verify. The result shows the conjecture is false for every dimension larger than 2, while the original two-dimensional form remains open. The writeup places the find alongside other recent AI-linked math results, including OpenAI's disproof of the unit distance conjecture and a proof of Erdos problem 1196 by amateur mathematician Liam Price, and notes AI's value in searching large spaces of candidate objects as well as constructing proofs. Implication: A verified higher-dimensional counterexample narrows a century-old algebra problem and highlights AI search as a tool for finding mathematical objects.