140-Year-Old Jacobian Conjecture Falls — AI Finds Counterexample in 3D

Levent Alpöge used Claude Fable 5 to disprove the Jacobian Conjecture for N > 2. Terrence Tao independently verified the counterexample using ChatGPT. The 2D case remains open.

One of mathematics' oldest unsolved problems has fallen to artificial intelligence.

The Jacobian Conjecture, first stated in 1884 and listed as Problem #16 in Stephen Smale's "Mathematical Problems for the Next Century," has been disproven for all dimensions greater than two. Anthropic mathematician Levent Alpöge presented an explicit counterexample in three-dimensional space on July 19, 2026, discovered using Claude Fable 5.

The conjecture asserts that if a polynomial function from N-dimensional space to itself has a constant non-zero Jacobian determinant, then the function must have a polynomial inverse. It was long suspected to be true — but the sheer number of published and withdrawn false proofs over the decades made it notorious in algebraic geometry circles.

Alpöge's counterexample, found through an extended dialogue with Anthropic's Claude model, definitively disproves the conjecture for N > 2. The two-dimensional case remains open.

In a striking parallel, UCLA mathematician and Fields Medalist Terrence Tao shared his own ChatGPT conversation independently analyzing the counterexample, demonstrating how multiple frontier AI systems are converging on the same mathematical territory. The Hacker News discussion drew over 500 points and more than 340 comments within hours.

A 140-year wait, ended in a single week of July.

Sources: Wikipedia — Jacobian Conjecture, ChatGPT — Terrence Tao's Jacobian Conjecture Counterexample