Terence Tao · 2026-07-21 · notable
Terence Tao — A digestion of the Jacobian conjecture counterexample
Fields medalist Terence Tao walks through the AI-discovered counterexample to the 87-year-old Jacobian conjecture on his 'What's new' blog, and shares the ChatGPT conversation he used to verify several of the calculations.
Terence Tao rebuilds the C^3 counterexample step by step and posts the ChatGPT session he used to double-check the algebra.
What is it?
Terence Tao's July 21 blog post is a mathematician's reading guide to the counterexample that broke the Jacobian conjecture — a claim open since 1939 that any polynomial map from C^n to C^n with a constant nonzero Jacobian determinant must be globally invertible. The original counterexample was produced by Anthropic's Claude Fable 5 in three variables; Tao rebuilds it from first principles.
How does it work?
The post treats the counterexample as a multiplication map on polynomials — linear × quadratic → cubic — that is locally injective but globally non-injective because one cubic can factor as three different products. Tao normalizes with resultants, exploits SL_2 equivariance, restricts to an affine hyperplane away from the origin, and does an asymptotic analysis near the degenerate limits. He notes the underlying variety is secretly isomorphic to C^3, which he calls the 'miracle' of the construction.
Why does it matter?
At the end of the post Tao discloses that he used a ChatGPT session to talk through the problem and confirm several calculations, and links the full transcript. Coming from arguably the best-known living mathematician, that public workflow — human intuition, AI verification, formalizer in the loop — is one of the clearest endorsements yet that top-of-field researchers are already integrating chatbots into their day-to-day proof work.
Who is it for?
Working mathematicians, AI-for-science researchers, and anyone tracking how frontier LLMs are being folded into research math.
Try it
Read: terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample