Terence Tao uses ChatGPT to digest AI's Jacobian counterexample
The Fields medalist shared his full ChatGPT session dissecting the AI-discovered counterexample that felled the 87-year-old Jacobian conjecture.
Terence Tao, the UCLA mathematician and Fields medalist, has publicly shared the complete ChatGPT conversation he used while working through the newly discovered counterexample to the Jacobian conjecture. The transcript circulated widely on July 22, reaching the top of Hacker News, and gives an unusually candid view of how one of the world's leading mathematicians actually works alongside a language model — probing definitions, testing hunches, and asking the system to check computations.
An 87-year-old conjecture falls
The Jacobian conjecture, posed by Ott-Heinrich Keller in 1939, asserts that any polynomial map from complex n-dimensional space to itself whose Jacobian determinant is a nonzero constant must be globally invertible. Last week the conjecture collapsed: a counterexample found with the help of Anthropic's Fable model, as Tao's post recounts, exhibits a degree-seven polynomial map on C^3 with constant Jacobian −2 that nevertheless sends three distinct points to the same output. The construction settles the question negatively for every dimension three and above, while the classical two-dimensional case remains open.
From machine artifact to human understanding
In a July 21 blog post titled "A digestion of the Jacobian conjecture counterexample," Tao set out to make the machine-found object humanly comprehensible. Rather than repeating the raw formulas, he rebuilds the example geometrically from symmetric powers of linear and quadratic polynomials, reducing verification to three checkable properties: local injectivity, global non-injectivity, and affine equivalence to C^3. He states plainly that he used an AI chatbot to discuss aspects of the problem and to confirm several of the calculations — and then linked the session itself.
Why it matters: this is a live demonstration of a new division of labor in mathematics. An AI system produced a counterexample that surprised the field; a top human mathematician then used another AI system to translate that artifact into structural insight, and published the process transparently. The episode reframes the debate about AI in research — the question is no longer whether models can contribute to frontier mathematics, but how quickly working practices, verification norms, and credit conventions will adapt.