Fields Medalist Terrence Tao verifies AI's counterexample to the Jacobian Conjecture, using a rival AI to check the work

Terence Tao used GPT-5 to check Claude's counterexample to the Jacobian Conjecture

When Terence Tao wanted to understand a new counterexample to the Jacobian Conjecture, he did what any mathematician might do with a hard result: he worked through it step by step. His working partner was GPT-5, OpenAI's model. The counterexample he was unpacking came from Claude Fable, built by Anthropic. Tao published both his detailed walkthrough and the full transcript of his GPT-5 conversation on his blog on July 21, 2026 1.

The Jacobian Conjecture poses a question simple enough for a napkin: if a polynomial map has a Jacobian determinant equal to a non-zero constant, must the map be invertible? A non-zero constant Jacobian means the map is locally invertible at every point. The conjecture asserts that local invertibility forces global invertibility. For decades, it resisted proof and counterexample 1.

Claude Fable broke the stalemate with an explicit construction: a degree-seven polynomial in three complex variables whose Jacobian determinant is a non-zero constant, yet which is not globally invertible. The cancellation is extreme. A degree-seven polynomial map in three variables should produce a Jacobian of degree up to eighteen. In this construction, every non-constant coefficient vanishes, collapsing the determinant through what Tao describes as a massive cancellation involving far more equations than the polynomial's degrees of freedom would suggest 1. The conjecture survives in two dimensions, where it remains open 1.

The interesting part is not the result. Mathematicians have since re-explained the counterexample in standard geometric terms 1. What drew attention on Hacker News was the method Tao chose to understand it 2. He opened a session with GPT-5, fed it the construction, and worked through the algebra step by step. He then published the entire exchange alongside his own written explanation 1.

Here is the signal. One frontier model produced a construction that survived scrutiny from a Fields Medalist. A second model, from a competing company, served as the reasoning partner that helped him work through it. The workflow is portable: generate, cross-check, verify. Any researcher with access to two frontier models can attempt the same loop.

A commenter on the Hacker News thread asked a question that hangs over the episode: what are the odds that this counterexample was already in Claude Fable's training data, rather than genuinely discovered 2? The question has no clean answer, and it exposes the central tension in AI-assisted mathematics: models can produce correct results through reasoning their users cannot fully inspect.

Tao's walkthrough and his GPT-5 conversation are both public. Anyone can read them and judge the quality of the reasoning for themselves. That openness is what separates this from a press release. The counterexample either holds or it does not, and two independent AI systems plus one Fields Medalist have now examined it from different angles. The trajectory points toward a near future where the limiting factor in frontier mathematics is not compute or model capability, but the thin supply of humans who can tell a valid proof from a fluent one.