The Clearest Explanation of an AI Math Proof Was the AI's Own Chat Log

An 87-year-old conjecture, and the AI that found the hole in it

The Jacobian conjecture makes a promise about polynomial maps: if F sends ℂⁿ to ℂⁿ and its Jacobian determinant (a single number describing how the map stretches space at each point) stays a nonzero constant everywhere, F must be globally invertible, with a polynomial inverse. Mathematicians have chased a proof or a counterexample since 1939; the two-dimensional case is still open. On July 20, Anthropic researcher Levent Alpöge announced that the company's model, Claude Fable 5, had produced a counterexample in three dimensions: a degree-seven polynomial map whose Jacobian determinant is a constant -2 everywhere, yet which sends three distinct points, (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2), to the exact same output. The premise holds. The conclusion doesn't. The claim moved fast on HN, with a follow-up thread dryly titled "Human mathematicians are being outcounterexampled".

The next day, Terence Tao, one of the most prominent living mathematicians, posted "A digestion of the Jacobian conjecture counterexample," walking through Alpöge and Fable's construction from scratch instead of presenting it as a finished miracle. He breaks it into three properties he builds up one at a time: local injectivity, global non-injectivity, and a polynomial isomorphism onto ℂ³. The map comes from multiplying symmetric powers of low-degree polynomials, and the decisive step was picking one specific slice of the space of cubic polynomials, the case where a certain differential operator has two repeated roots, which unexpectedly turns out to define a clean geometric surface rather than a random cross-section.

The scale of the coincidence is what makes the result strange: a degree-seven map like this one could, in principle, produce a Jacobian as complicated as a degree-eighteen polynomial, with 1,329 non-constant coefficients that would all have to cancel to zero. A generic degree-seven map in three variables has only 120 numbers you can actually tune. Brute-force search has no business landing on a cancellation that large; only real geometric structure explains it. At the very end of the post, Tao adds a line that undercuts any idea this was purely a human feat: he used an AI chatbot, by his account, to talk through parts of the argument and confirm several of the calculations himself.

Readers needed the chat log more than the write-up

On a 57-comment thread that stayed civil and mostly technical, the loudest reaction wasn't about whether the counterexample holds; outside mathematicians were already confirming the arithmetic by hand within a day. It was about whether anyone besides Tao could follow his own explanation of it.

Even one of Hacker News's most-read commenters said the write-up beat him:

"The introduction to this piece was easy to follow, but as soon as he got into recapitulating it with algebra he lost me (because I'm bad at math). But he includes the GPT5 prompts for his conversation, which are easier to follow" — tptacek

It's tptacek's comment, not Tao's own vague phrasing, that identifies the linked conversation as a GPT-5 session, and recommends it as the more legible route through the same algebra that stopped him cold in the prose.

The same admission showed up in stranger metaphors further down the thread.

"reading through this I eventually realized a situation similar to my experience of it is what my dog sees if I attempt to explain Python programming to him." — zzzeek, creator of SQLAlchemy and Alembic

Not everyone let the substance go

Not every reply gave up on the math. One commenter had worked all the way through Tao's own degree-count before conceding defeat too, and left behind the sharper question the thread never actually answered:

"Sounds like the most interesting part would be learning what approaches the LLM did use to see if that's reusable elsewhere… Because I also couldn't follow the maths any more." — a reader who'd worked through Tao's own coefficient count before losing the thread

Not every reaction was awed, either.

"Guys, this guy is so brilliant that his mathematical expressions are embedded as images on his Wordpress blog. Truly amazing. This guy should be everywhere and get nothing but praise." — a reader further down the thread

The question about reusability has a real answer the thread didn't get to: Tao's three-property decomposition is not a one-off trick to admire, it's a recipe, a way to go looking for other structured cancellations instead of stumbling into them. But nobody in 57 comments tried to generalize it by hand, and nobody who could actually audit the model's reasoning showed up either; one reader asked outright whether that chain-of-thought could ever be checked, and got no answer. Fifty-seven comments verified the destination. None of them checked the route.