An AI just helped disprove the Jacobian conjecture (an 85-year-old open problem)

In the early hours of 20 July 2026, while much of the world watched the World Cup final, a number theorist at Anthropic named Levent Alpöge posted three lines on X that close a problem open since 1939: the Jacobian conjecture.

It wasn’t a 40-page preprint. It was a single explicit polynomial map you can check on the back of an envelope, found with the help of Claude Fable. The arithmetic is elementary enough that thousands of people verified it within hours. We did too, and it holds.

This is worth slowing down for, because it’s a new kind of event. Not “AI writes a proof no one can read,” but a working mathematician using a frontier model as a research collaborator to hunt down a concrete object that decades of human effort had missed.

What the Jacobian conjecture actually says

Ott-Heinrich Keller posed it in 1939, and it’s deceptively simple to state.

Take a polynomial map F from complex n-dimensional space to itself: n output polynomials in n variables, built only from additions and multiplications. Form its Jacobian, the matrix of all first partial derivatives, and take the determinant. That determinant is itself a polynomial.

The conjecture claims: if that determinant is a nonzero constant, then F has to be invertible, and its inverse is also a polynomial.

A constant nonzero Jacobian says the map never locally folds or pinches anywhere. The conjecture asserts that this local property is enough to force the map to be one-to-one and onto everywhere. That jump from local to global is where the difficulty lives, and it’s why the problem is notorious. It has drawn dozens of “proofs” over eight decades, in both directions, almost all of which quietly collapsed under scrutiny. Whole sub-conjectures (Dixmier, Kontsevich’s equivalences, the Poisson case) hang off it.

The counterexample

To break the conjecture you need one polynomial map with a constant nonzero Jacobian determinant that is not one-to-one. Alpöge and Fable produced exactly that, in three dimensions:

F(x, y, z) = (
  (1+xy)^3 · z + y^2 · (1+xy) · (4+3xy),
  y + 3x · (1+xy)^2 · z + 3x·y^2 · (4+3xy),
  2x − 3x^2·y − x^3·z
)

Two things have to be true, and both are checkable by hand. First, the Jacobian determinant is the constant −2: nonzero everywhere, no variables left, exactly the hypothesis the conjecture is built on. Second, the map isn’t injective. Three distinct points all collapse onto the same image:

A map that sends three different points to the same place can’t be invertible. So here’s a polynomial map with a constant nonzero Jacobian that isn’t a bijection, which is precisely what the conjecture says can’t exist.

We ran the determinant and the three substitutions ourselves in a computer algebra system. The determinant simplifies to −2, and all three inputs land on (−1/4, 0, 0). No sleight of hand, no appeal to authority. That verifiability is the whole point, and it’s why the reaction was so fast.

levent

@__alpoge__

hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final

((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x − 3 x^2 y − x^3 z): \C^3\to \C^3, has jacobian determinant −2, and sends (0, 0, −1/4), (1, −3/2, 13/2), and (−1, 3/2, 13/2) to (−1/4, 0, 0)

2:19 AM · Jul 20, 2026 · 6M Views

724 Replies 2.7K Reposts 11K Likes View on X

The timeline

The compression is the story. A problem that survived 87 years was, at least on the public record, dispatched in a single evening and confirmed by strangers before breakfast.

What Claude Fable did here

It’s easy to over- or under-read this. What happened is that a strong human mathematician who knew exactly which object to look for used a frontier model to do the heavy algebraic search and construction, under real time pressure, and came out with a clean, hand-verifiable result. Alpöge framed the direction, asked the right question, and vetted the output. The reason the world could trust it within hours is that the answer is a concrete object anyone can check, not an argument you have to take on faith.

That’s the shape of the next year of headlines. These models are becoming extraordinary instruments for the parts of research that are search, construction, and bookkeeping over enormous algebraic spaces. The taste, the question, and the verification still sit with the mathematician.

There’s an interesting mathematical afterlife too. A counterexample doesn’t kill a subject, it sharpens it. The question mathematicians asked within the hour was whether a repaired version of the conjecture, with extra conditions ruling out the specific degeneracy this map exploits, still stands. The 1939 statement can be false while the spirit of the problem lives on. As of this writing there’s no journal publication or referee report yet, so on the official record the conjecture is “open” until the community certifies it. Unlike a long proof, though, this claim is a finite arithmetic check, which is about as close to self-certifying as mathematics gets.

Why this one is mind-blowing

We build with these models every day, so we’re not easily impressed by demos. This is different. A concrete, checkable mathematical object that eighty-seven years of human ingenuity didn’t find was surfaced in an evening with an AI in the loop, then verified by the crowd faster than any journal could convene. Human taste plus machine search plus instant public verification. That workflow is the thing to watch. It’s a preview of how a lot of hard technical work is about to get done.

Sources: Levent Alpöge on X, Jacobian conjecture (Wikipedia), OfficeChai coverage, Hacker News discussion.