The Jacobian conjecture stood for 87 years. It died in a tweet during the World Cup final.

On July 19, Levent Alpöge, a number theorist at Harvard, posted three polynomials on X. He credited Akhil Mathew for suggesting the question and Claude Fable 5 for the work. The whole counterexample fits in 216 characters:

P = (1+xy)^3 z + y^2 (1+xy)(4+3xy)
Q = y + 3x (1+xy)^2 z + 3x y^2 (4+3xy)
R = 2x - 3x^2 y - x^3 z

Keller conjectured in 1939 that a polynomial map with constant nonzero Jacobian determinant must be invertible, with a polynomial inverse. This map has Jacobian determinant equal to -2 at every point of C^3. And it sends three different inputs to the same output. So it can’t be inverted at all.

Conjecture false.

Verification took hours, not months. Mathematicians around the world ran exact symbolic checks in Macaulay2 and Singular and got the same answer. Paul Lezeau formalized it in Lean for DeepMind’s Formal Conjectures repo. Tim Gowers said it was the first time an LLM cracked a well-known problem he knew from outside his own area. Kevin Buzzard’s blog post title says the rest: “Human mathematicians are being outcounterexampled.”

Then it happened again. This week a counterexample landed for the Dinitz-Garg-Goemans conjecture, a flow-network problem that had been open for about 30 years. GPT-5.6 Pro found a graph where the fractional flow costs 58, but any unsplittable flow violating capacities by at most 15 costs at least 60. Earlier in the month the cycle double cover conjecture reportedly fell too. That one waited fifty years.

Math is being solved by tokens.

I keep coming back to the shape of these results. Both are counterexamples, and a counterexample is the perfect deliverable for a token machine. Finding one takes taste plus an enormous search. Checking one is mechanical. You paste the map into a computer algebra system and the determinant either comes out -2 or it doesn’t. No trust in the model required. The artifact is its own proof.

You can check this one yourself. Three points collide at (-1/4, 0, 0), and there’s an interactive explainer with the arithmetic, SymPy code, and a test suite. Grade-school multiplication settles an 87-year-old question.

Terry Tao has already digested it. Presented cold, the map looks like a massive miracle: a determinant that should be degree 18 collapses to the constant -2. His writeup shows the miracle is structure. The map is secretly polynomial multiplication, linear times quadratic equals cubic, and a generic cubic factors into three linear pieces in three different ways. That triple factorization is exactly the three-to-one collision. The cancellations were natural all along.

The transcripts are worth reading, too. After Alpöge’s post, people prompted ChatGPT to explain the geometry behind the map, and it produced a construction that working mathematicians took seriously. The Dinitz chat is just a raw log of a model doing frontier math in public. Five years ago these tools couldn’t multiply reliably.

The humans still matter, and it’s worth being precise about where. Mathew picked the target. Alpöge drove the model and recognized a real answer when he saw one. And Buzzard’s point stands: the next step is for humans to understand what is actually going on with this example, why it exists, what it says about dimension two, what else falls with it. The tokens found the map. The meaning is still our job.

I spend my days shipping agents that sit in meetings and open PRs. Watching the same machinery close an 87-year-old problem during a soccer game makes the ceiling feel a lot higher than it did last month.

Keller asked the question in 1939. A language model answered it in 2026, in a tweet.

— AM

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