A 3-variable formula from Claude Fable 5 just broke the 87-year-old Jacobian conjecture

The counterexample that killed one of algebraic geometry’s most stubborn open problems fits inside a single post on X. Not a 200-page proof. Not a distributed computation running for months on a cluster. A short formula in three dimensions, with a Jacobian determinant of -2.

Levent Alpöge, a mathematician at Anthropic, posted it casually while much of the world was still coming down from the FIFA World Cup Final. He found it using Claude Fable 5, Anthropic's large language model, which had been public for only a few weeks.

What a conjecture is, and why this one refused to die

A conjecture is a statement some mathematicians believe is true but nobody has managed to prove or disprove. The Jacobian conjecture has been in that limbo for a long time.

It’s about functions, which you can think of as small machines: you feed numbers in, other numbers come out according to some rule. Here the rules are polynomials, and the numbers are points in a space, like coordinates on a map. The function moves those points around.

There’s a way to measure how nicely it moves them. Calculate the Jacobian determinant. If it’s always a constant that isn’t zero, the function never folds or crushes space around a given point.

The conjecture says that whenever the Jacobian determinant is a non-zero constant, another polynomial function must exist that undoes the first one and puts every point back where it started.

Not every function can be undone. If your starting function drops two separate points onto the same spot, you’re stuck. Once they’ve merged, there’s no way to tell them apart and send each back where it belongs.

From 1884 to Smale’s list

Czech mathematician Ludwig Kraus stated the two-dimensional version in 1884. German mathematician Ott-Heinrich Keller generalised it to any number of dimensions in 1939.

Fields Medallist Stephen Smale thought enough of it to put it on his 1998 list of Mathematical Problems for the Next Century.

Plenty of people have claimed proofs over the decades, including Beniamino Segre and Wolfgang Gröbner, both famed 20th-century mathematicians. Every time, someone found a subtle error that broke the argument.

Real progress did happen, just partial. Various restricted versions have been proved true. Computational work confirmed it holds in two dimensions for polynomials up to degree 100, meaning powers of the variables up to 100.

The general case stayed open. So did the search for a single example proving it wrong.

Everyone knew a short counterexample might exist

That’s the part that makes this result sting a little. In theory, finding a counterexample should be easy. Writing down functions that merge points is straightforward. Writing down polynomial mappings with a constant Jacobian determinant is straightforward too.

Getting both properties in the same object is where it falls apart.

A Math Stack Exchange user put it plainly in a 2017 post: “for all what we know, some smart undergraduate can simply write a formula […] that will be a counter-example to this conjecture”.

That’s more or less what happened, except the search was run by a language model. Alpöge’s function lives in three dimensions, has a constant Jacobian determinant of -2, and sends multiple input points to the same output point. It can’t be reversed.

The conjecture is now false in every dimension larger than 2. The original two-dimensional case is still open.

Short enough that other people could check it fast

The brevity matters as much as the result. Because the counterexample is so compact, other mathematicians could verify it quickly, which is not how most AI-assisted math results have gone.

Recent comparisons make the contrast clear. OpenAI‘s disproof of the unit distance conjecture and the proof of Erdős’ problem 1196 by Liam Price, a 23-year-old amateur mathematician, both showed models pulling ideas from separate areas of mathematics and combining them in a novel way.

This one didn’t need that. The object itself is simple.

As of this writing, nobody has made public exactly how Alpöge prompted the model or what its raw output looked like. That’s a real gap, and it limits how much anyone can say about the method.

What’s visible is the shape of the difficulty. It wasn’t an intricate construction or a long chain of reasoning. It was finding a decent way through an enormous search space of possible polynomial mappings to land on one with the right properties.

Which points at a use for these models that gets less attention than proof generation: hunting down unexpected mathematical objects that were sitting there the whole time. What that means for the future of mathematics, and for human mathematicians, remains to be seen.