One of algebraic geometry’s most obstinate open problems was undone by something that fits in a single post on X. No 200-page manuscript. No cluster grinding away for months on a distributed computation. Just a compact formula in three dimensions whose Jacobian determinant equals -2.
Levent Alpöge, a mathematician at Anthropic, shared it almost offhandedly while much of the world was still winding down from the FIFA World Cup Final. The tool behind the find was Claude Fable 5, Anthropic's large language model, which at that point had been publicly available for just a few weeks.
What a conjecture is, and why this one refused to die
A conjecture is a claim that some mathematicians take to be true, yet nobody has succeeded in proving or disproving. The Jacobian conjecture has sat in that limbo for a very long time.
It concerns functions, which are worth picturing as little machines: numbers go in, and other numbers come out according to some rule. In this case the rules are polynomials, and the numbers are points in a space, much like coordinates on a map. The function shuffles those points around.
There is a way to gauge how gracefully it does that shuffling: compute the Jacobian determinant. When that value is always a non-zero constant, the function never folds or crushes space around a given point.
The conjecture asserts that any time the Jacobian determinant is a non-zero constant, there has to exist another polynomial function that reverses the first one and returns every point to its original location.
Reversal isn’t always possible. Should your original function drop two distinct points onto the same spot, you’re out of luck. After they have merged, nothing distinguishes them, so neither can be routed back to where it came from.
From 1884 to Smale’s list
The two-dimensional version was first stated in 1884 by Czech mathematician Ludwig Kraus. German mathematician Ott-Heinrich Keller extended it to arbitrarily many dimensions in 1939.
Fields Medallist Stephen Smale rated it highly enough to include it on his 1998 list of Mathematical Problems for the Next Century.
Proofs have been announced repeatedly over the decades, among them attempts by Beniamino Segre and Wolfgang Gröbner, both celebrated 20th-century mathematicians. On each occasion somebody located a subtle flaw that collapsed the argument.
Genuine progress was made, though only in pieces. A range of restricted versions have been established as true. Computational efforts verified that it holds in two dimensions for polynomials up to degree 100, that is, with powers of the variables reaching 100.
The general case remained unresolved, and so did the hunt for one example that would sink it.
Everyone knew a short counterexample might exist
That is what gives this result its sting. On paper, turning up a counterexample ought to be easy. Writing down functions that merge points is simple enough. So is writing down polynomial mappings whose Jacobian determinant is constant.
The trouble is packing both properties into one object.
A Math Stack Exchange user said as much 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”.
Broadly speaking, that is what unfolded, except the searching was done by a language model. Alpöge’s function sits in three dimensions, carries a constant Jacobian determinant of -2, and maps several different input points onto a single output point. Reversing it is impossible.
The conjecture is therefore false in every dimension above 2. The original two-dimensional question remains open.
Short enough that other people could check it fast
Its brevity counts for as much as the result itself. Because the counterexample is so small, other mathematicians were able to check it in short order, which is not how most AI-assisted mathematics has played out.
Recent comparisons sharpen the contrast. OpenAI’s disproof of the unit distance conjecture, along with the proof of Erdős’ problem 1196 by Liam Price, a 23-year-old amateur mathematician, both involved models drawing ideas out of separate mathematical fields and fusing them in a novel way.
None of that was required here. The object is simple on its own terms.
As of this writing, no one has disclosed how exactly Alpöge prompted the model or what its raw output contained. That is a genuine gap, and it caps how much can be said about the method.
What can be seen is the shape of the difficulty. There was no elaborate construction and no lengthy chain of reasoning involved. The challenge lay in navigating a vast search space of candidate polynomial mappings well enough to land on one with the right properties.
All of which highlights a use for these models that draws less notice than proof generation: tracking down unexpected mathematical objects that had been sitting there all along. What that implies for the future of mathematics, and for human mathematicians, is still an open question.