AI Solves Math Conjecture: Claude Fable 5 Breaks an 87-Year-Old Problem

AI solves math conjecture has been a cheap headline for years. On Sunday night it finally got specific. A number theorist and an Anthropic model posted a counterexample 216 characters long, and the Jacobian conjecture, open since 1939, finally fell.

The German mathematician Ott-Heinrich Keller posed it in 1939. The claim sounds almost modest. Take a polynomial map, the kind of formula that turns one set of coordinates into another. If its Jacobian determinant is a non-zero constant, the standard test for reversibility, then the map should be reversible by another polynomial.

Nearly everyone expected that to be true. It survived decades of attempts, collected a string of retracted proofs, and earned a place on Stephen Smale’s celebrated list of problems for the 21st century. Because of that reputation, most working mathematicians treated it as a question that would outlive them.

Part of the difficulty was psychological. Researchers built whole programmes on the assumption that the conjecture held, so effort flowed toward proving it rather than toward hunting for the object that would kill it. Most people also assumed any counterexample would be monstrous, something no human would ever stumble across by hand.

The counterexample maps three-dimensional complex space to itself. Its Jacobian determinant is a constant -2, so it sails through the standard check. Yet it cannot be reversed, because three separate inputs land on exactly the same output: (0, 0, -1/4), (1, -3/2, 13/2) and (-1, 3/2, 13/2) all arrive at (-1/4, 0, 0).

That is the whole refutation. If two roads lead to one destination, you cannot look at the destination and know which road you came from. The formula was short enough to fit in a single post, which is exactly where it appeared.

The brevity is the part that unsettled people. Mathematicians had spent decades assuming that any counterexample would be enormous, buried somewhere far out of reach. Instead the object was small enough to read in one breath, and it had been sitting inside the search space the whole time. Daniel Litt, who saw the post at two in the morning, called it incredible and stayed up.

Levent Alpoge, a number theorist who works at Anthropic and previously held a fellowship at Harvard, published the result on X late on Sunday, crediting the company’s Claude Fable 5 model. He was not a bystander. The model generated a candidate; he knew which question was worth aiming it at, and he knew how to confirm the answer was real.

That division of labour matters more than the headline suggests. Machines are now competent at searching enormous spaces for an object that fits a description. Deciding which object would be worth finding is still a human job.

It is also worth noting where the announcement landed. There was no press embargo and no staged launch event, just a post on a Sunday night during the World Cup final. The work was checkable, so it did not need packaging.

Plenty of AI proof claims have collapsed under review. This one did not, and the reason is structural rather than lucky. A counterexample is trivially cheap to verify: substitute the numbers and see whether the arithmetic holds. Nobody has to trust the model’s reasoning, or even read it.

Long proofs are the opposite. They can take referees months, and a single soft step buried on page 40 can sink the whole thing. As The Next Web reported, mathematicians confirmed this result within roughly a day, because anyone holding the formula could check it by hand.

This was not an isolated weekend. Three months of 2026 produced a run of results that would each have been remarkable on their own.

Fields medallist Timothy Gowers, who called the OpenAI result a milestone, said the Jacobian news was the first time an AI had settled a problem outside his own field that was big enough for him to have heard of it.

The reaction was not unanimous. Andrew Blumberg was blunt about it: “This did not cause me to update my priors.” His objection is reasonable. Hunting for a counterexample is precisely the shape of task machines are built for, a huge search space paired with cheap verification.

One detail supports that reading. OpenAI’s internal Codex model reportedly found essentially the same counterexample independently. When two unrelated systems trip over the same object in the same week, the fairer conclusion is that the problem had become reachable, not that one model is uniquely brilliant.

There is a wider caveat too. Disproving something takes one well-chosen object. Proving something takes an argument that holds for every case, forever, and that is where the bulk of mathematics actually lives. Breaking a conjecture is a genuine result, but it is the cheaper of the two directions, and nobody involved is claiming otherwise.

The scope is narrower than the headlines suggest, too. Terence Tao, who worked through the construction on his blog, points out that the counterexample settles the conjecture in three dimensions. In two dimensions the question is still open.

You will not be disproving conjectures this week. The useful signal is narrower and more practical: reasoning models have become genuinely strong at problems that are bounded and easy to check. That is the same shape as a failing test, a broken query or a function that returns the wrong value.

It also explains why the gap between demo and daily work stays wide. When a task has no cheap correctness check, the same models are far less reliable, which is roughly what the 2026 productivity data on AI coding tools keeps showing.

So the practical lesson is not that the machines got smarter overnight. It is that they are most useful wherever you can hand back a clear verdict on the answer. Give a model a target it can be measured against and it will grind through options faster than you can. Give it a vague brief and you are back to reading its output line by line.

The models behind these results are the same ones sitting in everyday developer tools. If you want to put that reasoning to work, our roundup of the best AI coding tools covers which assistants handle real problems and which fold. For the sober counterweight, our report on the productivity impact of AI coding tools looks at what the measured data says once the demos stop.

*Sources: Terence Tao, The Next Web, CoinDesk, OpenAI, TechCrunch, Forbes, Boing Boing.*