Three different points enter the same function and come out identical: (-1/4, 0, 0). The formula takes up a handful of lines; the problem that causes it to collapse had been open since 1939. The mathematician Levent Alpöge published it, attributing the discovery to Claude Fable 5Anthropic’s artificial intelligence model.
There Jacobian conjecture it had resisted attempts, alleged demonstrations and several failed accounts for 87 years. Fable 5 has found a counterexample: a function that respects the conditions predicted by the conjecture and produces the very result that, according to that same conjecture, it should not have produced.
Saying that AI has “solved” the problem gives an idea. The most mathematically precise verb is refuted. In fact, a single case that contradicts it is enough to dispel a conjecture. Fable 5 found it in three-dimensional space, making the higher-dimensional version also false. The one with two variables, however, continues to resist.
A function from which there is no going back
A function can be imagined as a mathematical machine: it receives one or more numbers, performs some operations and returns a result. In the case of the Jacobian conjecture, entire series of numbers enter and are returned.
A function is invertible when, knowing the result, we can go back to the starting point without doubt. If two different inputs produce the same output, the return road splits: it becomes impossible to know which of the two was the original one.
Fable 5 even found three:
F(0, 0, -1/4) = F(1, -3/2, 13/2) = F(-1, 3/2, 13/2) = (-1/4, 0, 0).
Three distinct starting points therefore end at the same point. This is enough to establish that the function is not invertible. The difficult part, evidently, was finding just that function among an enormous number of possibilities.
Because the conjecture seemed plausible
The conjecture was formulated in 1939 by the German mathematician Ott-Heinrich Keller. It concerns polynomial functions, those constructed using additions, subtractions, multiplications and integer powers.
To study how these functions transform space, mathematicians use the determinant of the Jacobian matrix. The name seems designed to scare away anyone without a degree in mathematics, but the basic idea is simpler: it allows you to check that, in the immediate vicinity of each point, the transformation does not crush different positions together.
In the function found by Fable 5 the Jacobian determinant always holds -2. The transformation therefore works well if we observe a small area of it. Looking at it in full, however, three points distant from each other are sent to the same place.
Keller had hypothesized that a constant, non-zero determinant guaranteed the operation of the inversion on the entire function. The counterexample shows that local control is not enough: up close, every stretch looks fine, while the full map leads multiple roads to the same destination.
The question was considered so important that in 1998 the mathematician Stephen Smale, winner of the Fields Medal, included it among the great mathematical problems proposed for the new century.
The result can be verified, the work of the AI much less
Alpöge published the counterexample on X, allowing other mathematicians to check both the value of the determinant and the three points with the same output. Terence Tao then dedicated a long mathematical analysis to the result, also reconstructing its geometric structure.
Public materials therefore allow the formula to be verified. They say much less about how it was obtained. The full conversation with Fable 5, the prompts used, the time required, or any computational tools used during the research were not released.
The attribution of the discovery to the model comes from Alpöge and was taken up in the first mathematical works dedicated to the counterexample. Without complete documentation of the process, however, it is not possible to precisely measure how much the AI did autonomously and how much it depended on the mathematician’s guidance. The result holds; behind the scenes remains closed.
The case, however, shows a useful ability for mathematical research: exploring many possible solutions until finding the rare object that contradicts a rule considered plausible. Hundreds of demonstration pages were not needed here. It served a specific function. She arrived with three different points and only one exit.
The general version of the Jacobian conjecture is therefore false from the third dimension upwards. The two-variable case, the smaller and more stubborn one, still remains on the board.
