Jacobian conjecture disproved with Claude Fable 5

Levent Alpöge's counterexample, found with Claude Fable 5

The result did not arrive as a paper or a press release. It came as a single casual post from a mathematician, and within a day other mathematicians had redone the arithmetic.

A short X post said the conjecture is false

Alpöge, a Princeton-trained number theorist, wrote on X that the Jacobian conjecture is false and pasted the formula underneath. The rest of the post was thanks: to his friend Akhil, the University of Chicago mathematician Akhil Mathew, for raising the problem, and to "fable" for working on it while the World Cup final was on. He also linked to Wolfram Alpha pages so readers could rerun the calculations.

Two details are easy to miss. The post names only "Fable", not a model version, and it was a personal announcement rather than anything Anthropic published. Claude Fable 5 had been available to the public for only a few weeks at that point.

Mathematicians checked the counterexample within a day

Because the formula fits in one post, verifying it took hours rather than months. By the following day other mathematicians had worked through it by hand and with software, and the claim held.

Reactions came fast. Qiaochu Yuan called it by far the most famous open problem a large language model has resolved to date, and Jared Duker Lichtman described it as a remarkable result. Checking in public is still not peer review, though, and no journal has refereed the result.

The Jacobian conjecture said polynomial maps can be undone

The conjecture is about maps built only from adding and multiplying, known as polynomial maps. It links a local measurement to a global promise, and that jump from local to global is where the counterexample breaks it.

A Jacobian determinant only measures the map near a point

Picture a map as something that takes a point in space and moves it somewhere else. The Jacobian determinant, a single number computed from the map's partial derivatives (the Jacobian matrix), tells you how the map stretches or squeezes space right around a given point. When it is a constant other than zero, the map never folds space onto itself in any small neighborhood.

The Jacobian conjecture says that for polynomial maps, this local good behavior is enough: there must be a polynomial inverse that reverses the whole map. That does not follow automatically, for a simple reason. If a map sends two different points to the same place, nothing can send them back, because the output no longer says which one you started from.

A well-known older result is sometimes confused with this one. In 1994 Sergey Pinchuk built a two-variable real map that is reversible near every point but not on the whole plane. His determinant is never zero but is not constant, and the map works only over the real numbers, so it answers a weaker question.

From Kraus in 1884 to Keller in 1939 and Smale in 1998

The Czech mathematician Ludwig Kraus wrote down the two-variable form in 1884, and the German mathematician Ott-Heinrich Keller stated it for any number of dimensions in 1939. Headlines count the 87 years from Keller. Stephen Smale later put it at number 16 on his 1998 list of problems for the next century.

The problem also has a long history of failed proofs, including ones by Beniamino Segre and Wolfgang Gröbner. Partial results kept narrowing the search: the degree-2 case was proved, so any counterexample had to be at least cubic.

Checking the Jacobian conjecture counterexample yourself

You only need to confirm two facts, and neither requires any theory beyond calculus. We ran both checks in SymPy 1.14.0 on September 26, 2026, using the formula exactly as posted.

The map's Jacobian determinant is −2 at every point

Take the partial derivatives of the three components with respect to x, y and z, arrange them in a 3-by-3 matrix, and simplify the determinant. The answer comes out as −2 with no variables left, so it holds at every point of complex three-space rather than at a few sampled ones.

The conjecture is often stated with determinant 1, but that is only a scaling choice. Multiplying the first output by −1/2 turns the determinant into 1, and the three inputs below still collide. In our run the three components came out with total degrees 7, 6 and 4, consistent with the older result that any counterexample had to be at least cubic.

Three inputs give the same output: (−1/4, 0, 0)

Now substitute the three points Alpöge listed. Each one produces the same output, coordinate for coordinate.

Given only the output (−1/4, 0, 0), there is no way to know which of the three inputs produced it, so no inverse can exist. The map meets the conjecture's condition and breaks its conclusion, and one such example is enough. Adding extra coordinates that the map leaves alone carries the same collision into four, five or any higher number of dimensions.

Some commenters online pointed out that the map is not "proper", meaning some points that run off toward infinity are sent back into a bounded region. That is true, but properness was never one of the conjecture's assumptions, so it does not disqualify the example.

The plane case and Fable 5's role are still open

A verified counterexample disproves the general statement, but it answers fewer questions than the headlines suggest. Three gaps remain.

The two-variable Jacobian conjecture is still unsolved

Padding a map with extra coordinates only moves a counterexample up in dimension, never down. The plane case, the version Kraus first wrote down and the one several mathematicians regard as the core of the problem, is untouched.

Computer searches have ruled out plane counterexamples of low degree. Tzuong-Tsieng Moh's 1983 work covered every degree up to 100, and Thuy Nguyen raised the bound to 104 in 2025.

No peer review yet, and no record of how Fable 5 was used

As of late July 2026 there was no journal paper, only the post and the public checks. There is also no transcript: outsiders do not know how the model was prompted, how many candidates failed, or how much Alpöge and Mathew guided it. Mathew has put it this way: the answer can be verified, but a story explaining it would be nicer.

That leaves a narrower lesson than "AI solves math problems." The formula itself is simple; the hard part was searching an enormous space of polynomial maps for one with both properties.

A model like Claude Fable 5 can be a useful search partner when every candidate can be checked exactly, and the checking still falls to people. To see how it compares with Anthropic's other models, read Claude models compared: Fable, Opus, Sonnet, Haiku.