Artificial Intelligence

87 Years Fell to an Answer Anyone Can Check

A polynomial map found with Claude Fable 5 disproves the Jacobian conjecture in three variables, and a stranger machine-checked it in a day. The procurement lesson is the story.

Artificial IntelligenceBy Servola Tech Desk2026-07-215 min read

AI-assisted, edited by humans. | Editor-in-Chief: Leon Soliman | Editorial standards

87 Years Fell to an Answer Anyone Can Check

Key takeaways

A polynomial map with a Jacobian determinant of -2 sends three different inputs to the same output, which disproves an 87-year-old conjecture, and the whole claim was independently machine-checked within about a day.
Ott-Heinrich Keller asked the question in 1939, and the counterexample settles it only for three or more variables, with the two-variable case still open.
The check was fast because Google DeepMind's formal-conjectures repository already held a human-agreed formal statement of the conjecture, written down before any counterexample existed.
An independent audit of the Lean proof confirmed it contains no sorry, no native_decide and no custom axioms, which is what made a stranger's verification worth anything.
Before your next AI purchase, write one acceptance criterion that can be scored without the vendor's cooperation, because a benchmark the seller computed is not evidence.

Sunday night, a formula and a football final

Late on Sunday 19 July 2026 the mathematician Levent Alpoge posted a single polynomial map and thanked two friends: one for asking the question, and one, he wrote, for working during the world cup final. The second friend was Claude Fable 5. The post carried a timestamp of 02:19 UTC on 20 July. The formula in it was a counterexample to the Jacobian conjecture, which had been open since 1939.

The map takes three complex variables to three complex outputs. Its Jacobian determinant is -2, a non-zero constant, which is precisely the condition the conjecture said should guarantee a polynomial inverse. It has no such inverse, because three different inputs land on the same point: (0, 0, -1/4), (1, -3/2, 13/2) and (-1, 3/2, 13/2) all map to (-1/4, 0, 0).

Alpoge works at Anthropic and credited the model with finding it. That credit is the part the industry will quote for the next month. It is not the part that matters most to anyone running a business.

What Keller wrote down in 1939

The conjecture was a promise about recoverability. The German mathematician Ott-Heinrich Keller asked whether a polynomial map whose Jacobian determinant is a non-zero constant must have an inverse that is itself a polynomial. In plainer terms: if a transformation never locally collapses information anywhere, can you always reconstruct the input from the output using the same kind of arithmetic you used going in.

For 87 years the answer was assumed to be yes and nobody could prove it. It sat in algebraic geometry as one of those problems that attracts partial results, special cases and, at intervals, proofs that had to be withdrawn.

The counterexample answers it in dimension three and above. The two-variable case is still open, and the pull request that formalised the disproof says so explicitly. That is a narrower result than the headline version, and honest reporting has to carry the difference.

The check took hours because someone wrote the question down first

Within a day, Paul Lezeau had formalised the counterexample in the Lean proof assistant and opened pull request 4474 on Google DeepMind's formal-conjectures repository, titled feat: add Jacobian disproof. Reviewers approved it. An independent audit posted to the same thread confirmed the proof contains no sorry, no native_decide and no custom axioms, which is the Lean community's way of saying nothing was assumed and nothing was waved through.

The speed came from preparation, not from the model. The repository already held a formal statement of the Jacobian conjecture, agreed by humans and written down before anyone had a counterexample to test against it. As the Xena Project blog put it, once humans agree the formal statement captures the conjecture accurately, checking that possibly AI-generated Lean code does comprise a proof or disproof of the conjecture is a triviality.

Read that sentence twice. The hard, slow, human part happened years earlier, when someone translated a sentence from 1939 into machine-checkable form. The claim arrived on a Sunday night and was settled by Monday because the acceptance test already existed.

Verified is not understood

The same blog is blunt about the limit. The next step, it says, is for humans to understand exactly what is going on with the example. A machine can certify that the three points collide. It cannot yet tell anyone why this map, out of all maps, was the one that broke an 87-year assumption. Journal peer review is not finished either, and a verification preprint is not a journal.

It is also not an isolated event. The same blog counts three counterexamples in three months: the Erdos unit distance conjecture in May, a question of Grothendieck's about group schemes in July, and now Keller's. Timothy Gowers described this one as the first time a language model had solved a well-known problem he had heard of outside his own area. Three data points is a pattern forming, not a pattern proven.

Write the acceptance test before you buy the model

The transferable lesson is procedural, not mathematical. Almost every AI capability claim an owner has been handed this year arrived as a benchmark score published by the party selling the model. This one arrived as an object a stranger could check in an afternoon, against a criterion written down before the claim existed, using a tool that answers to neither the lab nor the claimant.

That is a specification you can copy. Before the next AI procurement, write the acceptance criterion in a form that can be evaluated without the vendor's cooperation: a fixed test set you hold yourself, a scoring rule agreed in advance, an output format that a script you control can grade. If the only evidence of performance is a number the vendor computed, you have bought a press release.

Keep the second lesson next to the first. The counterexample is certified and still not explained, which is the exact shape of most AI output inside a business: correct in a way you can test, opaque in a way you cannot audit. Build for the first and staff for the second.

Frequently asked questions

Is the Jacobian conjecture now settled?

For three or more variables, yes, on the strength of an explicit counterexample that has been formalised in Lean and independently audited. The two-variable case remains open, and journal peer review of the result is not finished.

Did the model do the mathematics on its own?

Levent Alpoge posed the problem and published the result, and credited Claude Fable 5 with finding the counterexample. The Lean formalisation was done by hand by Paul Lezeau, and the audit was carried out by people who had no part in the original claim.

Why does this matter to a company that does no mathematics?

Because of how the claim was settled rather than what it was about. The acceptance criterion existed before the claim, in a form a third party could check without trusting the claimant. That is a template for evaluating any AI vendor.

The mathematics will be understood eventually. The procurement lesson is available now, and it costs nothing to adopt: no claim is worth more than the test you can run on it yourself.

AnthropicClaude Fable 5LeanGoogle DeepMindAI evaluationMathematics

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