Claude Claude Fable 5 produced a candidate counterexample to the Jacobian Conjecture, and mathematicians are checking it
A mathematician says Claude Fable 5 helped him construct a concrete counterexample to the Jacobian Conjecture, an open problem since 1939. The object is mechanically checkable, real mathematicians are taking it seriously, and it has not been through peer review.
Over the weekend the mathematician Levent Alpoge posted a concrete polynomial map and a claim that would, if it holds, close an 87-year-old open problem: the Jacobian Conjecture is false. What makes the post more than a provocation is that he credits Claude Fable 5, Anthropic’s current flagship, with constructing the candidate counterexample, and that the object he posted can be checked by arithmetic rather than trust.
What was claimed
The Jacobian Conjecture, posed in 1939, roughly says that any polynomial map from complex n-dimensional space to itself whose Jacobian determinant is a nonzero constant must be invertible. It has resisted proof for decades and sits on the short list of problems whose resolution would be a genuine event in algebraic geometry. Alpoge’s post gives a specific map in three complex dimensions with a Jacobian determinant of exactly negative two, a nonzero constant, that is nonetheless not one-to-one: he lists three distinct input points that all map to the same output. A map with a constant nonzero Jacobian that fails to be injective is precisely what the conjecture says cannot exist.
By Alpoge’s account the construction came out of a casual exchange, a question from the mathematician Akhil Mathew while a soccer final was on, run through Claude Fable 5, which produced the algebra. He shared Wolfram Alpha links so others could verify the intermediate steps. That is the important structural point. Unlike a prose proof that a reader has to trust, a claimed counterexample is a finite object: you can plug the three points into the map and confirm they collide, and you can compute the determinant and confirm it is constant. Other mathematicians, including Jared Duker Lichtman, called it a remarkable result and attributed it jointly to Alpoge, Mathew, and the model.
Why developers should care about a math result
This is not a coding story, but it is squarely a capability story, and the same model families sit behind both. For most of the last two years the pitch for frontier models in technical work has been assistance: autocomplete, refactoring, test generation, review. A model contributing the load-bearing step of a candidate disproof to a named open problem is a different claim about where the ceiling is. If it survives scrutiny, the useful framing for a working developer is not “AI does math now” but that these systems can occasionally propose a correct nonobvious object in a space where a human can verify it cheaply. That is the same shape as a good code suggestion: the model proposes, and a cheap check decides.
The caution is the same too, and it is why this is a beat and not a celebration. Nothing here is peer-reviewed. Wolfram Alpha catching arithmetic errors is a sanity check, not a certificate, and the history of this particular conjecture is littered with claimed proofs and disproofs that did not hold up. Until the construction is worked through carefully and published, the conjecture stays open in the official record, and a viral thread is not the record.
What’s worth watching
- Independent verification. The object is checkable, so verification should be fast if it is correct. Whether working algebraic geometers confirm the map does what the thread says, in a form that survives careful reading, is the whole story.
- The provenance of the construction. How much of the counterexample the model generated versus refined from a human sketch will shape how much this actually says about model capability. Expect that detail to get litigated.
- The pattern, not the headline. If frontier models start contributing verifiable objects to open problems on a regular basis, the interesting shift is methodological: humans steering models toward candidates and keeping the verification, rather than models producing unreviewable claims.
The honest read is that this is a striking, concrete, and still unconfirmed result that happens to sit at the intersection of pure mathematics and a question developers actually care about: how far these models can be trusted to reason. It is worth watching precisely because it is checkable. Stackmaven will follow up once the counterexample has been independently verified or refuted, on or around October 18.