Can AI search find what human proof cannot?

Monash Lens reports that Levent Alpöge, a mathematician at Anthropic, posted on X that he had found a counterexample to the Jacobian conjecture using Anthropic's large language model Fable 5, "released to the general public only a few weeks ago." The counterexample is a three-dimensional polynomial mapping with "a constant Jacobian determinant of -2" that nonetheless "moves multiple input points to the same output point," making it irreversible — which the conjecture says should be impossible when the determinant is a nonzero constant. Per the outlet, this is enough to show "the conjecture is false for every dimension larger than 2, with the original conjecture in two dimensions remaining open." The post itself was "short enough to fit into a single X post," and that brevity, the outlet says, "made it easy for other mathematicians to verify."

The piece frames what made the discovery notable as distinct from the mechanics of proof-writing: "the difficulty in finding it seems to have lain not in an intricate construction or a lengthy proof, but rather in finding a good way of navigating an enormous search space of possible polynomial mappings to find one with the right properties." The conjecture had resisted proof attempts for over a century — claimed proofs by Segre and Gröbner were later found to contain "subtle errors" — precisely because, as a 2017 Math Stack Exchange post the article quotes put it, a valid counterexample could in principle be written by "some smart undergraduate," yet nobody had found one. Monash Lens draws the implication directly: "this suggests AI may prove to be just as valuable for discovering unexpected mathematical objects as it is for constructing proofs."

This reframes what several nearest-note cases treat as the same category of event. What made OpenAI's unit distance counterexample succeed? attributes that breakthrough to a novel construction move layered onto existing number theory — difficulty sat in the idea, not the search. Alpöge's case, by the article's own account, inverts that: the idea (a constant-Jacobian, non-injective polynomial map) is simple, and the hard part was locating one instance inside a vast space of candidates. It also sidesteps the validation question Can opaque machine learning models help prove new mathematics? raises about opaque tools needing external checks — a counterexample short enough to fit in a social media post is, as reported, verified by direct inspection rather than by a separate reliability layer. And where Can automated scoring verify mathematical constructions without human understanding? relies on an automated score to certify search output at scale, verification here is reported as informal and immediate, done by other mathematicians reading the post.

The excerpt does not establish how Fable 5 was prompted or what the model's intermediate output looked like — the outlet states plainly that "details have not been made public regarding exactly how Alpöge prompted the AI model." Nor does it establish that search-over-construction is the general pattern for AI mathematics; it draws that conclusion from two cases (this one and the unit-distance disproof) and frames it as a suggestion, not a settled finding. The reporting also rests on a claim that, at the time of writing, had circulated only as a social media post rather than a peer-reviewed write-up, however brief and checkable the construction itself was said to be.