Claude Fable 5 finds a tiny formula that topples an 87-year-old math conjecture

I'll pull the ScienceDaily article and HN thread so the paragraphs stick to real names, numbers, and mechanics—not invented detail.Levent Alpöge, a mathematician at Anthropic, announced on X that he used Claude Fable 5 to find a counterexample to the Jacobian conjecture. The model had been released to the public only a few weeks earlier. The example is a three-dimensional polynomial map with constant Jacobian determinant -2 that sends distinct inputs to the same output, so it has no inverse. That kills the conjecture in every dimension three and higher. The two-dimensional case, first stated by Ludwig Kraus in 1884 and generalized by Ott-Heinrich Keller in 1939, is still open.

The Jacobian conjecture sits in algebraic geometry. It says that if a polynomial map from n-space to itself has a Jacobian determinant that is a non-zero constant, then the map should have a polynomial inverse: no folding or collapsing of space, and a clean reverse map also built from polynomials. Finding maps that collapse points is easy; finding maps with constant non-zero Jacobian is also easy. Finding both at once is hard. Alpöge’s counterexample is short enough to fit in a single social post, so other mathematicians could check it quickly. The hard part was not a long proof but searching a huge space of candidate polynomial maps until one hit both properties.

For engineers and builders, the result is a concrete search-and-verify win, not a chain-of-thought spectacle. An LLM helped surface an object that experts could audit by hand in minutes. That pattern matters for anyone building tools over combinatorial spaces: bug-finding, symbolic regression, configuration search, inverse design. The useful product shape is propose-candidate → independent check, not accept the model’s narrative as proof. Prompting details for this run have not been published, so the reusable process is still opaque.

The competitive frame is clear. OpenAI recently had a hand in disproving the unit distance conjecture, and Liam Price used models on Erdős problem 1196. Anthropic’s Fable 5 now sits in that same class of public math hits, with an employee-led result and a counterexample simple enough for peer verification. Labs are competing not only on chat benchmarks but on whether their models can place real objects into open research problems that humans then certify.

Two dimensions remain open; computational checks already support the conjecture there for polynomials up to degree 100. Watch for a formal write-up of the three-dimensional map, independent re-derivations with other models, and whether similar short counterexamples appear in related invertibility or birational-geometry questions. Until prompting and search procedure are documented, treat this as a strong existence proof of AI-assisted discovery, not a turnkey method you can copy into a pipeline.