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The Jacobian Conjecture survived 87 years, a pile of published proofs that later collapsed, and a spot as problem 16 on Smale’s list of math problems for the 21st century. On July 19, an Anthropic employee handed it to Claude Fable 5. The model found a concrete counterexample. The conjecture is false.

A refutation you can check by hand

Keller’s 1939 conjecture says any polynomial map with constant nonzero Jacobian determinant must have a polynomial inverse. Fable 5 — Anthropic’s Mythos-class frontier model, the tier above Opus — wrote down an explicit map from ℂ³ to ℂ³ with Jacobian determinant −2 that sends two different points, (0, 0, −1/4) and (1, −3/2, 13/2), to the same output. Two inputs, one output, no inverse. Verification takes pen and paper.

Third conjecture down in 2026

GPT-5.6 proved the cycle double cover conjecture. An OpenAI model overturned Erdős’s unit distance conjecture in May. Now Fable 5 kills the most notorious of the three — a problem famous for wrecking careers with subtly wrong proofs. The pattern: LLMs are winning at counterexample hunting first, generating strange candidates fast and checking them cheaply. The HN thread hit the front page within hours.

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