A mathematical conjecture that stood for 87 years has been overturned — by an AI paired with human mathematicians.

From the Jul 24, 2026 daily brief

The Jacobian conjecture, posed in 1939, is a famous problem in algebraic geometry. It says that if a polynomial map's "Jacobian determinant" is a nonzero constant, the map should be invertible. An Anthropic researcher announced that, in human-machine collaboration, Claude Fable found a concrete counterexample — the determinant is constant everywhere, yet the map sends distinct points to the same place, directly violating the conjecture. The counterexample was then independently verified by models from rival companies — Moonshot AI's Kimi and OpenAI's ChatGPT — as well as Anthropic's own Sonnet, and multiple counterexamples of the same type followed (theZvi's weekly, Jul 23). Our July 20 issue covered DeepMind's system autonomously solving 9 of 353 open Erdős problems; the evidence quality here is better: the counterexample is mechanically checkable mathematics, and the verification came from competing vendors' models rather than one company's self-report. Three boundaries belong in the record together. The human mathematicians' contribution was substantial — this was not AI working alone. The relayer himself adds a caveat: a problem's being solved also suggests it was comparatively easy. And in the comments, one reader questions whether the counterexample traces back to existing human results in the training data. Attribution and any "first" designation await the academic community. This item, too, is a single weekly relay; we have not obtained the original announcement post. One methodological aside: cross-verification by multiple competing models is worth borrowing for evaluation and acceptance-testing design. No other major new papers this week in the directions we track.

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