How Claude Fable 5 Helped Disprove the Jacobian Conjecture, an 87-Year-Old Mathematical Problem

On the night of July 19–20, 2026, while watching the FIFA Club World Cup final, mathematician Levent Alpöge — a number theorist who works at Anthropic after holding a Junior Fellowship at Harvard's Society of Fellows — was asked about the Jacobian conjecture by a friend named Akhil Mathew . Alpöge asked Anthropic's Claude Fable 5 model to work on the problem. The model returned a concrete 216-character polynomial map that satisfied the condition of having a constant nonzero Jacobian determinant but was not injective, thereby violating the conjecture's conclusion .

Alpöge posted the result on X on July 19, 2026, writing: "hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final" .

The Specific Counterexample

The counterexample is a polynomial map F: ℂ³ → ℂ³ defined explicitly as :

Two key properties hold :

• Jacobian determinant: det Jac F = −2 (a nonzero constant), so F is a Keller map.
• Non-injectivity: Three distinct input points all map to the same output (-1/4, 0, 0):
  • (0, 0, -1/4)
  • (1, -3/2, 13/2)
  • (-1, 3/2, 13/2)

Because a non-injective map cannot have a polynomial (or any) inverse, F contradicts the Jacobian conjecture. The construction works over any field of characteristic zero since all coefficients and witness points are rational .

A companion preprint (dated July 20, 2026) provides a full algebraic verification, determines the global geometry of F, and proves that the nonproperness set of F is a hypersurface — the mechanism by which the map fails to be a polynomial automorphism . The paper further constructs infinite families of such counterexamples .

Status of the Two-Variable Case

The Jacobian conjecture for n = 2 (two complex variables) remains an open problem as of July 2026. The counterexample only disproves the conjecture for dimensions n ≥ 3 .

Current Status of Verification and Peer Review

As of July 22, 2026, the following is known:

• Mathematically credible: The counterexample is explicit and mechanically checkable. Mathematicians are taking it seriously . A formal verification in Lean 4 has been produced and passes continuous integration builds . Mathematician Kevin Buzzard called the result "a big day" .
• No peer-reviewed publication yet: There is no published peer-reviewed paper or arXiv preprint as of the latest reports . The material exists as a social media post and a companion PDF on ulam.ai, but has not been through formal journal review.
• Wikipedia has updated: The Wikipedia entry for the Jacobian conjecture now lists the counterexample as disproving the conjecture for N > 2 .
• Wolfram MathWorld updated: The MathWorld entry now also lists the counterexample .
• Community assessment: The consensus among commentators is that the mathematics is almost certainly correct, but formal acceptance and citation in the official record are still pending as of this date . The result is credited to a human–AI collaboration, not to Claude acting alone .

Bottom line: The Jacobian conjecture appears to be genuinely false in dimensions ≥ 3. The two-variable case remains open. The counterexample is mathematically sound and has been verified independently, but has not yet completed formal peer review.