A counterexample to the Jacobian conjecture
On 19–20 July 2026 the mathematician Levent Alpöge posted an explicit polynomial map of complex 3-space with Jacobian determinant −2 that sends three different points to one, refuting the Jacobian conjecture in dimension three, open since 1939. He credited Claude Fable 5 with finding it. State of the claim: made in a public post, checkable by hand, and confirmed by independent expositions.
Why it matters
A problem open since 1939 fell to three short polynomials whose correctness a reader can check by direct calculation instead of trusting the model. The model’s role is described only in the words of the author and of Anthropic; there is no independent account of how the model worked.
State of the claim. Claimed: Alpöge’s post on X with the map F(x, y, z) = ((1+xy)^3 z + y^2 (1+xy)(4+3xy), y + 3x(1+xy)^2 z + 3xy^2 (4+3xy), 2x − 3x^2 y − x^3 z), which has Jacobian determinant −2 and sends (0, 0, −1/4), (1, −3/2, 13/2) and (−1, 3/2, 13/2) to (−1/4, 0, 0). Checked by people: David Speyer’s post of 20 July on the Secret Blogging Seminar blog calls the map generically three to one, not bijective; Terence Tao’s post of 21 July gives an exposition of the example; Shuhong Gao’s preprint of 31 July (arXiv 2608.00222) restates it, says all identities were verified in exact rational arithmetic, and builds counterexamples in every dimension above two. The pass also checked in a computer algebra system that the determinant of the Jacobian matrix is −2 and that the three points have the same image. No document read reports a formalisation in Lean. No dispute was found in what was read. Dimension two stays open (Tao). Dates. The post is stamped 20 July 02:19 UTC (by the viewing browser’s time zone), which is late on 19 July in US Eastern time; Gao and Speyer say 19 July. The record keeps both days. The model’s role. In the post Alpöge thanks "my other close friend fable for working during the world cup final". Speyer writes that Alpöge tweeted that Fable had found a counterexample; Gao’s preprint names Claude Fable 5. Anthropic’s own page of 10 August (the record on the zeta zeros) says a similar prompt was used to help Claude disprove the Jacobian conjecture, and calls Alpöge one of its mathematicians. No document read describes the process: no prompts, no division of work between the person and the model. Tao states in his post that he used an AI chatbot to discuss the problem and to confirm calculations. What the record does not claim: that the model was the sole author, that dimension two is settled, or how long the model worked.