The Jacobian conjecture has swallowed careers for 87 years. Posed in 1939 by Ott-Heinrich Keller, it asks a deceptively simple question: if a polynomial map from n-dimensional complex space to itself has a Jacobian determinant that is a nonzero constant everywhere, must the map have a polynomial inverse? Generations of mathematicians have claimed proofs, watched them collapse, and in at least one famous case, watched a promising career derail entirely. On the night of the World Cup final in July 2026, a mathematician at Anthropic posted a 216-character formula that ends the conjecture for every dimension above two.
Levent Alpoge’s X post, written in the informal register of a group chat, announced that the Jacobian conjecture is false. The counterexample is a map from three-dimensional complex space to itself with three polynomial components of degrees 7, 6, and 4. Its Jacobian determinant is identically -2, a nonzero constant, satisfying the conjecture’s hypothesis at every point. Yet three distinct inputs, (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2), all map to the same output (-1/4, 0, 0). A map that is not injective cannot have an inverse, polynomial or otherwise. The conjecture is false in three dimensions, and by padding with extra variables, in every dimension above three. The two-dimensional case remains open.
The machine did the searching
The twist is how the counterexample was found. Alpoge credits Claude Fable 5, Anthropic’s model, with doing the work during the World Cup final, prompted by a question from his friend Akhil Mathew of Stanford. The exact prompts were never made public, and mathematicians have noted the search required genuine insight, not a single well-phrased question.
What has been verified is the mathematics itself, and with unusual thoroughness. Within a day, independent checks appeared in SymPy, Wolfram Alpha, and two formal proof assistants, Isabelle/HOL and Lean 4. The Archive of Formal Proofs published an Isabelle/HOL proof that the determinant is -2, that the three points collide, and that no polynomial inverse exists. Multiple Lean 4 formalizations followed the same day. Because all coefficients are rational, the counterexample works over every field of characteristic zero.
A conjecture with a complicated legacy
The Jacobian conjecture has been called the most notorious open problem for attracting wrong proofs. Yitang Zhang, later famous for his breakthrough on prime gaps, had his PhD derailed when his advisor’s proof of the conjecture turned out to be flawed. The problem appears on Stephen Smale’s 1998 list of mathematical problems for the next century and is equivalent to the Dixmier conjecture and connected to the Mathieu and Gaussian-moments conjectures. The two-dimensional case had been verified computationally only up to degree 100.
Mathematicians’ reactions have been notably split. Fields medalist Timothy Gowers called the result the first AI-solved problem big enough that he had heard of it. Others were more measured, noting that a counterexample ends an argument but, unlike a proof, reveals little about the structure that made the conjecture plausible. The discovery method also remains opaque: the community can verify the output, but not yet the reasoning that produced it.
What survives
The mathematical landscape after the announcement is subtler than a simple demolition. The map is not proper, its image misses a smooth curve, and a refined theorem survives: proper Keller maps are polynomial automorphisms. The counterexample fails exactly by non-properness. Researchers have already begun building on the result, with papers appearing on graded Keller maps and the failure of related conjectures in higher dimensions. One OpenAI researcher reported that an internal model independently found essentially the same counterexample, suggesting the structure is not a fluke of a single system.
Alpoge has promised a full write-up, which had not appeared as of early August 2026. The result exists as an X post, independently verified by formal proof assistants but not peer-reviewed in the traditional sense. Whether that distinction matters may be the deeper question the episode poses about how mathematics will be done.
Sources
- Levent Alpoge (@__alpoge__), X post, July 20, 2026: https://x.com/__alpoge__/status/2079028340955197566
- The Conversation, “hello there the jacobian conjecture is false thanx…”, July 22, 2026
- New Scientist, July 20, 2026
- Archive of Formal Proofs, “Jacobian Counterexample” (Isabelle/HOL), July 20, 2026
- arXiv:2607.20210 (Shaska), arXiv:2607.18186, 2607.19012 (Long)
- ScienceDaily release, August 5, 2026