Timothy Gowers, Fields Medal laureate and one of the most influential mathematicians alive, called it "remarkable": for the first time, a language model had disproven a conjecture that extended beyond his own area of specialization. The Jacobian conjecture, formulated in 1939 and unsolved for 87 years, fell to a 216-character formula generated by Claude Fable 5. Twelve days later, OpenAI published a blog post with ten additional solved decades-old problems from eight mathematical fields, solved for a combined $2,000 by an unreleased model called Astra.
The longer of the two questions
The Jacobian conjecture traces to German mathematician Ott-Heinrich Keller, who formulated it in 1939. The question sounds technical but is deceivingly simple: If a polynomial map from n-dimensional complex space into itself has a constant, non-zero Jacobian determinant everywhere, must it be globally invertible? Intuitively, the answer seemed yes—a constant determinant means the map neither contracts nor expands locally. But proving it never succeeded. Fields Medal winner Steve Smale included the conjecture in his list of 18 most important open mathematical problems of the 21st century.
Levent Alpöge, a mathematician at Anthropic, found the counterexample using Claude Fable 5, the company's AI model launched in June 2026. The result is a polynomial map from C³ to C³—three-dimensional complex space to itself—where the Jacobian determinant is constantly −2 everywhere, yet the map is not globally invertible: three different input points land at the same output. The entire formula spans 216 characters. On July 20, Alpöge posted on X at 2:19 UTC: "the jacobian conjecture is false."
AI as research partner, not oracle
Alpöge described the process directly: Claude Fable 5 generated candidates, he handled mathematical verification. Other mathematicians verified the result within hours using SymPy, a Python library for symbolic computation. Gowers called the finding remarkable, noting it was the first time a language model solved a conjecture outside his specialization.
On Hacker News, mathematicians reacted with wonder mixed with healthy skepticism. "You tweeted a disproof of an 87-year-old conjecture?" was a common response. An independently created verification preprint confirmed correctness days later. The result holds for n = 3, the three-dimensional complex space. The Jacobian conjecture remains open in two dimensions.
OpenAI sneaked its next model launch into a math blog post
On August 1, OpenAI published a blog post titled "Ten advances in mathematics and theoretical computer science." Almost offhandedly, the company mentioned results from an internal version of its "next major model," called Astra. Tech magazine Gizmodo reported that OpenAI smuggled the announcement of its ChatGPT successor into a math blog post.
The blog post itself was no side note. Ten open problems from eight fields, some unsolved for decades. In group theory, Astra constructed objects proving the existence of non-sofic groups, a question open since 1999. In von Neumann algebra theory, the model disproved Connes' rigidity conjecture. In high-dimensional geometry, it improved the upper bound for sphere packing density, unchanged since 1978. From Paul Erdős' catalog of open problems, Astra solved three questions. Thomas Bloom, who maintains the Erdős catalog, called it major news.
The cost was around $2,000 for all ten solutions. For verification, Astra produced machine-checkable Lean-4 certificates, which OpenAI published on GitHub under Apache-2.0 license, together with a 249-page technical manuscript. Gowers and nine other mathematicians authored supporting publications. Gowers said he would have recommended the proofs for publication in top-tier journals without hesitation.
Answers without explanations
Both breakthroughs share a striking weakness: the AI systems deliver answers, not explanations. Alpöge's counterexample to the Jacobian conjecture is mathematically correct but offers no conceptual intuition for why such a map exists. Mathematics has had a dual mission for centuries: find and understand proofs. AI reliably accomplishes the first half.
Additionally, none of these results have yet appeared peer-reviewed in classical journals. The Jacobian disproof exists as a preprint, verified symbolically by independent mathematicians. Astra's Lean-4 certificates enable machine verification far exceeding classical peer review in speed, but not replacing its conceptual depth. The journal system faces a structural question: if machine certificates guarantee proof correctness, do human referees still need verification roles?
249 pages and no release date
OpenAI announced Astra as its next major model but named no exact launch date. Polymarket bets had traded August 2026 release as unlikely before the blog post. The 12 days between Alpöge's July post and the Astra announcement are no accident. They show a repeatable pattern: AI-driven foundational research, where the model explores the search space and humans handle mathematical verification. Whether Astra solves more open problems at public launch, or hits boundaries current models cannot overcome, will appear at release. One thing is already clear: the tools for mathematicians have fundamentally changed.