Anthropic's AI agent disproved a 1939 mathematical hypothesis: breakthrough or another fake?
On July 20, 2026, Anthropic researcher Levent Alpoge announced a sensational result: the Claude Fable 5 language model found a counterexample to the Jacobian conjecture — one of the central unsolved problems in algebraic geometry, formulated by German mathematician Otto-Heinrich Keller in 1939.
The essence of the conjecture is simple but profound. Imagine a formula that transforms one set of numbers into another. Mathematicians can check such a formula "locally" — at each individual point, it does not lose information or "glue together" different inputs. The Jacobian conjecture stated that if a formula behaves perfectly on a small scale, its action can be reversed globally: reconstructing the original numbers from the result. This property is called "invertibility."
A Counterexample That Breaks Everything
Claude Fable 5 constructed a formula that passes the local check, yet three different sets of numbers are mapped to the same result. This is a direct refutation of the conjecture: reconstructing the original data from the result is impossible. Alpoge published links to Wolfram Alpha, where anyone can independently verify the example.
Notably, the question about the conjecture was posed by a friend of Alpoge, while Claude itself worked on the problem during the World Cup final. Over its 87 years of existence, the conjecture became infamous due to numerous erroneous proofs. Mathematician Qiaochu Yuan noted that this is the most famous open problem ever solved by a language model and cited an analysis by T.T. Moh, who pointed out in 2008 that even luminaries such as Beniamino Segre, Claude Chevalley, and Igor Shafarevich had made mistakes in their assessments of it.
Mathematician Jared Duker Lichtman called the result "outstanding," recalling that a special case of the conjecture was the subject of Yitang Zhang's dissertation — the future author of a breakthrough in prime number theory.
Verifiability vs. Hype
The key difference between this result and typical AI company claims is its absolute verifiability. Usually, we hear about records on benchmarks or internal tests. Here, there is a concrete formula that can be plugged in and recalculated manually. While the result has not yet undergone formal peer review, its very nature makes it accessible for verification by any mathematician.
Alpoge is no newcomer to science. He earned his PhD at Princeton under the supervision of Fields Medalist Manjul Bhargava, and in 2015 he received the Morgan Prize — the highest US award for undergraduate research in mathematics.
AI vs. Open Problems: The Context
This is not the first time AI has helped solve mathematical problems. In October 2025, OpenAI Vice President Kevin Weil claimed that GPT-5 had solved ten Erdős problems, but the claim was not confirmed — the model had found already known solutions. In January 2026, GPT-5.2 Pro solved Erdős problem #397, which was confirmed by Terence Tao himself. In May, an OpenAI model disproved Erdős's 1946 conjecture on unit distances — the first case of this level for AI. The proof was verified by nine external mathematicians.
However, as Tao rightly noted, AI is still only picking "low-hanging fruit" accessible to standard techniques. Nevertheless, progress is evident: from unsubstantiated claims to formally verified refutations of 80-year-old conjectures.
My comment: The result by Alpoge and Claude Fable 5 is not just another AI record, but a demonstration that language models can generate fundamentally new mathematical objects, rather than merely compiling known ones. If verification confirms its correctness, this will be a watershed moment: AI ceases to be merely a tool for calculations and begins to participate in fundamental discoveries. However, the history of OpenAI's errors reminds us: for now, each such result requires Occam's razor and the hand of a living mathematician.