On July 20, Anthropic researcher Levent Alpoge announced that he found a counterexample to the Jacobian conjecture using Claude Fable 5, marking a significant breakthrough in algebraic geometry.
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
((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3,…
— levent (@__alpoge__) July 20, 2026
The conjecture was proposed by German mathematician Ott-Heinrich Keller in 1939. In simple terms, it posits that there exists a formula that transforms one set of three numbers into another. Mathematicians can verify such a formula at every point, ensuring it neither merges nor loses information nearby. The conjecture claimed that since the formula does not lose information locally, its action can always be fully reversed—allowing the original numbers to be recovered from the result.
Claude Fable 5 constructed a formula that breaks this rule. It passes local checks at every point but maps three different sets of numbers to the same result. Therefore, it is impossible to recover the original data from it, proving the conjecture false. Alpoge provided links to Wolfram Alpha, where the calculations can be verified.
According to the researcher, his friend posed the question about the conjecture. Claude worked on the problem during the World Cup final.
Over the past 87 years, the conjecture has become one of the central unresolved problems in its field, gaining notoriety for erroneous proofs.
Mathematician Qiaochu Yuan noted that this is the most famous open problem solved by a language model. He quoted mathematician T. T. Moh, who analyzed one of the erroneous works in 2008. Moh stated that Benjamino Segre published three incorrect proofs, Claude Chevalley accepted a false proof as valid, and Igor Shafarevich used the conjecture as a proven theorem.
for some background,
1) the jacobian conjecture is by far the most famous open problem resolved by an LLM so far
2) it's also infamous for attracting wrong proofs so this is quite funny in a specific way. a comment from a 2008 paper by t.t. moh debunking such a proof:
> The… https://t.co/PymlXIho2P pic.twitter.com/2lwERRGHGP— QC (@QiaochuYuan) July 20, 2026
Mathematician Jared Duker Lichtman called the result "outstanding." He reminded that a special case of the conjecture was the topic of Ethan Zhang's dissertation, who is expected to make a breakthrough in number theory.
This is quite a remarkable result:
Posed in 1939, the Jacobian conjecture is one of the central open problems in algebraic geometry, but was just disproved by Alpoge, Matthew, and Claude Fable 5.
The Jacobian conjecture roughly says that a multivariable polynomial F has an… https://t.co/P9A1ZLocai— Jared Duker Lichtman (@jdlichtman) July 20, 2026
So far, Alpoge's result has not undergone peer review. However, its uniqueness lies in its verifiability. AI companies' claims of new records typically rely on benchmarks and internal tests. The counterexample is different: it is a formula that anyone can substitute and recalculate.
Alpoge earned his PhD at Princeton under Fields Medalist Manjul Bhargava. In 2015, he received the Morgan Prize, the highest award in the U.S. for student research in mathematics.
AI vs. Open Problems
The counterexample to the Jacobian conjecture is not the first result obtained using language models in mathematics.
In October 2025, OpenAI's Vice President Kevin Vail claimed that GPT-5 solved ten Erdős problems. Hungarian mathematician Paul Erdős is one of the most renowned mathematicians of the 20th century, having authored around 1500 papers. After his death, a collection of hundreds of unsolved problems remained. They are formulated briefly, and their answers can be verified—hence AI developers use the list as a testing ground for models.
However, Vail's statement was not substantiated. Shortly after its publication, it became clear that the model had found solutions that already existed in the literature. Following criticism, Vail deleted the post.
In January 2026, GPT-5.2 Pro solved Erdős problem #397 concerning central binomial coefficients. The model's query was made by engineer Neil Somani. The answer was formalized by Aristotle, an automatic prover from the startup Harmonic.
The correctness was confirmed by Fields Medalist Terence Tao. Previously, he noted the autonomous solution of problem #728, which has no parallels in the literature. However, the mathematician warned that AI is currently only tackling "low-hanging fruit" accessible to standard techniques.
In May, the OpenAI model disproved the 1946 Erdős hypothesis about unit distances in the plane. Unlike most problems on the list, this hypothesis is a famous problem that mathematicians have worked on for 80 years. The company called the result the first of its kind for AI.
The proof was verified by nine external mathematicians and described in a supplementary paper. Shortly thereafter, Google DeepMind introduced the AlphaProof Nexus system, which solved nine smaller Erdős problems. Its proofs were verified by the Lean system.
In July, the project Star Fleet Math presented solutions to 20 Erdős problems, obtained by 20 parallel Codex agents. Each proof was verified by the Lean 4 core.
As a reminder, in February, Google unveiled the AI mathematician Aletheia, which autonomously solved four problems from the Erdős list.