Mathematician Levent Alpoge of Anthropic used the Claude Fable 5 model to successfully disprove the Jacobian conjecture, a problem that had puzzled the mathematical community for 87 years. Originally proposed by Keller in 1939 and equivalent to the Dixmier and Poisson conjectures, it was once listed as one of the major unsolved mathematical challenges of the 21st century. Alpoge refuted the conjecture by constructing a polynomial mapping counterexample that satisfies the premises but yields non-unique outputs. Although a formal paper has not yet been published, the counterexample has been verified by tools such as Wolfram Alpha and ChatGPT and has already been added to Wikipedia. This breakthrough not only resolves a long-standing mathematical problem but also highlights the immense potential of AI in fundamental scientific research.Article author, source: AI Information Gap
On the night of the World Cup final, a mathematician at Anthropic posted a tweet.
The tone of this tweet is as simple as thanking a friend for a small favor—basically saying thanks to my good friend Akhil for asking me a question, and to my other good friend Fable for working overtime during the World Cup final.
Good friend Fable. He calls Anthropic's own most powerful model Claude Fable 5 his good friend.
The tweet claimed that the Jacobian conjecture, unsolved since 1987, had been disproven, with his friend Fable helping to find a counterexample. On July 19, Spain beat Argentina 1-0. A purely mathematical tweet, it garnered 14.3 million views and went viral across the tech community overnight.
The mathematician is Levent Alpoge, who posted a polynomial mapping formula in his tweet, along with a verification link to Wolfram Alpha—clicking it allows you to verify it.
You don't need to understand the formula—its core consists of just two things. The formula fully satisfies the prerequisites of the Jacobian conjecture; plugging it into Wolfram Alpha yields a Jacobian determinant of -2, a non-zero constant, meeting all required conditions. Yet, when three entirely different sets of inputs are fed in, the output is identical. For example, inputting (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) all produce (-1/4, 0, 0).
According to the logic of the hypothesis, different inputs should produce different outputs. Therefore, the only conclusion is that the hypothesis itself is flawed.
A netizen threw the formula to the dramatic ChatGPT, which thought for 5 minutes and 22 seconds before replying, “Okay, holy shit. Wow, this is actually true.” It specifically noted that exact symbolic algebra was used, not floating-point approximation.
Deeply impressed, I also had the recently viral guy Kimi K3 verify it, and got the same result.
The Jacobian conjecture was first proposed in 1939 by German mathematician Ott-Heinrich Keller and has remained unsolved for 87 years. The core issue is not complicated to state.
Imagine a machine that takes a set of numbers as input and outputs another set of numbers, with the operation governed by a polynomial. Mathematicians want to know whether this machine can run “in reverse”—that is, given the output, can you uniquely determine the original input?
There is a criterion called the Jacobian determinant; if the determinant is everywhere equal to a nonzero constant, the conjecture states that the machine must be reversible. This holds true when examining each small region, and the conjecture suggests that if the system can run backward locally everywhere, it can also run backward globally.
For 87 years, researchers have continuously published purported "proofs" of the Jacobian conjecture in academic journals, all of which were later found to contain errors. Fields Medalist Smale included it in his list of the "18 Great Mathematical Problems for the 21st Century," ranking it 16th, alongside the Riemann Hypothesis and P vs NP.
Zhang Yitang, a 1978 graduate of Peking University’s Department of Mathematics and the renowned mathematician who gained fame in 2013 for his breakthrough on the twin prime conjecture, also suffered a major setback on this very conjecture. Few know that during his PhD at Purdue University, his thesis topic was the Jacobian conjecture. Due to a flawed lemma provided by his advisor, his paper could not be published, leaving Zhang unable to secure an academic position—he worked at Subway for several years until, more than a decade later, at age 58, he achieved widespread acclaim with his breakthrough on the twin prime problem.
The conjecture he had hoped to prove true was disproven thirty years later by a mathematician from Anthropic and Claude Fable 5.
Levent Alpoge is not a pseudoscientist. In 2015, he received the Morgan Prize, the highest research award in mathematics given to undergraduates; at the time of receiving the award, he had not yet begun graduate school and had already published seven papers. He graduated from Harvard, earned his PhD from Princeton, was a member of the Harvard Society of Fellows, and is now at Anthropic.
His bio states, "1 hilbert problem so far," having solved one Hilbert problem to date.
Mathematically, the Dixmier conjecture, the Poisson conjecture, and the Jacobian conjecture have been proven equivalent—when one falls, all three fall. Alpoge confirmed this in a reply to the tweet. This single tweet effectively overturns all three conjectures.
Algebraic geometer Daniel Litt is still awake at 2 a.m. "It's already 2 a.m. here, but I can't stop laughing. Amazing."
Wikipedia has included this counterexample in the entry for the Jacobian conjecture.
As of now, Alpoge has not yet published a formal paper or undergone peer review; he added in a tweet that the PDF will follow. However, this counterexample has one key feature: you don’t need to understand the proof—just plug the formula and the three sets of numbers into the calculation and check whether the Jacobian determinant equals -2 and whether the three inputs produce identical results. Wolfram Alpha can compute it, ChatGPT can compute it, and Kimi can too. Independent researchers have already verified the results, and they match.
The counterexample is right there, and anyone can verify it.
Claude Fable 5 Doing research, this time they went big.