While the whole world was watching the World Cup on Sunday evening, mathematician Levent Alpöge casually announced on X that he had used Anthropic’s Fable 5 to disprove the Jacobian conjecture.
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The Jacobian conjecture is a long-standing open problem in algebraic geometry that has been solved by the most successful mathematicians for nearly 90 years. Included in “Younger Problems,” a list of unsolved mathematical problems put forward by mathematician Stephen Smale in 1998.
But that hasn’t been an obstacle for Fable 5, Anthropic’s latest AI model. (You may remember Fable 5 as the public version of the Claude Mythos Preview, an Anthropic AI model with such advanced cyber security capabilities that it was too dangerous to be released publicly.)
Alpöge is part of Harvard’s Society of Fellows, and his LinkedIn profile notes his affiliation with Anthropic. We reached out to Anthropic with questions about Alpöge’s work.
Meanwhile, for those of us who don’t understand polynomial functions from n-dimensional space, the question is: Does this represent a major breakthrough in artificial intelligence, or mathematics, or something else?
“This hasn’t made me reevaluate my assumptions about what AI can and can’t do,” professor and mathematician Andrew Blumberg told Mashable. “That’s exactly the kind of thing I would expect AI to be able to do. If there was a counterexample that was short and simple to say that humans haven’t found because it’s a pain to search through all this stuff, AI will find it.”
Blumberg, who holds a joint position in mathematics and computer science at Columbia University, is involved in the First Evidence project, an effort to test the capabilities of large-scale linguistic models in solving research-level calculations. So, it’s safe to say that you understand math and AI better than most of us.
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‘You want to learn something from the answer’
Providing a counterexample to the Jacobian conjecture is still an important achievement of AI in mathematics, Blumberg notes. But he also says that there is a big difference between giving a good proof of a big mathematical problem and giving a single counter-example that disproves it. Here’s a metaphor that Blumberg used to explain it.
“Let’s say Moses came down from the mountain carrying the tablets, and it was written on the tablet that cancer can be cured. Would you care? You don’t just want an answer to the question. You want learn something from the answer. And the reason Smale thought this problem was important was because he thought that if we solved it, we would understand a lot of things about how nature is made.”
In other words, Blumberg says, “this counterexample doesn’t tell us anything fundamentally. It’s just, you know, there are a lot of polynomials, and it’s hard for humans to evaluate them all, but it’s not hard for machines.”
Jacobian conjecture is not the only mathematical problem that AI has faced in recent months. OpenAI announced in May an “internal model” disproved the Erdős unit distance assumption, a central assumption in abstract geometry.
However, in that case, the result was more productive, Blumberg said. That’s because OpenAI didn’t just provide a counterexample — but proof of the problem itself.
“The disagreement there has led to interesting things because experts in the field are uncovering what happened in the release of evidence and then using it to do other things,” said Blumberg.
And that’s not the case with the Jacobian conjecture example, he adds – while admitting he’s not an expert enough to see if there’s anything special about the counterexample’s structure that we can learn from. “But it is not in and of itself interesting.”
Disclosure: Ziff Davis, Mashable’s parent company, in April 2025 filed a lawsuit against OpenAI, alleging that it infringes Ziff Davis’s copyright in training and using its AI programs.
