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While the rest of the 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.
The Jacobian conjecture is a long-standing open problem in algebraic geometry that's bedeviled highly accomplished mathematicians for almost 90 years. It was included in "Smale's problems," a list of unresolved math problems put forward by mathematician Stephen Smale in 1998.
But that proved no obstacle for Fable 5, Anthropic's latest frontier AI model. (You may remember Fable 5 as the public version of Claude Mythos Preview, the AI model that Anthropic says has such advanced cybersecurity capabilities that it was too dangerous for public release.)
Alpöge is part of Harvard's Society of Fellows, and his LinkedIn profile notes an affiliation with Anthropic. We've reached out to Anthropic with questions about Alpöge's work, and we'll update this story if we receive more information.
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 neither?
"This did not cause me to update my priors about what AI can and can't do," professor and mathematician Andrew Blumberg told Mashable. "This is exactly the kind of thing I would expect AI to be able to do. If there was a counterexample that was concise and easy to state that people haven't found because it's a pain to search through all this stuff, AI will find it."
Blumberg has a joint appointment in mathematics and computer science at Columbia University and is on the board of directors of the First Proof project, an effort to test the capabilities of frontier large language models in solving research-level mathematics. So, it's safe to say he understands mathematics and AI better than most of us.
You want to learn something from the answer.
Blumberg affirmed that providing a counterexample to the Jacobian conjecture is still a significant achievement for AI in mathematics. But he also told us there's a big difference between providing a positive proof for a major math problem and providing a single counterexample that disproves it. Here's the metaphor Blumberg used to explain.
"Suppose that Moses came down from the mountain with tablets, and on the tablet was written, 'Cancer can be cured.' Would you care? You don't just want the answer to the question. You want to learn something from the answer. And the reason Smale thought this problem was important is because he thought that if we solved it, we would understand more things about the about the way nature is structured."
In other words, Blumberg says, "this counterexample tells us essentially nothing. It's just, you know, there are a lot of polynomials, and it's hard for people to check them all, but it's not hard for machines."
The Jacobian conjecture isn't the only math problem AI has cracked in recent months. OpenAI announced in May an "internal model" had disproved the Erdős unit distance conjecture, a central conjecture in discrete geometry.
However, in that case, the result was more productive, Blumberg says. That's because OpenAI didn't just provide a counterexample — but a disproof of the problem itself.
"The disproof there already has led to interesting things because experts in the area unpacked what was going on in the disproof and then used it to do other things," Blumberg says.
And that's not necessarily the case with the Jacobian conjecture counterexample, he adds. "I'm not an expert in that area, and so I can't say if there's something special about the structure of this counterexample that people will learn something from, but it's not in and of itself interesting."
Disclosure: Ziff Davis, Mashable’s parent company, in April 2025 filed a lawsuit against OpenAI, alleging it infringed Ziff Davis copyrights in training and operating its AI systems.
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