Scott Kominers has taught Robert Aumann’s 1976 theorem dozens of times. He’s assigned it in economics courses at Harvard. He’s built on it in his own research. So when Axiom Math’s formal verification system flagged a gap in the proof’s foundations earlier this year — an assumption Aumann stated but never actually proved — Kominers did what any rigorous economist would do. He called his colleagues.
“They all sort of said, ‘Oh, well, that makes sense. Aumann knows this,'” Kominers told Fortune in a recent interview. The problem: Aumann never proved all of the underlying structures underneath. And almost every theorem built on top was resting on foundations no one had formally examined. Until now. Kominers said there was “something that was under the surface and people hadn’t noticed was relevant, and these structures were never taught … the proof was implicitly relying on more than people realized.”
That finding is the first public result of EconLib, a project Fortune can exclusively reveal that Axiom Math is building — and which its founders believe could reshape how economic theory is used in American law.
To understand why this matters, you have to understand what Axiom is actually building. Axiom Math was founded by its CEO, Carina Hong, an MIT and Oxford-trained mathematician who dropped out of Stanford to launch it, raising $200 million at a $1.6 billion valuation in March, led by Menlo Ventures.
Axiom is building what Hong calls “verified AI.” The system doesn’t just generate mathematical proofs — it writes them in Lean, an open-source formal programming language that behaves like code: either every logical step compiles, or the program won’t run. No hallucination tucked into step 47. No drift between premises and conclusions. Just true or false, with no wiggle room in between. It is, as Axiom sees it, the only honest way to let AI near mathematics — and, as it turns out, near economics.
courtesy of Axiom
Hong joined my interview with Kominers and told me about a saying in the Lean community: “Every time a proof cannot be formalized, the proof is wrong.” When Axiom ran Aumann’s theorem through it, the debugger flagged something.
Hong is a winner of the Morgan Prize, the most prestigious award in undergraduate mathematics, and she had a simple, audacious premise for her start-up: the bottleneck holding back science isn’t human intelligence, but human time. “The Ramanujans, the John Nashes — that is a scarce resource,” she said, referring to two eerily gifted mathematicians whose minds anticipated the scary processing power of AI.
Each genius was the subject of a Hollywood film, The Man Who Knew Infinity and A Beautiful Mind, respectively. Hong said the next time a person knows infinity or has a beautiful mind, “we want to have AI superintelligence to collaborate with them, to compound and scale their impact.”
Besides Ono and Hong, other members of the founding Axiom team include Chief Technical Officer Shubho Sengupta, who spent eight years at Meta’s AI research lab (FAIR) working on mathematical reasoning and deep learning, and lead AI researcher François Charton, who recently trained LLMs to solve a 132-year-old math problem at Meta. The company has 40 people and has customers in industries where mathematical precision is essential (e.g., finance, cybersecurity, software and hardware development), and gives them access to its AI systems and formal verification infrastructure.
Axiom also has Ken Ono, who has mentored both 10 Morgan Prize winners including both Hong and Kominers, and who was devastated when he realized AI was ruining his life’s work—at first.
As a number theorist at the University of Virginia and one of the most decorated mathematicians of his generation, with a Guggenheim fellowship and a presidential early career award from President Clinton among his owners, Ono was shocked that AI could instantly do what used to take him hours. “I almost retired last summer, saying the hell with it,” he told Fortune. “My entire mathematical career amounts to nothing!” he exclaimed.
courtesy of Ken Ono
Instead, he rethought his assumptions about what AI actually was. “For most of my career, I never really understood what math actually is,” he said, pointing to his head. “I misunderstood math for having all of this information crammed in my brain.” Before, you needed years of training and practice to call yourself a mathematician, and others without a PhD just couldn’t. “I now realize that we’re at an era where knowledge has become cheap, but verification has become more expensive, by contrast. And also the value of creativity is much higher than ever before.”
That reframe is also, it turns out, the business thesis of the company he joined.
In December, Ono took a leave from UVA to join Axiom. The signs that AI was crossing a threshold had been accumulating around him. In May, OpenAI solved the unit distance problem — one of Erdős’s most famous open questions in combinatorial geometry, unsolved since 1946. OpenAI hailed it as “the first time that a prominent open problem, central to a subfield of mathematics, has been solved autonomously by AI,” and prominent mathematicians were impressed. Fields medalist Timothy Gowers was impressed, and Thomas Bloom, who runs erdosproblems.com, predicted “similar successes in many other areas of mathematics” in the coming months and years.
In December, the proprietary model AxiomProver achieved a perfect score on the Putnam Competition — the notoriously brutal math exam that has humbled Fields medalists. Only five humans have ever done the same. Hong was not one of them — a fact she raised herself, unprompted.
“AxiomProver scored three times more than my Putnam score,” Hong said, before offering what can only be described as mathematician humor: “We were joking in the office that we should have a ‘beat Carina party.'” She and Ono laughed. But they insist that AxiomProver and EconLib are no joke.
Axiom is quietly building — and Fortune can exclusively reveal — what it calls EconLib. To understand what it is, it helps to know about Mathlib — a community-maintained library of over 210,000 formally verified theorems written in Lean, built over nearly a decade by hundreds of mathematicians around the world. It’s something like the open-source building-block layer of modern mathematics: researchers import what they need, trust that it’s correct, and build on top of it.
The idea, which Kominers is co-leading with Axiom’s team, is to do for economic theory what Lean’s Mathlib project has done for mathematics: build a formal, machine-verifiable library of foundational economic results. EconLib will be a public project, open for other economists to contribute to, extend and build on. At least one outside researcher has already asked to help build the library. EconLib is set to go live soon, Axiom told Fortune.
What excites Kominers about it is the kind of thing that sounds modest until you realize what it implies.”Economic theory is a century at least of ideas and methods and models,” he said. “We’re hoping to produce something that right out of the gate is useful to economists.”
Take the aforementioned 1976 theorem by Robert Aumann on common knowledge — the famous result established that rational agents who share a common prior cannot “agree to disagree.” It’s foundational to information economics and to how we think about reputation systems in online marketplaces, like how platforms like Airbnb decide what information to surface about hosts and guests. But it was never mathematically proven until Axiom tackled it with AI. Consider how shaky the foundations of antitrust law were before this, Kominers said. “All the stuff that’s built on top of it — almost nobody interacts with the foundations. And so making the foundational assumptions precise is, first of all, incredibly valuable as a scholar of economics to understand, but also downstream, this has lots of implications,” explaining that mergers have been approved and businesses reshaped based on theorems based on economic models that have never been fully interrogated.
“In formal economic antitrust analysis, there are often literally arguments about the foundations of the models used to derive the metrics written down in the merger guidelines,” Kominers said. “These are all economic theory models that deliver the statistics people use in the cases,” he said. “And my hope is that we can use these tools to put far more precision and clarity into what world model you’re assuming before you choose your metric.”
Brian Albrecht, a theoretical economist at the International Center for Law and Economics who works at the intersection of formal proofs and antitrust policy, has no affiliation with Axiom but uses Lean often in his work, especially with regard to antitrust law. The rise of AI tools has made it an “exciting time for economic theory,” he said, adding that something based on Lean for economic theory would “absolutely” be useful for his work. “There are a few people who have started to play around with it,” he said, adding that it would “allow people to build up on the work of others” and provide another angle to tackle big economic problems.
At the same time, Albrecht drew a sharp line between tightening theoretical foundations and winning real cases. “The big disputes in antitrust … aren’t about proving results in a model,” he said. They’re more about what’s the right market definition or who are the right competitors, and he didn’t think AI can be used to formalize this yet. He pointed to the ongoing Google search appeal as an example: “We’re parsing through the Microsoft decision and what exactly the words mean.” That’s not really about proving the foundations of Aumann’s theorem correct, in other words.
Also, Albrecht said the verification burden is real — the “hallucination” problem. Even as AI helps fill in the steps, Albrecht said there is “just no way around needing different people to approve and verify” new results — though “Lean helps get us a long way there.”
courtesy of Brian Albrecht
Since joining Axiom in March, Ono has started eight new papers and finished six using generative AI tools — many in fields he has never formally studied. He’s blown away by the company he’s keeping. “I’m writing a paper with Scott Kominers, one of the most famous mathematical economists in the world!” Ono said excitedly.
“There’s a huge missed opportunity in academia,” Kominers said, “where people are not thinking, or not enough people are thinking of the LLM as enabling them to scale their intellectual capabilities.” He said he lists his AI tools in acknowledgment footnotes — as collaborators, not minions — and is unapologetic about it. “It enables me to see beyond what I could see just myself, in the same way as talking with students and collaborators provides that same expansion of vision and expansion of scope.” He said he’s never been so productive in his entire life, calling it “surreal.”
In his papers, Kominers lists AI tools in detailed acknowledgment footnotes, saying that he thinks of them as collaborators. These acknowledgments typically give specific details about how the tool was used, including directly crediting them for methods and discoveries where appropriate.
Kominers cited advice that he gives to students: when you’re an expert in a field, you go through moments of discovering that you don’t know anything. Every two to three years, he returns to the foundational texts of matching theory — his primary specialty — and teaches it to himself from scratch, and comes away with a deeper understanding, yielding new papers and ideas. “This project,” he said of EconLib, “is like that for all of economic theory. I’m realizing I didn’t understand any of it. I have to reorganize it in my brain, rediscover all of the ideas.”
He looked almost embarrassed by how excited this made him. “It’s what people don’t normally talk about, like discovering that you know nothing as a superpower, but to me it is.”
Albrecht landed in almost exactly the same place. AI has turned him, he said, “a little bit more into a mathematician” — letting him formalize things he couldn’t pin down before because he lacked the pure-math training of a PhD. “That’s one more mathematician out in the world solving problems,” he said. “In a lot of ways it’s an exciting time.”
[This report has been updated with additional context from Scott Kominers on the implications of this breakthrough.]
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。