When Ukrainian mathematician Maryna Viazovska received a Fields Medal—widely regarded as the Nobel Prize for mathematics—in July 2022, it was big news. Not only was she the second woman to accept the honor in the award’s 86-year history, but she collected the medal just months after her country had been invaded by Russia. Nearly four years later, Viazovska is making waves again. Today, in a collaboration between humans and AI, Viazovska’s proofs have been formerly verified, signaling rapid progress in AI’s abilities to assist with mathematical research.

“These new results seem very, very impressive, and definitely signal some rapid progress in this direction,” says AI-reasoning expert and Princeton University postdoc Liam Fowl, who was not involved in the work.

In her Fields Medal–winning research, Viazovska had tackled two versions of the sphere-packing problem, which asks: How densely can identical circles, spheres, et cetera, be packed in n-dimensional space? In two dimensions, the honeycomb is the best solution. In three dimensions, spheres stacked in a pyramid are optimal. But after that, it becomes exceedingly difficult to find the best solution, and to prove that it is in fact the best.

In 2016, Viazovska solved the problem in two cases. By using powerful mathematical functions known as (quasi-)modular forms, she proved that a symmetric arrangement known as E₈ is the best 8-dimensional packing, and soon after proved with collaborators that another sphere packing called the Leech lattice is best in 24 dimensions. Though seemingly abstract, this result has potential to help solve everyday problems related to dense sphere packing, including error-correcting codes used by smartphones and space probes.

The proofs were verified by the mathematical community and deemed correct, leading to the Fields Medal recognition. But formal verification—the ability of a proof to be verified by a computer—is another beast altogether. Since 2022, much progress has been made in AI-assisted formal proof verification.

Serendipity leads to formalization project

A few years later, a chance meeting in Lausanne, Switzerland, between third-year undergraduate Sidharth Hariharan and Viazovska would reignite her interest in sphere-packing proofs. Though still very early in his career, Hariharan was already becoming adept at formalizing proofs.

“Formal verification of a proof is like a rubber stamp,” Fowl says. “It’s a kind of bona fide certification that you know your statements of reasoning are correct.”

Hariharan told Viazovska how he had been using the process of formalizing proofs to learn and really understand mathematical concepts. In response, Viazovska expressed an interest in formalizing her proofs, largely out of curiosity. From this, in March 2024 the Formalising Sphere Packing in Lean project was born. Lean is a popular programming language and “proof assistant” that allows mathematicians to write proofs that are then verified for absolute correctness by a computer.

A collaboration bringing in experts Bhavik Mehta (Imperial College London), Christopher Birkbeck (University of East Anglia, England), Seewoo Lee (University of California, Berkeley), and others, the project involved writing a human-readable “blueprint” that could be used to map the 8-dimensional proof’s various constituents and which of them had and had not been formalized and/or proven, and then proving and formalizing those missing elements in Lean.

“We had been building the project’s repository for about 15 months when we enabled public access in June 2025,” recalls Hariharan, now a first-year Ph.D. student at Carnegie Mellon University. “Then, in late October we heard from Math, Inc. for the first time.”

The AI speedup

Math, Inc. is a startup developing Gauss, an AI specifically designed to automatically formalize proofs. “It’s a particular kind of language model called a reasoning agent that’s meant to interleave both traditional natural-language reasoning and fully formalized reasoning,” explains Jesse Han, Math, Inc. CEO and cofounder. “So it’s able to conduct literature searches, call up tools, and use a computer to write down Lean code, take notes, spin up verification tooling, run the Lean compiler, et cetera.”

Math, Inc. first hit the headlines when it announced that Gauss had completed a Lean formalization of the strong prime number theorem (PNT) in three weeks last summer, a task that Fields Medalist Terence Tao and Alex Kontorovich had been working on. Similarly, Math, Inc. contacted Hariharan and colleagues to say that Gauss had proven several facts related to their sphere-packing project.

“They told us that they had finished 30 “sorrys,” which meant that they proved 30 intermediate facts that we wanted proved,” explains Hariharan. A proportion of these sorrys were shared with the project team and merged with their own work. “One of them helped us identify a typo in our project, which we then fixed,” adds Hariharan. “So it was a pretty fruitful collaboration.”

From 8 to 24 dimensions

But then, radio silence followed. Math, Inc. appeared to lose interest. However, while Hariharan and colleagues continued their labor of love, Math, Inc. was building a new and improved version of Gauss. “We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss,” says Han. “This new version reproduced our three-week PNT result in two to three days.”

Days later, the new Gauss was steered back to the sphere-packing formalization. Working from the invaluable preexisting blueprint and work that Hariharan and collaborators had shared, Gauss not only autoformalized the 8-dimensional case, but also found and fixed a typo in the published paper, all in the space of five days.

“When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised,” says Hariharan. “But at the end of the day, this is technology that we’re very excited about, because it has the capability to do great things and to assist mathematicians in remarkable ways.”

A laptop with sphere packing code in the foreground, with an autumn sunset at Carnegie Mellon in the background. Hariharan was working on sphere-packing proof verification as the sun was setting behind Carnegie Mellon’s Hamerschlag Hall.Sidharth Hariharan

The 8-dimensional sphere-packing proof formalization alone, announced on February 23, represents a watershed moment for autoformalization and AI–human collaboration. But today, Math, Inc. revealed an even more impressive accomplishment: Gauss has autoformalized Viazovska’s 24-dimensional sphere-packing proof—all 200,000+ lines of code of it—in just two weeks.

There are commonalities between the 8- and 24-dimensional cases in terms of the foundational theory and overall architecture of the proof, meaning some of the code from the 8-dimensional case could be refactored and reused. However, Gauss had no preexisting blueprint to work from this time. “And it was actually significantly more involved than the 8-dimensional case, because there was a lot of missing background material that had to be brought on line surrounding many of the properties of the Leech lattice, in particular its uniqueness,” explains Han.

Though the 24-dimensional case was an automated effort, both Han and Hariharan acknowledge the many contributions from humans that laid the foundations for this achievement, regarding it as a collaborative endeavor overall between humans and AI.

But for Han, it represents even more: the beginning of a revolutionary transformation in mathematics, where extremely large-scale formalizations are commonplace. “A programmer used to be someone who punched holes into cards, but then the act of programming became separated from whatever material substrate was used for recording programs,” he concludes. “I think the end result of technology like this will be to free mathematicians to do what they do best, which is to dream of new mathematical worlds.”

Articles tagged with "AI, mathematics, collaboration"

Watershed Moment for AI–Human Collaboration in Math

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