AI Proof Verification: Gauss Tackles 24D

When the Ukrainian mathematician Maryna Viazovska received a Fields Medal—widely considered the Nobel Prize for mathematics—in July 2022, it was big news. Not only was she the second woman to receive this honor in the award’s 86-year history, but she also received the medal just months after Russia invaded her country. Nearly four years later, Viazovska is making waves again. Today, in a collaboration between humans and AI, Viazovska’s evidence has been verified before, signaling rapid progress in AI’s abilities to help in mathematicsic research.

“These new results look very, very impressive and definitely signal rapid progress in this direction,” says Liam Fowl, an AI reasoning expert and postdoctoral fellow at Princeton University, who was not involved in the work.

In her Fields Medal-winning research, Viazovska addressed two versions of the sphere packing problem, which asks: at what density do circles, spheres, etc. fit? identical can they 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 extremely difficult to find the best solution and prove that it is indeed the best.

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

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

Serendipity leads to a formalization project

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

“Formal verification of a proof is like a rubber stamp,” Fowl explains. “It’s a kind of certification of good faith that you know your reasoning is correct.”

Hariharan explained to Viazovska how he used the process of formalizing proofs to learn and truly understand mathematical concepts. In response, Viazovska expressed interest in formalizing her proofs, largely out of curiosity. From there, the Formalizing Sphere Packing in Lean project was born in March 2024. 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 between experts Bhavik Mehta (Imperial College London, UK), Christopher Birkbeck (University of East Anglia, UK), Seewoo Lee (University of California, Berkeley) and others, the project involved writing a human-readable “blueprint” that could be used to map the various constituents of 8-dimensional evidence and which of them had and had not been formalized and/or proven, then proving and formalize these missing elements in Lean.

“We had been building the project repository for around fifteen months when we made public access in June 2025,” recalls Hariharan, now a first-year doctoral student at Carnegie Mellon University. “Then, in late October, we heard Math, Inc. for the first time.”

The acceleration of AI

Math, Inc. is a startup developing Gauss, an AI specially designed to automatically formalize proofs. “It’s a special type of language model called a reasoning agent, intended to interweave both traditional natural language reasoning and fully formalized reasoning,” explains Jesse Han, CEO and co-founder of Math, Inc. “So it’s capable of doing literature searches, calling tools, and using a computer to write Lean code, take notes, run verification tools, run the Lean compiler, and more.” »

Math, Inc. first made headlines when they announced that Gauss had completed a Lean formalization of strong systems. prime number theorem (PNT) in three weeks last summer, a task that Fields Medalist Terence Tao and Alex Kontorovich had worked on. Similarly, Math, Inc. contacted Hariharan and his colleagues to tell them that Gauss had proven several facts related to their sphere packing project.

“They told us they had completed 30 ‘sorry’, which meant they had proven 30 intermediate facts that we wanted to prove,” says Hariharan. Part of this apology was 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 corrected,” adds Hariharan. “So it was quite a fruitful collaboration.”

From 8 to 24 dimensions

But then radio silence followed. Math, Inc. seemed to lose interest. However, while Hariharan and his colleagues continued their labor of love, Math, Inc. was building a new and improved version of Gauss. “We had a research breakthrough in mid-January that produced a much more powerful version of Gauss,” says Han. “This new version reproduced our three-week PNT result in 2-3 days.”

A few days later, the new Gauss was brought back to the formalization of sphere packing. Building on the invaluable pre-existing plan and work shared by Hariharan and his collaborators, Gauss not only self-formalized the 8-dimensional case, but also found and corrected a typo in the published paper, all within the span of five days.

“When they contacted us at the end of January to tell us that they had completed it, to say the least, we were very surprised,” says Hariharan. “But ultimately, it’s a technology that we’re very excited about, because it has the ability to do great things and help mathematicians in remarkable ways.”

Hariharan worked on verifying the evidence of the sphere’s packaging as the sun set behind Carnegie Mellon’s Hammerschlag Hall.Sidharth Hariharan

The formalization of the proof of packing of 8-dimensional spheres, announced on its own on February 23, represents a watershed moment for self-formalization and AI-human collaboration. But today, Math, Inc. revealed an even more impressive achievement: Gauss self-formalized Viazovska’s 24-dimensional sphere packing proof (over 200,000 lines of code) in just two weeks.

There are commonalities between the 8- and 24-dimensional cases in terms of fundamental theory and overall proof architecture, meaning that some of the code from the 8-dimensional case could be refactored and reused. However, Gauss had no pre-existing plans to work from from this time on. “And it was actually much more complex than the 8-dimensional case, because it was missing a lot of the background material that needed to come online regarding many properties of the Leech lattice, especially its uniqueness,” says Han.

Although the 24-dimensional case is an automated effort, Han and Hariharan acknowledge the many contributions of humans who laid the foundation for this achievement, viewing it as an overall collaborative effort between humans and AI.

But for Han, it represents much more than that: the start of a revolutionary transformation in mathematics, where very large-scale formalizations are commonplace. “A programmer used to be someone who punched holes in boards, but the act of programming then became separated from the hardware substrate used to record the 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.”