Maryna Viazovska has set incredible breakthroughs in the field of mathematics. In particular, she solved the sphere-packing problem in 8-dimensional and 24-dimensional space. This complex problem seeks to determine how densely identical circles, spheres, and other shapes can be packed in n-dimensional space. Her innovative contributions to maths previously made her the first woman to receive a Fields Medal in July 2022. This highly sought after accolade is referred to as the Nobel Prize for mathematics.
The sphere-packing problem has long intrigued mathematicians. In two dimensions, like that of honeybee harbor, a hexagonal structure yields the most efficient answer. In three dimensions, the densest packing of spheres occurs when they’re stacked in a pyramid formation. Yet, Viazovska’s contributions go beyond these classical dimensions, leading to new understandings in higher-dimensional packing.
February 23, 2016, the official publication of Viazovska’s 8-dimensional proof. That win served to underscore the deep, unprecedented collaboration possible between human mathematicians and artificial intelligence. This milestone not only highlights the advancements in mathematical understanding but underscores the potential of AI in formalizing complex proofs.
The Role of AI in Formalization
To formalize Viazovska’s proof that 8-dimensional spheres are optimal at packing space, they used the Lean programming language and theorem prover. Lean has emerged as an exciting tool, increasingly popular in recent years, for formalizing mathematical theories. Sidharth Hariharan, who is now working on a Ph.D. at Carnegie Mellon University, was instrumental in this effort. He and his collaborators produced a human-readable “blueprint” that helped tremendously in the formalization of the proof.
This innovative collaboration is exemplified in Math, Inc.‘s announcement that Gauss, an AI developed by the company, successfully autoformalized Viazovska’s 24-dimensional proof in just two weeks. This formal proof is more than 200,000 lines of code. It demonstrates the incredible potential of AI in addressing complex mathematical challenges.
“When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised.” – Sidharth Hariharan
Jesse Han, CEO and co-founder of Math, Inc. described how technical and substantial challenges existed in formalizing the 24-dimensional proof. He said restoring what’s been lost with the background papers about the idiosyncratic nature of the Leech lattice would be crucial. These materials have been indispensable in getting a grasp on this highly complex mathematical object.
“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.” – Jesse Han
This new partnership between mathematicians and generative AI is a historic turning point in the field, expanding horizons beyond what we once believed was possible. This partnership has garnered recognition and accolades — and even envy — from the academic world. It’s a celebration of exciting recent advances in the field of mathematical proof verification.
Liam Fowl, an expert leader in the field, was effusive in his praise for the new findings. He stated, “These new results seem very, very impressive and definitely signal some rapid progress in this direction.” This recognition is indicative of the overall buzz around AI’s transformational potential to help mathematicians work on new, more complex problems.
The reasoning agent technology behind Gauss is characterized as interleaving traditional natural-language reasoning with fully formalized reasoning. This exceptional ability to create new realities greatly enhances the depth and breadth of understanding mathematical concepts, unlocking potential for the future.
“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.” – Jesse Han
The ramifications of these developments reach far beyond just generating proofs. They have the power to fundamentally change the way mathematicians interact with their work. Han hopes that the technology Gauss represents will one day allow mathematicians to ascend to new worlds of thought.
“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.” – Jesse Han
The Future of Sphere Packing and Mathematics
The sphere-packing problem is still an amazing story within mathematics. Theoretical impact Viazovska’s accomplishments have laid the groundwork for raising expectations on what could be achieved in higher dimensions. Despite this bureaucratic clunkiness, formalization efforts thus far have underscored AI’s powerful potential to augment human intelligence in addressing monumental challenges.
Harian also reflected that it was a lengthy 15 months working to create the project’s repository. It didn’t receive any public access until June 2025 when it finally opened. Providing this level of transparency ideally will help combat those challenges and allow for collaboration and innovation with the greater mathematical community. The success this exceptional collaboration has realized should be an encouragement not just for other efforts in the field of mathematics, but those across every shared opportunity.
“We had been building the project’s repository for about 15 months when we enabled public access in June 2025.” – Sidharth Hariharan