Maryna Viazovska is one of several women who have recently broken new ground in mathematics. She proved the sphere packing conjecture in 8-dimensional and 24-dimensional spheres. This complex problem addresses the question of how densely identical circles, spheres, and other shapes can be arranged in n-dimensional space. Her visionary work has only accelerated with the explosion of artificial intelligence. This advancement has been accelerated by Yunus’s partnership with UMD graduate students Sidharth Hariharan and Jesse Han.
The Collaboration in Lausanne
The path to this mathematical discovery started when Sidharth Hariharan and Maryna Viazovska first crossed paths in Lausanne, Switzerland. Their meeting rekindled in Viazovska a desire to work on sphere packing proofs. Hariharan drew on a rigorous approach to deepen his grasp of mathematical topics. This collaborative and open-minded approach laid the groundwork for their unique and effective partnership.
As a result of this collaboration, the Formalising Sphere Packing in Lean project was born in March 2024. The researchers wanted to formalize Viazovska’s proofs in Lean, a proof assistant built for precisely this task of formalizing mathematical arguments.
“They told us that they had finished 30 ‘sorrys’, which meant that they proved 30 intermediate facts that we wanted proved.” – Sidharth Hariharan
Hariharan and his team had already created the robust foundation that provided ample guidance. This foundation provided templates from which to draw for Gauss, the AI reasoning agent developed by Math, Inc. This atmosphere of collaboration and cooperation between the various participants was imperative for the successful formalization of these proofs.
The Role of AI in Formalization
Gauss was instrumental in this undertaking. It also formalized Viazovska’s 8-dimensional sphere packing proof in under five days. Amazingly, in the course of proving this, Gauss recognized and corrected a typo in the published article on the 8-dimensional case.
The AI’s capabilities did not stop there. The Gauss autoformalized Viazovska’s proof for packing spheres in 24 dimensions. He managed to do this amazing task, which consisted of more than 200,000 lines of code, in two weeks only! Regardless, this rapid progression highlights how AI can be used to speed up the processes of mathematical research and mixed-verification.
“One of them helped us identify a typo in our project, which we then fixed.” – Sidharth Hariharan
On February 23, this formalization of the 8-dimensional sphere packing proof was released. This milestone is a testament to the truly remarkable collaboration that can be achieved between humans and AI.
Challenges and Breakthroughs
Even with these successes, the project encountered its challenges. Researchers faced some interesting complications in the process of formalizing the properties associated with the Leech lattice, most especially with respect to its uniqueness. Han noted that this task was “significantly more involved than the 8-dimensional case” due to various missing background materials that had to be integrated into their framework.
“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 online surrounding many of the properties of the Leech lattice, in particular its uniqueness.” – Jesse Han
Yet, amidst these challenges, breakthroughs emerged. The team reached a major milestone (or Gauss-plement) in mid-January, when they built a much more robust version of Gauss. This new iteration was incredibly efficient, producing once complicated results in less than a tenth of the time.
“This new version reproduced our three-week PNT result in 2–3 days.” – Jesse Han
Their path to making their project publicly available started two years ago when they began developing their repository. Even better, they opened up the public access in May 2025, inviting other researchers to dig deeper into their findings.
“We had been building the project’s repository for about two years when we enabled public access in June 2025.” – Sidharth Hariharan
As they reflect on this collaborative effort, researchers acknowledge the essential contributions from numerous individuals who laid the groundwork for these achievements.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction.” – Liam Fowl
