On that day, an unassuming mathematician named Maryna Viazovska made a splash heard ‘round the mathematical world. She won the coveted Fields Medal for her pioneering research on the sphere-packing problem. This complex mathematical puzzle investigates how densely identical shapes, such as circles and spheres, can be packed within n-dimensional space. What Viazovska accomplished was truly incredible—she found an exact solution to this problem in both the 8-dimensional and 24-dimensional cases. This achievement is a historic landmark in the world of math.
Viazovska’s proof that the symmetric arrangement, called E8, gives the best possible packing in 8 dimensions. She collaborated with other scientists to prove that the Leech lattice maximizes packing in 24-dimensional space. Together, their shared labor of love produced this revolutionary theorem. These groundbreaking discoveries didn’t just lead to a profound new mathematical concept—they unlocked an entire new world of study in multidimensional geometry.
The Role of Formal Verification
Following Viazovska’s monumental discoveries, Sidharth Hariharan, a first-year Ph.D. student at Carnegie Mellon University, began working on formalizing her proofs using the Lean programming language and proof assistant. The ultimate aim here was to make sure that a computer could check Viazovska’s sophisticated conclusions with 100% accuracy. Mathematics has changed so much in the last few hundred years that this task has become essential.
Hariharan and his collaborators posted their progress on formalizing the 8-dimensional sphere-packing proof formalization, which was a big milestone. Miraculously, Math, Inc.’s reasoning agent, Gauss, was able to autoformalize this proof in only five days’ time. In the course of this process, Gauss even found and corrected a typo in the published paper describing the 8-dimensional proof.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction.” – Liam Fowl
The work to formalise these proofs was officially started on the project Formalising Sphere Packing in Lean on March 2024. This RFC marks a new step in a larger movement that uses automated verification techniques to augment and improve mathematical rigor.
Collaboration and Progress
Artist Jesse Han collaborated with Hariharan to develop the ideas of the Formalising Sphere Packing project. Collectively, the three of them digitized Viazovska’s proofs step by step. Han noted that while the 24-dimensional sphere-packing proof was an automated effort, it was built upon extensive groundwork laid by human researchers. Han and Hariharan’s collaboration demonstrated the power of combining human intuition and automated reasoning systems to create an AI-powered tool suited to their specific workflow.
For reference, Gauss took a remarkable two weeks to autoformalize the 24-dimensional sphere-packing proof, which itself was over 200,000 lines of code. This accomplishment was evidence of the effectiveness of automated technology. In addition, it highlighted the importance of human contributions to the mathematical research enterprise.
“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
The pair’s collaboration continued to yield fruitful results, as they worked diligently to improve Gauss’s capabilities.
“We made a research breakthrough sometime mid-January that produced a much stronger version of Gauss.” – Jesse Han
The Future of Mathematical Research
As technology advances, the use of technology in mathematics must adapt to those advances as well. Agents of automated reasoning such as Gauss are already transforming how mathematical research is done. They overhaul the process of producing and validating proofs. These innovations have the potential to liberate mathematicians from mundane verification chores. In doing so, they’ll be freed up to focus on the more creative and innovative aspects of their work.
“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
Active research in formal verification increases assurance in existing proofs. It ignites discovery into new frontiers of math. As Hariharan mentioned, working together has no doubt been the secret sauce to their execution success.
“So it was a pretty fruitful collaboration.” – Sidharth Hariharan
During the next 15 months, researchers diligently worked to curate an inclusive repository rich with resources. It took them until June 2025 to make it publicly available. This commitment to transparency highlights the spirit of collaboration and pursuit of knowledge which inspires the contemporary age of mathematics.