A Milestone in Formalization: The Sphere Packing Problem in Dimension 8
For mathematicians and formal verification researchers, this demonstrates a milestone in formalizing complex mathematical proofs, though the result is incremental as it builds on existing work.
The paper reports the formal verification of Viazovska's solution to the sphere packing problem in dimension 8 using the Lean Theorem Prover, with the final stages completed by an autoformalization model. The verification was achieved in February 2026.
In 2016, Viazovska famously solved the sphere packing problem in dimension $8$, using modular forms to construct a 'magic' function satisfying optimality conditions determined by Cohn and Elkies in 2003. In March 2024, Hariharan and Viazovska launched a project to formalize this solution and related mathematical facts in the Lean Theorem Prover. A significant milestone was achieved in February 2026: the result was formally verified, with the final stages of the verification done by Math, Inc.'s autoformalization model 'Gauss'. We discuss the techniques used to achieve this milestone, reflect on the unique collaboration between humans and Gauss, and discuss project objectives that remain.