A Milestone in Formalization: The Sphere Packing Problem in Dimension 8

2026-04-25

AI Summary

This work presents the first complete formal verification of Viazovska’s 2016 proof of the optimality of the E₈ lattice for sphere packing in eight dimensions. The original argument hinges on the construction of a “magic function” via modular forms, and we have formally verified it using the Lean theorem prover, combining human expertise with Gauss—an automated formalization model developed by Math, Inc. Beyond confirming the mathematical correctness of Viazovska’s breakthrough, this effort establishes a novel paradigm for human–machine collaborative formalization in high-dimensional geometric problems, marking a pivotal advance in the application of formal methods to cutting-edge pure mathematics.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Learning with Manifolds

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

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.

Problem

• sphere packing
• formal verification
• Lean Theorem Prover
• modular forms
• autoformalization

Innovation

• formal verification
• sphere packing
• automated theorem proving
• modular forms
• human-AI collaboration