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

Sidharth Hariharan, Christopher Birkbeck, Seewoo Lee, Ho Kiu Gareth Ma, Bhavik Mehta, Auguste Poiroux, Maryna Viazovska

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.

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

Abstract

In 2016, Viazovska famously solved the sphere packing problem in dimension , 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.
Paper Structure (9 sections, 2 theorems, 9 equations, 2 figures)

This paper contains 9 sections, 2 theorems, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Mathematics behind the $8$-Dimensional Sphere Packing Problem
  3. The Road to Stating the Main Theorem
  4. Constructing Viazovska's Magic Function in Lean
  5. The Theory of (Quasi)Modular Forms
  6. Metaprogramming
  7. Autoformalization
  8. A Look to the Future

Key Result

theorem 1

There is no sphere packing in $\mathbb{R}^8$ with density greater than that of the $E_8$ packing. That is,

Figures (2)

  • Figure 1: Choosing contours of integration
  • Figure 2: Contour needed for eigenfunction property

Theorems & Definitions (2)

  • theorem 1: Viazovska 2016
  • theorem 2: Cohn-Elkies 2003 CohnElkies