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This work aims to solve the sphere packing problem in dimensions 8 and 24 through novel mathematical techniques.

August 2, 2026Open Access

Viazovska's Modular Form Proof Solves Sphere Packing in Dimensions 8 and 24 — E8 Intelligence Research

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Randomized trial finds optimal sphere packing densities in dimensions 8 and 24, indicating significant mathematical advancements.

This work aims to solve the sphere packing problem in dimensions 8 and 24 through novel mathematical techniques.
Developed an auxiliary function using modular forms and Fourier transform positivity.
Applied linear programming bounds to establish optimal packing density in ℝ⁸ and ℝ²⁴.
Explored connections to the E₈ and Leech lattices through mathematical constructions.
Optimal sphere packing density in ℝ⁸ found to be π⁴ / 384 (≈ 0.253669) using E₈ lattice.
Optimal density in ℝ²⁴ determined as π¹² / 12! (≈ 0.001929) with the Leech lattice.
Established relationships between modular forms and the structure of the E₈ root system and Leech lattice.

Andrew Stewart Caldin (2026) studied this question.

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