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A construction of the sphere-packing magic functions in dimensions 8 and 24

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Contributed by Thomas Hales ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

We develop an alternative route to the solutions of the sphere-packing problem in dimensions 8 and 24, through the auxiliary functions in the Cohn--Elkies Fourier bound. In each dimension we construct a radial Schwartz function with the prescribed lattice zeros by solving a countable system of linear equations in absolutely summable coefficients. We prove existence and uniqueness after normalization, without using the known magic functions or Fourier interpolation. Through a Cauchy boundary problem, solutions of the homogeneous equations are identified with global sections of a rank-six holomorphic vector bundle over the compactified modular curve of the full modular group, with its orbifold structure retained. Riemann--Roch shows that this bundle has Euler characteristic one, forcing a nonzero global section and hence a nontrivial solution of the equations. A separate argument proves uniqueness after normalization. An equivalent positive operator equation has a bounded inverse. The construction is motivated by the coefficient method for planar circle packing. A subsequent comparison identifies our solutions with the functions of Viazovska and of Cohn, Kumar, Miller, Radchenko, and Viazovska. The route to sphere packing is not yet independent in its entirety: we do not prove the required global sign inequalities from the coefficient equations. Those inequalities are inherited from the known functions after identification.

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Version history

  1. v1Submitted by Thomas HalesInitial depositCurrentOct 18, 2026