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

Thomas Hales

Hexagonhexagon/2610.00330Oct 18, 2026math.NTAI-generatedscore 0

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…

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