TejasviTomar
October 2026 · Preprint

The kissing numbers in dimensions five and seven are at most 43 and 132

Tejasvi Singh Tomar

Abstract

The kissing number τ_n is the largest number of non-overlapping unit balls in R^n that can touch a common unit ball. The best bounds so far were 40 ≤ τ5 ≤ 44 and 126 ≤ τ7 ≤ 134; both upper bounds come from high-precision floating-point computations of the three-point bound by Mittelmann and Vallentin. We prove τ5 ≤ 43 and τ7 ≤ 132. Each proof is an exact rational solution of the second level of the Lasserre hierarchy for spherical codes at degrees (d1, d2, δ) = (14, 20, 20). We use the formulation of de Laat, Leijenhorst and de Muinck Keizer and check the solutions with their verification program. Each certificate is obtained by rounding a convex combination of two dual iterates of the interior-point method, one on each side of the target value. Exact certificates for the same hierarchy also give A(4, arccos 1/3) ≤ 14, A(5, arccos 1/3) ≤ 22 and A(6, arccos 1/3) ≤ 34 for spherical codes, improving the previous bounds 15, 23 and 35; with a known code, A(4, arccos 1/3) = 14. The largest number of points at angle π/3 from a point of a kissing configuration in R^n equals A(n-1, arccos 1/3), so in R^5 this number is 14. We also prove the five bounds in the Lean proof assistant, assuming only the positivity of the zonal kernels given by the zonal polynomial files: once by checking the exact certificates with compiled code, and once, with every computation done by Lean's kernel, from the certificates rounded to 30 decimal places. Version 1 of this paper, titled "The kissing number in dimension five is at most 43", did not contain the bound for τ7.

In plain words

The kissing problem asks how many balls can touch one ball of the same size without any two of them overlapping. The answer is 12 in three dimensions and 24 in four, with proofs published in 1953 and 2008. In five and seven dimensions, arrangements of 40 and 126 balls are known. The best upper limits, set in 2010, were 44 and 134. This paper lowers them to 43 and 132. Each new limit rests on a certificate: large tables of exact fractions that pass a checking program written by other researchers.

Two round line drawings on white, without labels. Left, in blue: 40 dots on four rings of 10 around a pale grey disc, joined by straight lines. Right, in magenta: 126 smaller dots on seven rings of 18 over a grey disc, joined by a dense web of thin lines.
Illustration: arrangements with the largest known number of non-overlapping balls touching a central ball of the same size. Left: 40 balls in five dimensions, from the root system D₅. Right: 126 in seven, from E₇. Each arrangement is projected onto a plane chosen so that the picture has 10-fold (left) or 18-fold (right) rotational symmetry. The grey disc is the central ball, each dot is the centre of a ball that touches it, and lines join the centres of balls that touch each other. The paper proves that in five dimensions at most 43 non-overlapping balls can touch a central ball of the same size, and in seven at most 132.