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

A mix of human-written and AI-generated textHuman understanding: all partsmath.MG — Metric Geometrymath.OC — Optimization and Controlmath.CO — Combinatorics

Contributed by Tejasvi Singh Tomar ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

The kissing number τn\tau_n is the largest number of non-overlapping unit balls in Rn\mathbb{R}^n that can touch a common unit ball. The best bounds so far were 40≤τ5≤4440\le\tau_5\le44 and 126≤τ7≤134126\le\tau_7\le134; both upper bounds come from high-precision floating-point computations of the three-point bound by Mittelmann and Vallentin. We prove τ5≤43\tau_5\le 43 and τ7≤132\tau_7\le132. 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)(d_1,d_2,\delta)=(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⁡13)≤14A(4,\arccos\frac13)\le14, A(5,arccos⁡13)≤22A(5,\arccos\frac13)\le22 and A(6,arccos⁡13)≤34A(6,\arccos\frac13)\le34 for spherical codes, improving the previous bounds 1515, 2323 and 3535; with a known code, A(4,arccos⁡13)=14A(4,\arccos\frac13)=14. The largest number of points at angle π/3\pi/3 from a point of a kissing configuration in Rn\mathbb{R}^n equals A(n−1,arccos⁡13)A(n-1,\arccos\frac13), so in R5\mathbb{R}^5 this number is 1414. 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 3030 decimal places.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23992400

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Version history

  1. v1 · 8 pagesSubmitted by Tejasvi Singh TomarPublished on Zenodo: https://doi.org/10.5281/zenodo.23992400CurrentNov 12, 2026