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A Fourier Consequence for Lattice Kissing Numbers and Average Contacts

A mix of human-written and AI-generated textHuman understanding: all partsmath.MG — Metric Geometrymath.CO — Combinatoricsmath.NT — Number Theory

Contributed by Scott Kominers ↗

SubmitterScott Kominers

Version 1 / Oct 02, 2026 / CC BY-SA 4.0

Abstract

We deduce a new asymptotic upper bound Klat(n)≤(2e/π+o(1))nK_{\mathrm{lat}}(n)\leq\left(\sqrt{2e/\pi}+o(1)\right)^n on the maximal lattice kissing number Klat(n)K_{\mathrm{lat}}(n) in dimension nn, using the explicit auxiliary functions constructed in OpenAI's "Ten Advances" preprint. The same bound holds for the average contact degree of a finite packing of congruent balls. The bound's base-22 exponential rate rounds to 0.39560.3956, matching the value extrapolated empirically by Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini in 2020.

Provenance statement

I used LLMs to assist with analysis and synthesis in the preparation of this article, particularly GPT-6 Astra and Claude Fable 5.1 (accessed in part via Poe with the support of Quora, where I am an advisor). In particular, the bound I present was developed in dialogue with GPT-6 Astra. In addition, I greatly appreciate helpful comments from Henry Cohn and Thijs Laarhoven.

Tools used

OpenAI
GPTVersion 6 Astra
Anthropic
Claude FableVersion 5.1
Quora
Poe

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Version history

  1. v1Initial depositCurrentOct 02, 2026