Kissing numbers

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open.

Problem
Problem 8 in the AlphaEvolve repository: Kissing numbers
Credit
AlphaEvolve (Google DeepMind) · LLM-guided evolutionary code search, with Georgiev, Gómez-Serrano, Tao and Wagner
Checked
Construction can be checked directlyPreprint (arXiv, 2025)
Timeline
Paper submitted 2025-11-03
Sources
Problem page (AlphaEvolve repository)Paper (arXiv:2511.02864)
PARTIALImproved bound · new construction

The problem

For a dimension n≥1n \geq 1, define the kissing number C(n)C(n) to be the maximum number of non-overlapping unit spheres that can be arranged to simultaneously touch a central unit sphere in nn-dimensional space. Establish upper and lower bounds on C(n)C(n) that are as strong as possible.

Statement quoted from problem 8 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open. Status in the repository: New result. See Section 6.4 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

Notes from the repository

Henry Cohn maintains a list of record upper and lower bounds for kissing numbers.

Some new kissing number constructions in medium dimensions were recently constructed (with AI assistance) in this recent paper.

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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