Finding Kissing Numbers with Game-theoretic Reinforcement Learning

TL;DR

We address the Kissing Number Problem $K(n)$ in high dimensions, where traditional approaches struggle due to combinatorial explosion. PackingStar reformulates the search as a two-player cooperative matrix-completion game on Gram matrices, with a Matrix Filler and a Matrix Corrector guided by cosine-constraint sets; computations operate entirely in the Gram-matrix domain to enable large-scale sampling. The approach yields state-of-the-art lower bounds for dimensions $n=25$–$31$, discovers a rational configuration in $13$D with $K_r(13)=1146$, and uncovers thousands of new configurations in $14$D and beyond, revealing geometric patterns linked to Leech lattice substructures. This AI-driven exploration demonstrates a new paradigm for high-dimensional geometry problems, providing novel objects for mathematical study and potential pathways to optimal constructions.

Abstract

Since Isaac Newton first studied the Kissing Number Problem in 1694, determining the maximal number of non-overlapping spheres around a central sphere has remained a fundamental challenge. This problem represents the local analogue of Hilbert's 18th problem on sphere packing, bridging geometry, number theory, and information theory. Although significant progress has been made through lattices and codes, the irregularities of high-dimensional geometry and exponentially growing combinatorial complexity beyond 8 dimensions, which exceeds the complexity of Go game, limit the scalability of existing methods. Here we model this problem as a two-player matrix completion game and train the game-theoretic reinforcement learning system, PackingStar, to efficiently explore high-dimensional spaces. The matrix entries represent pairwise cosines of sphere center vectors; one player fills entries while another corrects suboptimal ones, jointly maximizing the matrix size, corresponding to the kissing number. This cooperative dynamics substantially improves sample quality, making the extremely large spaces tractable. PackingStar reproduces previous configurations and surpasses all human-known records from dimensions 25 to 31, with the configuration in 25 dimensions geometrically corresponding to the Leech lattice and suggesting possible optimality. It achieves the first breakthrough beyond rational structures from 1971 in 13 dimensions and discovers over 6000 new structures in 14 and other dimensions. These results demonstrate AI's power to explore high-dimensional spaces beyond human intuition and open new pathways for the Kissing Number Problem and broader geometry problems.

Figures (2)

• Figure 1: AI breakthroughs in the Kissing Number Problem. Grey markers show the best previously known lower bounds. The blue triangle marks the 11D improvement achieved by AlphaEvolve (2025) novikov2025alphaevolve. In 13D, PackingStar (red star) uncovers a new rational configuration with kissing number $K_r(13)=1146$, distinct from the current best non-rational lower bound $K(13)=1154$. For $25 \le n \le 31$, the red stars indicate new lower bounds $K_{\mathrm{new}}(n)$ discovered by PackingStar, with the inset reporting the corresponding increments $\Delta K(n)=K_{\mathrm{new}}(n)-K_{\mathrm{prev}}(n)$ compared to the previous best $K_{\mathrm{prev}}(n)$.
• Figure 2: Overview of PackingStar system. The workflow consists of three steps for $n$ dimensions: Step 1. New spheres tangent to $n-1$ existing spheres are iteratively solved from equations and added to the search space for simulation. Cosine frequencies are recorded until they converge to a discrete set to capture the geometric features. Step 2. The cosine set from Step 1 populates an initial matrix (red region) for the matrix completion game. Initialization can leverage geometric priors or start from scratch. Step 3. The first agent, Filler, fills the matrix with candidates, while the second agent, Corrector, removes previously added entries that are not good enough. After each round, the matrix is decomposed and reassembled for the next iteration based on geometric feature analysis.