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 that can be fully parallelized at large scale, 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, discovers over 6000 new structures in 14 and other dimensions, and establishes new records for generalized kissing configurations under various angular constraints. 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.

翻译：自艾萨克·牛顿于1694年首次研究球体最大接触数问题以来，确定围绕中心球体可排列的非重叠球体最大数量始终是一项基础性挑战。该问题是希尔伯特第18问题（球体堆积问题）的局部对应，连接了几何学、数论和信息论。尽管通过格点与编码理论已取得重要进展，但高维几何的不规则性以及超过8维后呈指数增长的组合复杂度（其复杂度已超过围棋），限制了现有方法的可扩展性。本文将该问题建模为可大规模并行化的双玩家矩阵补全博弈，并训练博弈论强化学习系统PackingStar以高效探索高维空间。矩阵元素表示球心向量间的两两余弦值；一方填充元素，另一方修正次优元素，共同最大化矩阵尺寸（对应球体最大接触数）。这种协作机制显著提升了样本质量，使极端庞大的空间变得可处理。PackingStar复现了已有构型，并在25至31维中超越了所有已知人类记录——其中25维构型几何对应Leech格点，暗示其可能的最优性。该系统首次在13维突破了1971年以来有理结构的局限，在14维及其他维度发现了6000余种新结构，并在多种角度约束下建立了广义接触构型的新记录。这些成果证明了人工智能探索超越人类直觉的高维空间的能力，为球体最大接触数问题及更广泛的几何问题开辟了新路径。