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 is the local analogue of Hilbert's 18th problem, bridging geometry, number theory, and information theory. Although significant progress has been made through lattices and codes, the irregularities of high-dimensional geometry, dimensional structure variability, and combinatorial explosion beyond Go limit the scalability and generality of existing methods. Here we model the problem as a two-player matrix completion game and train the reinforcement learning system, PackingStar, to play the games. The matrix entries represent pairwise cosines of sphere center vectors. One player fills entries while another corrects suboptimal ones to improve exploration quality, cooperatively maximizing the matrix size, corresponding to the kissing number. These matrices are decomposed into representative substructures, providing diverse bases and structural constraints that steer subsequent games and make extremely large spaces tractable. PackingStar surpasses records from dimensions 25 to 31 and sets new lower bounds for generalized kissing numbers under various angular constraints. It achieves the first breakthrough beyond rational structures from 1971 in 13 dimensions and discovers over 6000 new structures in other dimensions. Notably, some configurations challenge long-held antipodal paradigms, revealing algebraic correspondences with finite simple groups as well as geometric relationships across dimensions. Inspired by these patterns, humans devised further improved constructions. These results demonstrate AI's power to explore high-dimensional spaces beyond human intuition via extreme-scale reinforcement learning and open new pathways for the Kissing Number Problem and broader geometry research.

翻译：自艾萨克·牛顿于1694年首次研究接触数问题以来，确定中心球体周围可容纳非重叠球体的最大数量始终是一个基础性难题。该问题是希尔伯特第18问题的局部类比，连接了几何学、数论与信息论。尽管通过格与编码理论已取得重要进展，但高维几何的不规则性、维度结构可变性以及超越围棋的指数级组合爆炸限制了现有方法的可扩展性与普适性。本研究将该问题建模为双玩家矩阵补全博弈，并训练强化学习系统PackingStar进行博弈。矩阵元素表示球心向量间的成对余弦值。一名玩家填充矩阵元素，另一名玩家修正次优元素以提升探索质量，双方协作最大化矩阵规模——即对应接触数。这些矩阵被分解为代表性子结构，提供多样化的基与结构约束，从而引导后续博弈并使极大空间可处理。PackingStar在25至31维度上突破现有记录，并为多种角度约束下的广义接触数设定了新的下界。该系统在13维度上实现了自1971年以来首次超越有理结构的突破，并在其他维度发现了6000余种新结构。值得注意的是，部分构型挑战了长期存在的对径范式，揭示了与有限单群的代数对应关系以及跨维度的几何关联。受这些模式启发，研究者进一步设计了改进构造。这些成果证明了人工智能通过超大规模强化学习探索超越人类直觉的高维空间的能力，为接触数问题及更广泛的几何学研究开辟了新路径。