Finding Kissing Numbers with Game-theoretic Reinforcement Learning

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 defining challenge in discrete geometry. As the local analogue of Hilbert's 18th problem, it has profound implications across geometry, number theory and information theory. Although lattices and codes have achieved significant progress, the field is confined to isolated extremal configurations, leaving underlying geometric principles obscured. Here we shift the object to the broader extremal configuration space, thereby opening a new path for the Kissing Number Problem. Accordingly, we recast this problem as a cooperative matrix-completion game, and train a reinforcement learning system, PackingStar, to solve it. One player fills cosine entries while the other corrects suboptimal ones, making explosive geometric complexity tractable. Working within extremal configuration spaces, PackingStar discovers new interpretable geometric structures that improve 15 strong bounds held for decades in kissing numbers and their generalizations, several of them provably optimal under natural inner products. These findings reveal the first explicit spherical-code realization of the Fischer group Fi22, extend the classical Euclidean representation of subgroup structure, and directly inspire subsequent breakthroughs by mathematicians. Overall, the work provides an early example of AI-driven progress on a Hilbert-calibre problem, showing how reinforcement learning advances mathematical discovery by unlocking more expressive objects.