Title: Leveraging Game-Theoretic Reinforcement Learning to Solve the Kissing Number Problem

Since Isaac Newton first examined the Kissing Number Problem in 1694, identifying the maximum count of non-overlapping spheres that can surround a central sphere has stood as a pivotal challenge in discrete geometry. Recognized as the local counterpart to Hilbert’s 18th problem, this issue carries significant weight across number theory, information theory, and geometry. While lattices and coding theory have yielded notable advancements, research has historically been restricted to isolated extremal configurations, which has kept the fundamental geometric principles largely hidden.

This study proposes a shift toward the broader extremal configuration space, creating a novel avenue for tackling the Kissing Number Problem. The researchers reformulate the challenge as a cooperative matrix-completion game and employ a reinforcement learning framework, dubbed PackingStar, to address it. In this game-theoretic setup, one agent is tasked with filling in cosine entries, while another works to rectify suboptimal values, thereby rendering the explosive geometric complexity manageable.

By exploring extremal configuration spaces, PackingStar uncovers new, interpretable geometric structures. These discoveries successfully improve upon 15 long-standing bounds in kissing numbers and their generalizations, many of which have held firm for decades. Several of these updated bounds are provably optimal under standard inner products. The results mark the first explicit spherical-code realization of the Fischer group Fi22 and expand the classical Euclidean representation of subgroup structures. Furthermore, these insights have directly motivated subsequent breakthroughs by mathematicians. Ultimately, this work serves as a pioneering example of AI-driven progress on a problem of Hilbert-calibre, demonstrating how reinforcement learning can propel mathematical discovery by accessing more expressive mathematical objects.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC