Sphere packing, Hilbert's eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains unresolved: beyond a few special dimensions, neither optimal packings nor tight upper bounds are known. Even a major breakthrough in dimension $n=8$, later recognised with a Fields Medal, underscores its difficulty. A leading technique for upper bounds, the three-point method, reduces the problem to solving large, high-precision semidefinite programs (SDPs). Because each candidate SDP may take days to evaluate, standard data-intensive AI approaches are infeasible. We address this challenge by formulating SDP construction as a sequential decision process, the SDP game, in which a policy assembles SDP formulations from a set of admissible components. Using a sample-efficient model-based framework that combines Bayesian optimisation with Monte Carlo Tree Search, we obtain new state-of-the-art upper bounds in dimensions $4-16$, showing that model-based search can advance computational progress in longstanding geometric problems. Together, these results demonstrate that sample-efficient, model-based search can make tangible progress on mathematically rigid, evaluation limited problems, pointing towards a complementary direction for AI-assisted discovery beyond large-scale LLM-driven exploration.

翻译：球体堆积问题，即希尔伯特第十八问题，旨在寻找n维欧几里得空间中全等球体的最密堆积方式。尽管该问题与密码学、晶体学和医学成像等领域密切相关，但其仍未得到完全解决：除少数特殊维度外，既未发现最优堆积方式，也未获得紧致的上界。即使在维度$n=8$上取得的一项重大突破（后获菲尔兹奖认可）也凸显了该问题的难度。用于上界求解的主流技术——三点法——将问题转化为求解大规模高精度半定规划（SDP）。由于每个候选SDP可能需要数天时间评估，传统数据密集型AI方法难以适用。我们通过将SDP构建形式化为序列决策过程（即SDP博弈）来应对这一挑战，其中策略从一组可容许的组件中组装SDP公式。采用结合贝叶斯优化与蒙特卡洛树搜索的样本高效模型化框架，我们在维度$4-16$中获得了新的最先进上界，表明基于模型的搜索能够推动长期几何问题的计算进展。这些结果共同证明，样本高效的模型化搜索能在数学结构严谨、评估受限的问题上取得实质性进展，为超越大规模LLM驱动探索的AI辅助发现指明了互补方向。