{"ID":2833011,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.04829","arxiv_id":"2512.04829","title":"Model-Based and Sample-Efficient AI-Assisted Math Discovery in Sphere Packing","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2512.04829","url_pdf":"https://arxiv.org/pdf/2512.04829v2","authors":"[\"Rasul Tutunov\",\"Alexandre Maraval\",\"Antoine Grosnit\",\"Xihan Li\",\"Jun Wang\",\"Haitham Bou-Ammar\"]","published":"2025-12-04T14:11:52Z","proceeding":"cs.AI","tasks":"[\"cs.AI\"]","methods":"[\"Large Language Model\",\"LoRA\"]","has_code":false}