# Model-Based and Sample-Efficient AI-Assisted Math Discovery in Sphere Packing

arXiv: https://arxiv.org/abs/2512.04829 · HTML: https://mlsanity.com/paper/2512.04829/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 21 references that are in the map.
- First posted on arXiv 4 Dec 2025.

## Builds on (most cited first, 6 of 6)

- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [DeepSeek-Prover-V2: Advancing Formal Mathematical Reasoning via Reinforcement Learning for Subgoal Decomposition](https://mlsanity.com/paper/2504.21801/) (2025)
- [DeepSeek-Prover-V1.5: Harnessing Proof Assistant Feedback for Reinforcement Learning and Monte-Carlo Tree Search](https://mlsanity.com/paper/2408.08152/) (2024)
- [Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2](https://mlsanity.com/paper/2502.03544/) (2025)
- [HEBO Pushing The Limits of Sample-Efficient Hyperparameter Optimisation](https://mlsanity.com/paper/2012.03826/) (2020)
- [A formal proof of the Kepler conjecture](https://mlsanity.com/paper/1501.02155/) (2015)

## Built on by (most cited first, 3 of 3)

- [Decoding as Optimisation on the Probability Simplex: From Top-K to Top-P (Nucleus) to Best-of-K Samplers](https://mlsanity.com/paper/2602.18292/) (2026)
- [Large Discovery Models: Empirically-grounded Model-Based Open-Ended Search](https://mlsanity.com/paper/2608.15669/) (2026)
- [iSDFT: Information-Proximal Self-Distillation for Continual Learning in LLMs](https://mlsanity.com/paper/2609.24646/) (2026)

## Abstract (arXiv)

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.