用8个超平面切10维超立方体，改进了高维几何切割的上界。
Improved Upper Bounds for Slicing the Hypercube
- 构造8个超平面完成10维超立方体切割，借助智能搜索工具生成
- 证明n维超立方体最少需超平面数不超过⌈4n/5⌉（奇倍数5时加1）
- 首次结合大模型与自动调参实现数学构造发现，适合几何与算法研究者
一组超平面集合 $/mathcal{H}$ 若能与 $n$ 维超立方体 $Q_n$ 的每条边在内部相交，则称其为切割集。令 $S(n)$ 表示切割 $Q_n$ 所需的最小超平面数。本文证明 $S(n) \≤ \lceil \frac{4n}{5} \rceil$，当 $n$ 为奇倍数5时，有 $S(n) \≤ \frac{4n}{5} +1$。该结果改进了1971年Paterson给出的旧上界 $\lceil \frac{5n}{6} \rceil$。同时，本文还获得使用少于 $n$ 个超平面时能切割的最大边数的新下界。关键构造基于一个名为 CPro1 的新型自动工具——它融合推理型大模型与自动化超参数调优，用于生成数学构造的搜索算法，并成功在 $Q_{10}$ 上实现8个超平面的切割。
原文摘要 · Abstract (English)
A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its interior. Let $S(n)$ be the minimum number of hyperplanes needed to slice $Q_n$. We prove that $S(n) \leq \lceil \frac{4n}{5} \rceil$, except when $n$ is an odd multiple of $5$, in which case $S(n) \leq \frac{4n}{5} +1$. This improves upon the previously known upper bound of $S(n) \leq \lceil\frac{5n}{6} \rceil$ due to Paterson reported in 1971. We also obtain new lower bounds on the maximum number of edges in $Q_n$ that can be sliced using $k<n$ hyperplanes. We prove the improved upper bound on $S(n)$ by constructing $8$ hyperplanes slicing $Q_{10}$ aided by the recently introduced CPro1: an automatic tool that uses reasoning LLMs coupled with automated hyperparameter tuning to create search algorithms for the discovery of mathematical constructions.
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