AI 中文总结
本文将强非周期单瓷砖的构造从三维推广到高维，引入框架标记形式使匹配规则搜索有限，给出三维证书并验证Chair44，同时指出四维问题仍开放。
AI 中文摘要
Chair44（Tsiokos，2026）的发现解决了三维爱因斯坦问题，在$\mathbb{R}^3$中给出了一个强非周期的多面体单瓷砖。本文将该机制——具有角/插座标记的rep-$2^N$椅子$C_N = [0,2]^N \setminus (1,2]^N$——推广到$\mathbb{R}^N$。除了阐述性材料（rep-$2^N$分割和在格点配准与层次强制下的条件性强非周期定理），本文还做出了新的计算贡献。我们引入了一种框架标记形式，其中瓷砖的标记是其完整的方向框架，匹配规则是由替换本身生成的接触语言；这使得在每一维度中搜索匹配规则变得有限。我们给出了一个有限证书（粗化封闭性、紧致性和两壳包围分析），其有效性意味着由标记瓷砖的每个格点配准平铺都是唯一层次的，因此是强非周期的。对于$N=3$，该证书通过：它在$C_3$的24个面板上产生了显式的面匹配规则（135个可允许的面接触三元组），并从第一原理出发，独立于已发表的构造，重现了Chair44的统计量2388→44个可允许接触（其中30个发生），33个单壳簇，15个可扩展，每个都强制一个唯一的超瓷砖。在具有平移中心子代的3D替换的2187个同手性框架分配中，经过认证的那个在共轭意义下是唯一的。对于$N=4$，相同的流程在几个结构化的框架分配族（典型、$D_4$-、$Z_2\times Z_2$-和$Z_4$-对称，以及3D解的提升）上运行；没有一个满足粗化封闭性，我们报告了失败数据。因此，$C_4$的自相似标记仍然是一个明确有限的开放计算问题，我们精确地陈述了它。代码：此https URL
英文摘要
The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in $\mathbb{R}^3$. This note extends the underlying mechanism---the rep-$2^N$ chair $C_N = [0,2]^N \setminus (1,2]^N$ with corner/socket markings---to $\mathbb{R}^N$. Besides expository material (the rep-$2^N$ dissection and a conditional strong-aperiodicity theorem under lattice registration and hierarchical enforcement), the note makes a new computational contribution. We introduce a frame-marking formalism in which the marking of a tile is its full orientation frame and the matching rule is the contact language generated by the substitution itself; this makes the search for matching rules finite in every dimension. We give a finite certificate (coarsening closure, tightness, and a two-shell enclosure analysis) whose validity implies that every lattice-registered tiling by the marked tile is uniquely hierarchical, hence strongly aperiodic. For $N=3$ the certificate passes: it yields explicit facet matching rules on the 24 panels of $C_3$ (135 admissible facet-contact triples) and reproduces, from first principles and independently of published constructions, the Chair44 statistics 2388 $\to$ 44 admissible contacts (30 occurring), 33 one-shell clusters, 15 extendable, each forcing a unique supertile. Among the 2187 homochiral frame assignments of the 3D substitution with a translated central child, the certified one is unique up to conjugation. For $N=4$ the same pipeline is run on several structured families of frame assignments (canonical, $D_4$-, $Z_2\times Z_2$- and $Z_4$-symmetric, and a lift of the 3D solution); none is coarsening-closed, and we report the failure data. A self-similar marking of $C_4$ thus remains an explicitly finite, open computational problem, which we state precisely. Code: https://github.com/dimkadimon/Monotile-RN
Comments13 pages, 3 tables, 4 figures
