均匀权重线性元胞自动机中二元轨道的单位系数充分性
Sufficiency of Unit Coefficients for Binary Orbits in Uniformly Weighted Linear Cellular Automata
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中文总结 AI 辅助
本文针对环$\mathbb{Z}/n\mathbb{Z}$上的均匀权重线性元胞自动机，引入二元投影算子，证明了与$n$有公共素因子的系数$c$的二元轨道可归约为单位系数轨道，明确了二元轨道类型数量，简化了其动力学分类框架。
中文摘要 AI 辅助
本文研究了环$\mathbb{Z}/n\mathbb{Z}$上均匀权重线性元胞自动机（LCA-UW）生成的时空模式分类。这类系统由状态规模$n$和转移系数$c$共同决定，二者的综合影响产生了大量难以通过穷举观察梳理的模式。我们引入二元投影算子$\mathcal{B}$，聚焦于这些自动机的基础结构演化（无限二元轨道）。核心结果证明了一个基础归约原理：对于任何与$n$有公共素因子的系数$c$，其生成的无限二元轨道最终会与具有归约后状态规模和单位系数$c=1$的LCA-UW的轨道重合。我们证明，对于固定的$n$，存在恰好$2^m - 1$种不同类型的二元轨道，其中$m$是$n$的不同素因子的数量。该定理将二维参数空间$(n,c)$归约为对$n$的一维搜索，为LCA-UW动力学的拓扑和分形分类提供了简化框架。
英文摘要
This paper investigates the classification of spatio-temporal patterns generated by linear cellular automata with uniform weights (LCA-UW) over the ring ${\mathbb Z} / n {\mathbb Z}$. While these systems are governed by the state size $n$ and a transition coefficient $c$, their combined influence produces a vast array of patterns that are difficult to organize through exhaustive observation. We introduce a binary projection operator $\mathcal{B}$ to focus on the fundamental structural evolution (infinite binary orbits) of these automata. Our main result demonstrates a fundamental reduction principle. For any coefficient $c$ that shares prime factors with $n$, the generated infinite binary orbit eventually coincides with the orbit of an LCA-UW with some reduced state size and a unit coefficient $c=1$. We prove that for a fixed $n$, there exist exactly $2^m - 1$ distinct types of binary orbits, where $m$ is the number of distinct prime factors of $n$. This theorem effectively collapses the two-dimensional parameter space $(n, c)$ into a one-dimensional search over $n$, providing a streamlined framework for the topological and fractal classification of LCA-UW dynamics.
