本文研究定义在环 $\mathbb{Z} / n \mathbb{Z}$ 上的均匀加权线性元胞自动机（LCA-UW）所生成的时空模式的分类问题。此类系统由状态规模 $n$ 和转移系数 $c$ 共同决定，二者耦合产生大量难以通过穷举观察加以系统归类的模式。我们引入一个二进制投影算子 $\mathcal{B}$，以聚焦于这些自动机的基本结构演化——即无限二进制轨道。我们的主要结果揭示了一条基本约化原理：对任意与 $n$ 具有公共素因子的系数 $c$，其所生成的无限二进制轨道最终将与某个具有约化后状态规模及单位系数 $c=1$ 的 LCA-UW 所生成的轨道重合。我们证明，对固定 $n$，恰好存在 $2^m - 1$ 种互异的二进制轨道类型，其中 $m$ 为 $n$ 的不同素因子个数。该定理实质上将二维参数空间 $(n, c)$ 约化为仅关于 $n$ 的一维搜索，从而为 LCA-UW 动力学的拓扑与分形分类提供了简明的框架。

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.