Sufficiency of Unit Coefficients for Binary Orbits in Uniformly Weighted Linear Cellular Automata

Abstract

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.