H5 高维循环群两步覆盖系统:形式化理论 High-Dimensional Cyclic Group Two-Step Covering Systems: A Formal Theory
Year: 2026
Type: preprint
Abstract: This paper studies two-step covering systems over high-dimensional cyclic groups. Let the state space be \[ \Omega(n,d)=\mathbb Z_n^d, \] with total number of states \(N=n^d\). The one-step displacement set is \[ S_1=\{0,1,2\}^d, \] and the two-step displacement set is \[ S_2=S_1+S_1. \] Full coverage requires \(S_2=\mathbb Z_n^d\). We introduce non-degeneracy conditions N1 and N2, together with an additional condition N3 that is necessary only in the one-dimensional case, and a closed-loop diversity condition C3: the system admits at least two topologically non-equivalent closed loops. The main result is: among all dimensions \(d\ge1\) and all numbers \(n\ge2\) of points per dimension, the feasible systems satisfying full coverage, non-degeneracy, and closed-loop diversity C3 are exactly \[ \{(n=5,d\ge1)\}\cup\{(n=3,d\ge2)\}. \] The global minimum of the total number of states \(N=n^d\) is \[ \boxed{N_{\min}=5}, \] and the unique realization is \[ \boxed{(d=1,\ n=5)}. \] Thus one-dimensional five-state is the unique realization attaining the global minimum number of states. Higher-dimensional extensions are possible, such as two-dimensional three-state and three-dimensional three-state systems, but their total numbers of states are strictly larger. We further explain that this theory is a combinatorial equivalent version of Theorem 0. Theorem 0 gives, at the spectral optimization level for \(n=5\), the optimal parameters \[ c_0=-\frac{3}{11},\qquad c_1=c_2=\frac{2}{11},\qquad \rho=\frac{5}{11}. \] The present theory proves at the combinatorial covering level that \(n=5\) is the unique dimension attaining the global minimum number of states. The two levels jointly lock \(n=5\), neither replacing nor implying the other.
Author: 夏祯
Language: csl
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214912948