发表机构
The Hong Kong University of Science and Technology（香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对近方形格点区域，提出稀疏 Moore 局部实现方法，将任意 N 次不可约二元多项式作为特征多项式，依赖数至多 3N-1，合成复杂度 O(N^3)，并给出解析构造及紧下界。
AI 中文摘要
对于每个 $N\ge36$，我们在恰好包含 $N$ 个细胞的近方形区域上，将任意给定的 $N$ 次首一不可约二元多项式实现为线性混合元胞自动机的特征多项式。其转移矩阵是 Moore 局部的，具有零边界，且至多有 $3N-1$ 个有向非自依赖。依赖图保留了一条双向哈密顿路径；其入度、出度以及底层无向度均至多为六。底层图包含一个边长与 $\sqrt N$ 成正比的显式方格网格子图。确定性合成需要 $O(N^3)$ 比特运算。该构造结合了局部相似变换、输运势以及跨连续行间隙的联合路由。对有限证书的精确验证，随后通过归纳法，证明了对于每个容许宽度所需的路由不等式。我们还给出了一种完全解析的构造，其依赖数少于 $7N/2$，并证明了保留路径和完整矩形网格子图的下界为 $5N/2-O(\sqrt N)$。当特征多项式不受限制时，该下界是紧的。
英文摘要
For every $N\ge36$, we realise any prescribed monic irreducible binary polynomial of degree $N$ as the characteristic polynomial of a linear hybrid cellular automaton on a near-square region of exactly $N$ cells. The transition matrix is Moore-local with a null boundary and has at most $3N-1$ directed nonself dependencies. The dependency graph retains a bidirectional Hamilton path; its indegree, outdegree and underlying undirected degree are at most six. The underlying graph contains an explicit square grid minor of side proportional to $\sqrt N$. The deterministic synthesis takes $O(N^3)$ bit operations. The construction combines a local similarity transformation with a transport potential and joint routing across consecutive row gaps. Exact verification of finite certificates, followed by induction, proves the required routing inequalities for every admissible width. We also give an entirely analytic construction with fewer than $7N/2$ dependencies and prove a lower bound of $5N/2-O(\sqrt N)$ for the retained path and full rectangular grid minor. This lower bound is sharp when the characteristic polynomial is unrestricted.
Comments35 pages, 3 figures
