AI 中文总结
该研究围绕Cayley图观测问题中信息与局部性的张力，定义三类分离参数，通过碳环面、Heawood环面及有限单群示例验证参数差异与构造方法。
AI 中文摘要
de Bruijn序列是Cayley图观测问题的循环原型：平移窗口$gY$上的有序标记词何时能确定顶点$g$？我们区分了三个参数。无约束数$\operatorname{sep}_q(G)$最小化任意分离模式；连通数$\operatorname{csep}_q(G,S)$要求包含$Y_S=\{1\}\cup S$的连通Cayley窗口；单步数$χ_1(G,S)$固定$Y_S$并最小化字母表大小。因此$\operatorname{sep}_q$是群层面的基线，$\operatorname{csep}_q$衡量局部性的代价，$χ_1$检验最小的指定局部窗口。\n 研究的核心主题是信息与局部性之间的张力。碳环面用于检验$\operatorname{sep}_q$与$\operatorname{csep}_q$之间的差距：对于广义二面体群$\mathbb{F}_{\ell^d}^{\times}\rtimes C_2$，我们证明了对于奇素数幂$\ell$，精确基线$\operatorname{sep}_\ell=d+1$，并构造了连通的锯齿形窗口；而阶为14的Heawood环面满足$\operatorname{sep}_4=2$且$\operatorname{csep}_4=4$。球面$A_5$示例和有限单群比较用于检验固定单步窗口：显式对称三次生成元组给出$χ_1(A_5,S)=3$和$χ_1(\operatorname{PSL}_2(\mathbb{F}_7),S)=4$，两者均达到计数界，并配有结构化的矩阵系数证明。循环陪集填充、有限域坐标和受限矩阵系数仅作为这两个示例所需的构造工具使用。
英文摘要
A de Bruijn sequence is the cyclic prototype of a Cayley-graph observation problem: when does the ordered label word on a translated window $gY$ determine the vertex $g$? We distinguish three parameters. The unrestricted number $\operatorname{sep}_q(G)$ minimizes an arbitrary separating pattern; the connected number $\operatorname{csep}_q(G,S)$ requires a connected Cayley window containing $Y_S=\{1\}\cup S$; and the one-step number $χ_1(G,S)$ fixes $Y_S$ and minimizes the alphabet. Thus $\operatorname{sep}_q$ is a group-level baseline, $\operatorname{csep}_q$ measures the cost of locality, and $χ_1$ tests the smallest prescribed local window. The organizing theme is the tension between information and locality. Carbon tori test the gap between $\operatorname{sep}_q$ and $\operatorname{csep}_q$: for generalized dihedral groups $\mathbb{F}_{\ell^d}^{\times}\rtimes C_2$ we prove, for odd prime powers $\ell$, the sharp baseline $\operatorname{sep}_\ell=d+1$ and construct connected zig-zag windows, while the order-$14$ Heawood torus satisfies $\operatorname{sep}_4=2$ and $\operatorname{csep}_4=4$. The spherical $A_5$ example and a finite simple-group comparison test the fixed one-step window: explicit symmetric cubic generating tuples give $χ_1(A_5,S)=3$ and $χ_1(\operatorname{PSL}_2(\mathbb{F}_7),S)=4$, both at the counting bound, with structured matrix-coefficient certificates. Cyclic-coset packings, finite-field coordinates, and restricted matrix coefficients are used only as the construction tools these two examples require.
Comments15 pages, 1 figure. Computational certificates are available in the companion GitHub repository linked in the paper
