发表机构
Department of Mathematics, College of Engineering, American University of the Middle East（中东美国大学工程学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明有限图表达式在等式推导中可保持宽度有界，通过受保护核心和路由窗口构造编码典范代表，并给出宽度与树宽的关系及团值不变量。
AI 中文摘要
等式表示的完备性保证了存在一条相等路径，但不需要控制沿该路径所使用的资源。对于有限图表达式，我们通过中间原始项的最大输入-输出接口来衡量推导空间。对于每个有限的双重排序边字母表$\Sigma$，我们证明模式宽度至多为$k$的相等闭表达式可以通过一个推导连接，其中每一步在任一方向应用一个等式，且每个中间宽度至多为一个可计算的$B_\Sigma(k)$，与图的大小无关。该推导仅使用结构magmoid定律和十五个有限图胚方案。该构造通过受保护核心和有限路由窗口将每个表达式编译为宽度至多$4k+4$的编码典范代表$\operatorname{NF}_k$。我们将此性质称为有界等式相干性。我们还证明当底层简单图至少具有两条边时，$\operatorname{bw}(F)\le\operatorname{patw}(F)\le4(\operatorname{tw}(F)+1)$，将团上的分层线性表达式与分支表达式区分开来，并为前五个非平凡团值提供机器可检查的见证和有限不变量。
英文摘要
Completeness of an equational presentation guarantees an equality path but need not control the resources used along it. For finite graph expressions we measure derivational space by the largest input-output interface of an intermediate raw term. For every finite doubly ranked edge alphabet $Σ$, we prove that equal closed expressions of pattern width at most $k$ are joined by a derivation in which every step applies an equation in either direction and every intermediate width is at most a computable $B_Σ(k)$, independently of graph size. The derivation uses only the structural magmoid laws and the fifteen finite-graphoid schemes. The construction compiles each expression through protected cores and finite routing windows to an encoding-canonical representative $\operatorname{NF}_k$ of width at most $4k+4$. We call this property bounded equational coherence. We also prove $\operatorname{bw}(F)\le\operatorname{patw}(F)\le4(\operatorname{tw}(F)+1)$ whenever the underlying simple graph has at least two edges, separate layered linear from branching expressions on cliques, and provide machine-checkable witnesses and finite invariants for the first five nontrivial clique values.
Comments71 pages. Includes complete supplementary proofs. Reproducibility package: https://doi.org/10.5281/zenodo.22297410
