发表机构
Hebei University of Technology（河北工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过配对自动机证明，确定了修正后图3族的精确压缩阈值，并给出复位构造上界，覆盖所有参数。
AI 中文摘要
我们给出了一个自包含的配对自动机证明，针对最近关于同步自动机开放问题综述中修正后的图3族的精确“与另一压缩”阈值。对于每个 $p \geq 3$，该自动机有 $n = 3p$ 个状态，且 $\mu(q_0) = 4p = 4n/3$。单词 $(ba)^p(ab)^p$ 达到该值。两个入口势能和一个排除区域给出了下界，其中 $p = 3$ 的端点例外被显式处理。一个独立的复位构造证明了对于每个参数，$\operatorname{rt}(A_p) \leq 3p^2 + 4p - 1$。可复现的计算验证了转移和入口恒等式；所有参数的结果由显式证明得出。
英文摘要
We give a self-contained pair-automaton proof of the exact compress-with-another threshold of the corrected Figure 3 family from a recent survey of open problems in synchronizing automata. For every $p \geq 3$, the automaton has $n = 3p$ states and $μ(q_0) = 4p = 4n/3$. The word $(ba)^p(ab)^p$ attains this value. Two entrance potentials and an excluded region yield the lower bound, with the endpoint exception at $p = 3$ treated explicitly. A separate reset construction proves $\operatorname{rt}(A_p) \leq 3p^2 + 4p - 1$ for every parameter. Reproducible computations verify the transition and entrance identities; the all-parameter results follow from the explicit proofs.
Comments8 pages. Reproducible computational verification included in the source
