AI 中文总结
研究同步单簇自动机的切尔尼猜想，通过特定方法证明对于相关条件下存在特定单词，给出复位单词长度上界，构造示例说明界的紧性，证明借助与OpenAI Codex交互并经作者验证。
AI 中文摘要
我们证明了同步单簇自动机的切尔尼猜想。具体而言，设具有状态集\(Q\)（\(|Q| = n\)）的同步自动机有一个字母\(a\)，其功能图有唯一长度为\(m\)的循环\(C\)，设\(\ell\)是使得\(a^\ell\)将\(Q\)映射到\(C\)上的最小非负整数，且\(\ell\geq1\)。对于\(C\)的每个非空真子集\(S\)，我们证明存在长度至多为\(n\)的单词\(w\)，使得\(wa^\ell\)将\(C\)中超过\(|S|\)个状态映射到\(S\)中。这证明了Kisielewicz、Kowalski和Szykuła关于单簇自动机相对扩展单词猜想的正水平部分。得到的复位单词长度至多为\((m - 1)(n - 1)+m\ell\leq(n - 1)^2\)。对于每个\(n\geq4\)，我们构造了一个\(m = 2\)，\(\ell = n - 2\)且复位阈值为\(3n - 5\)的强连通二元示例，所以依赖参数的界\((m - 1)(n - 1)+m\ell\)是紧的。上界证明使用有限维线性代数；紧性下界是组合性的。该证明是通过与OpenAI Codex（GPT - 5.6 Sol，超模式）交互获得并由作者验证的。
英文摘要
We prove the Černý conjecture for synchronizing one-cluster automata. More precisely, let a synchronizing automaton with state set $Q$, $|Q|=n$, have a letter $a$ whose functional digraph has a unique cycle $C$ of length $m$, and let $\ell$ be the least nonnegative integer for which $a^\ell$ maps $Q$ onto $C$. Assume $\ell\ge1$. For every nonempty proper subset $S\subset C$, we prove that there is a word $w$ of length at most $n$ such that $wa^\ell$ maps more than $|S|$ states of $C$ into $S$. This proves the positive-level part of a conjecture of Kisielewicz, Kowalski, and Szykuła concerning relative extending words for one-cluster automata. The resulting reset word has length at most \[(m-1)(n-1)+m\ell\le(n-1)^2. \] For every $n\ge4$, we construct a strongly connected binary example with $m=2$, $\ell=n-2$, and reset threshold $3n-5$, so the parameter-dependent bound $(m-1)(n-1)+m\ell$ is sharp. The upper-bound proof uses finite-dimensional linear algebra; the sharpness lower bounds are combinatorial. The proof was obtained through interaction with OpenAI Codex (GPT-5.6 Sol, ultra mode) and verified by the author.
