这篇论文用线性代数给出了Černý猜想的一个紧上界，并构造了例子证明最优，对自动机理论很关键。
论文证明了同步单簇自动机的Černý猜想：若自动机有n个状态，字母a的功能有向图有唯一长度为m的圈C，且a^ℓ将全部状态映射至C（ℓ≥1），则对任意非空真子集S⊂C，存在长度不超过n的词w使得wa^ℓ将C中多于|S|个状态映射到S。由此得到复位词长度上界(m-1)(n-1)+mℓ≤(n-1)²。对任意n≥4，构造了m=2、ℓ=n-2、复位阈值为3n-5的强连通二进制例子，证明该上界是紧的。证明使用了有限维线性代数，下界构造为组合方法。
The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent
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.