The \v{C}ern\'y Conjecture for One-Cluster Automata via Annular Spectral Descent

We prove the \v{C}ern'y conjecture for synchronizing one-cluster automata. More precisely, let a synchronizing automaton with state set $Q$, $\mathrm{\mid }Q\mathrm{\mid }=n$, have a letter $a$ whose functional digraph has a unique cycle $C$ of length $m$, and let $\mathrm{\ell }$ be the least nonnegative integer for which $a^{\mathrm{\ell }}$ maps $Q$ onto $C$. Assume $\mathrm{\ell }\ge 1$. For every nonempty proper subset $S\subset C$, we prove that there is a word $w$ of length at most $n$ such that $w{a}^{\mathrm{\ell }}$ maps more than $\mathrm{\mid }S\mathrm{\mid }$ states of $C$ into $S$. This proves the positive-level part of a conjecture of Kisielewicz, Kowalski, and Szyku\l a concerning relative extending words for one-cluster automata. The resulting reset word has length at most $$(m-1)(n-1)+m\mathrm{\ell }\le (n-1{)}^2\mathrm{.}$$ For every $n\ge 4$, we construct a strongly connected binary example with $m=2$, $\mathrm{\ell }=n-2$, and reset threshold $3n-5$, so the parameter-dependent bound $(m-1)(n-1)+m\mathrm{\ell }$ 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.