Černý conjecture for one-cluster automata

The problem asks whether every synchronizing one-cluster automaton on $n$ states has a reset word of length at most $(n-1)^2$. The new paper claims an affirmative answer for the entire class, extending the previously settled prime-cycle cases.

Known results

• Pin proved the circular case when the cycle length is prime.
• Dubuc proved the conjectured bound for all circular automata.
• Steinberg proved the one-cluster case with prime cycle length; the earlier bound is at most $(n-1)^2$.
• Before 2026, arbitrary cycle lengths remained open.

July 2026 claimed proof

The paper “The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent” claims the sharper bound $\operatorname{rt}(A)\leq(m-1)(n-1)+m\ell\leq(n-1)^2$, where $m$ is the cycle length and $\ell$ the level. It also claims a stronger relative extending-word result and reports examples attaining the refined bound; the proof remains unverified.

Current status (as of July 2026): The full one-cluster conjecture has a published-on-arXiv affirmative claim, while independent verification of the proof is still outstanding.