Černý synchronization conjecture

3 selected milestones · Source coverage checked 2026-10-14 · Historical annotation and mathematical review remain separate.

1. 1964 publication

   Foundation

   Černý’s reset-word question

   The source traces the quadratic synchronization conjecture to Černý’s original work.

   Statements & sources

   • The general quadratic reset-word conjecture →

   • The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent ↗Introduction; definitions of synchronization and one-cluster level

   Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.

2. 1998 publication

   Special case proved

   Circular automata satisfy the bound

   The source recalls Dubuc’s theorem for a letter acting as a full cycle.

   Statements & sources

   • The older level-zero circular case →

   • The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent ↗Introduction and cited Dubuc (1998)

   Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.

3. Jul 22, 2026 preprint

   Special case proved

   Positive-level spectral proof is published

   Zhu publishes the restricted theorem, relative-extension argument and sharp binary family.

   Statements & sources

   • The bound for positive-level one-cluster automata →
   • Relative extending words on the unique cycle →
   • A family attaining the parameter-dependent bound →

   • The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent ↗Main theorem and relative-extending-word result
   • The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent ↗Section 10, Definition 10.1, Lemma 10.2 and Corollary 10.8

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

Scope & review: This is a selected source-supported milestone, not an exhaustive history.