Proved
Older circular case recalled ↗
A letter acts as a cycle on all n states, so ℓ = 0.
Source dated Jul 22, 2026 · Observed Oct 14, 2026
Theorem · version 1
The older level-zero circular case
Statement
A complete synchronizing automaton with a letter that cycles through all n states has a reset word of length at most (n−1)².
Assumptions
- This is the level-zero case m = n, established in the older circular-automaton literature recalled by the source.
Evidence
Source reports
Connections
In this problem · publication v2 · Selected context
Černý synchronization conjecture →
Supporting result · Proved
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v4
Černý synchronization conjecture →
Supporting result · Proved
Relationship not independently reviewed. This concerns its use in this problem.
Sources
The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent →
- Introduction and cited Dubuc (1998) → · Original passage ↗
Recorded wording
The level-zero circular case is already covered by older results. It is not the positive-level advance claimed by the 2026 proof.
paraphrase · Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.
Yinfeng Zhu · Preprint · Publication date Jul 22, 2026
Source-reported results and external checks are separate from independent OpenMath assessment.
Revision & attribution
Source revision: v1. Observed Oct 14, 2026.
Zhu reports obtaining the proof through interaction with OpenAI Codex using GPT-5.6 Sol in ultra mode and verifying the argument himself.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Older circular case recalled ↗
A letter acts as a cycle on all n states, so ℓ = 0.
Source dated Jul 22, 2026 · Observed Oct 14, 2026 · Source passage →
Record details
- Claim ID
- claim-cerny-circular-case
- Statement version
- 1
- SHA-256
- a346bef0267ed83ac267365b5726a33c63dda8a4cef9de575dd23548d4baca19