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.

This is the level-zero case m = n, established in the older circular-automaton literature recalled by the source.

Proved

A letter acts as a cycle on all n states, so ℓ = 0.

Source dated Jul 22, 2026 · Observed Oct 14, 2026

In this problem · publication v4

Supporting result · Proved

Relationship not independently reviewed. This concerns its use in this problem.

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.

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

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