Černý Conjecture for One-Cluster Automata

Precise statement

Does every synchronizing one-cluster automaton on $n$ states admit a reset word of length at most $(n-1)^2$? The new bound $(m-1)(n-1) + m\ell \le (n-1)^2$ settles the one-cluster case of the Černý conjecture.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

OpenAI Codex (GPT-5.6 Sol Ultra)

The annular spectral descent argument was obtained in interaction with OpenAI Codex running GPT-5.6 Sol Ultra and verified by the author; the paper also proves the positive-level relative-extending-word conjecture of Kisielewicz, Kowalski and Szykuła.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

Author-verified arXiv preprint. Not yet peer-reviewed.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-07-25

   arXiv:2607.19675 - The Černý conjecture for one-cluster automata via annular spectral descent

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.