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Černý Conjecture for One-Cluster Automata

Theoretical computer science · posed by 2016 · solved

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Statement

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

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Yinfeng Zhu, using OpenAI Codex (GPT-5.6 Sol Ultra)

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    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.

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