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  "problem_id": "vibemathed:cerny-one-cluster",
  "title": "Černý Conjecture for One-Cluster Automata",
  "canonical_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.",
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      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-25T00:00:00.000Z",
      "title": "arXiv:2607.19675 - The Černý conjecture for one-cluster automata via annular spectral descent",
      "summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
      "ai_contribution": "ai_co_developed",
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      "attempted_at": "2026-07-25T00:00:00.000Z",
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      "prompt_public": null,
      "outcome": "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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