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Other letters may act differently.","source_entry_ids":["cerny-synchronization:entry:positive-level:v1"],"formal_declarations":[]},{"statement":"Under the positive-level one-cluster hypotheses, for every nonempty proper S ⊂ C there is a word w with |w| ≤ n such that |C ∩ S(wa^ℓ)⁻¹| > |S|.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Under the positive-level one-cluster hypotheses, for every nonempty proper S ⊂ C there is a word w with |w| ≤ n such that "},{"kind":"math","text":"|C ∩ S(wa^ℓ)⁻¹| > |S|","latex":"|C\\cap S(wa^\\ell)^{-1}|>|S|"},{"kind":"text","text":"."}]},"id":"claim-cerny-relative-extension","slug":"cerny-relative-extension","version":1,"kind":"theorem","title":"Relative extending words on the unique cycle","assumptions":["Letters act on the right; (wa^ℓ)⁻¹ denotes a set preimage, not an inverse transition function. The empty and whole-cycle subsets are excluded."],"scope":"Letters act on the right; (wa^ℓ)⁻¹ denotes a set preimage, not an inverse transition function. The empty and whole-cycle subsets are excluded.","source_entry_ids":["cerny-synchronization:entry:positive-level:v1"],"formal_declarations":[]},{"statement":"For each integer n ≥ 4 there is a strongly connected binary synchronizing one-cluster automaton F_n with m = 2, ℓ = n−2 and reset threshold 3n−5 = (m−1)(n−1)+mℓ.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For each integer n ≥ 4 there is a strongly connected binary synchronizing one-cluster automaton F_n with m = 2, ℓ = n−2 and reset threshold "},{"kind":"math","text":"3n−5 = (m−1)(n−1)+mℓ","latex":"3n-5=(m-1)(n-1)+m\\ell"},{"kind":"text","text":"."}]},"id":"claim-cerny-one-cluster-sharp-family","slug":"cerny-one-cluster-sharp-family","version":1,"kind":"theorem","title":"A family attaining the parameter-dependent bound","assumptions":["The example is sharp for the parameter-dependent theorem. It does not establish (n−1)² as the threshold for all n in this family."],"scope":"The example is sharp for the parameter-dependent theorem. It does not establish (n−1)² as the threshold for all n in this family.","source_entry_ids":["cerny-synchronization:entry:sharpness:v1"],"formal_declarations":[]},{"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.","id":"claim-cerny-circular-case","slug":"cerny-circular-case","version":1,"kind":"theorem","title":"The older level-zero circular case","assumptions":["This is the level-zero case m = n, established in the older circular-automaton literature recalled by the source."],"scope":"This is the level-zero case m = n, established in the older circular-automaton literature recalled by the source.","source_entry_ids":["cerny-synchronization:entry:circular:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"unresolved","extent":"general_case","summary":"The general conjecture remains open. The cited 2026 proof covers synchronizing one-cluster automata with positive least level, with a sharper parameter-dependent bound.","report_ids":["cerny-synchronization:report:general:v1"],"progress_claim_ids":["claim-cerny-positive-level-one-cluster"],"remaining":"Synchronizing automata without a one-cluster letter remain outside this result."},"statement_kind":"conjecture","problem_id":"problem-cerny-synchronization","slug":"cerny-synchronization","title":"Černý synchronization conjecture","question":"How long a word is needed to bring every state of a synchronizing automaton to one state?","scope":"A complete deterministic finite automaton has n ≥ 2 states and a finite alphabet; each letter acts as a function on states. It is synchronizing if some word maps every initial state to the same state. The reset threshold is the shortest such length. Černý’s conjecture asks for a bound of (n−1)² for every synchronizing automaton. The 2026 theorem assumes a one-cluster letter with positive level.","aliases":["Cerny conjecture","Reset word","One-cluster automata","Synchronizing automata"],"fields":["Computer science","Combinatorics"],"msc":["68Q45","20M35"],"topics":[],"background":"undergraduate","scale":"major conjecture","depth":"maintained","source_entry_ids":["cerny-synchronization:entry:question:v1","cerny-synchronization:entry:positive-level:v1","cerny-synchronization:entry:sharpness:v1","cerny-synchronization:entry:circular:v1"],"status_report_ids":["cerny-synchronization:report:general:v1"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"open","scope":"Source comparison and presentation; independent mathematical review remains pending.","rationale":"The general conjecture remains open. The cited 2026 proof covers synchronizing one-cluster automata with positive least level, with a sharper parameter-dependent bound.","decided_on":"2026-10-14","review_due_on":"2027-01-14","report_ids":["cerny-synchronization:report:general:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent","url":"https://arxiv.org/html/2607.19675v1","note":"Introduction; main positive-level results; Section 10, Definition 10.1, Lemma 10.2 and Corollary 10.8."}],"formalizations":[],"related":[],"tasks":[],"insights":[{"id":"cerny-synchronization-impact","title":"A sharper bound can have its own equality examples","summary":"The family F_n attains the parameter-dependent one-cluster bound 3n−5 for every n ≥ 4, even though that bound is often smaller than the universal conjectural quadratic value.","review":"pending","kind":"impact","source_entry_ids":["cerny-synchronization:entry:sharpness:v1"],"claims":[{"id":"claim-cerny-one-cluster-sharp-family","version":1}],"scenario":{"id":"claim-cerny-positive-level-one-cluster","version":1},"limitation":"Sharpness here is a claim about the specified one-cluster parameters, not a resolution of the general synchronization conjecture."}],"coverage_gaps":["Independent OpenMath mathematical review and external proof execution remain pending."],"checked_on":"2026-10-14"},"sources":[{"work":{"authors":["Yinfeng Zhu"],"kind":"research_paper","publication":"preprint","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"self_reported","ai_note":"Zhu reports obtaining the proof through interaction with OpenAI Codex using GPT-5.6 Sol in ultra mode and verifying the argument himself.","artifacts":[],"published":{"value":"2026-07-22"}},"id":"cerny-synchronization:source:paper:v1","source_id":"cerny-synchronization-paper","title":"The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent","owner":"Yinfeng Zhu","url":"https://arxiv.org/html/2607.19675v1","subject_scope":"Introduction; main positive-level results; Section 10, Definition 10.1, Lemma 10.2 and Corollary 10.8.","revision":"v1","content_sha256":"e4c9662c870b7a57340632830c50c88de9264fb345ef23792b160cfcd66ef803","update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://arxiv.org/html/2607.19675v1","attribution":"Yinfeng Zhu","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90}],"entries":[{"id":"cerny-synchronization:entry:question:v1","source_version_id":"cerny-synchronization:source:paper:v1","item_id":"question","url":"https://arxiv.org/html/2607.19675v1","wording":"The general quadratic conjecture is distinct from a result for a letter whose functional graph has exactly one cycle. Level is the least power mapping the state set onto that cycle.","wording_kind":"paraphrase","locator":"Introduction; definitions of synchronization and one-cluster level","supersedes":null},{"id":"cerny-synchronization:entry:positive-level:v1","source_version_id":"cerny-synchronization:source:paper:v1","item_id":"positive-level","url":"https://arxiv.org/html/2607.19675v1","wording":"For a synchronizing one-cluster automaton with n states, cycle length m and least level ℓ ≥ 1, the source proves the bound (m−1)(n−1)+mℓ ≤ (n−1)² and the stated relative-extension property.","wording_kind":"paraphrase","locator":"Main theorem and relative-extending-word result","supersedes":null},{"id":"cerny-synchronization:entry:sharpness:v1","source_version_id":"cerny-synchronization:source:paper:v1","item_id":"sharpness","url":"https://arxiv.org/html/2607.19675v1","wording":"A binary family F_n has m = 2, ℓ = n−2 and exact reset threshold 3n−5; a^(n−2)b^(n−1)a^(n−2) resets it to state 1.","wording_kind":"paraphrase","locator":"Section 10, Definition 10.1, Lemma 10.2 and Corollary 10.8","supersedes":null},{"id":"cerny-synchronization:entry:circular:v1","source_version_id":"cerny-synchronization:source:paper:v1","item_id":"circular","url":"https://arxiv.org/html/2607.19675v1","wording":"The level-zero circular case is already covered by older results. It is not the positive-level advance claimed by the 2026 proof.","wording_kind":"paraphrase","locator":"Introduction and cited Dubuc (1998)","supersedes":null}],"reports":[{"id":"cerny-synchronization:report:general:v1","entry_id":"cerny-synchronization:entry:question:v1","state":"reported_open","source_label":"General Černý conjecture remains open","scope":"All complete synchronizing deterministic finite automata.","source_date":"2026-07-22","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2607.19675v1","supersedes":null},{"id":"cerny-synchronization:report:positive-level:v1","entry_id":"cerny-synchronization:entry:positive-level:v1","state":"resolved","source_label":"Positive-level one-cluster theorem reported","scope":"Synchronizing, unique a-cycle, least level ℓ ≥ 1.","source_date":"2026-07-22","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2607.19675v1","supersedes":null},{"id":"cerny-synchronization:report:sharpness:v1","entry_id":"cerny-synchronization:entry:sharpness:v1","state":"resolved","source_label":"Exact reset threshold for the binary family","scope":"For every n ≥ 4, rt(F_n) = 3n−5.","source_date":"2026-07-22","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2607.19675v1","supersedes":null},{"id":"cerny-synchronization:report:circular:v1","entry_id":"cerny-synchronization:entry:circular:v1","state":"resolved","source_label":"Older circular case recalled","scope":"A letter acts as a cycle on all n states, so ℓ = 0.","source_date":"2026-07-22","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2607.19675v1","supersedes":null}]},"status":{"state":"open","label":"Open","extent":"general_case","summary":"The general conjecture remains open. 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Its finite trace explains synchronization; the all-n optimality theorem needs the separate lower-bound argument.","source_entry_ids":["cerny-synchronization:entry:sharpness:v1"]}],"contributors":null,"scopes":[]},"problem_id":"problem-cerny-synchronization","slug":"cerny-synchronization","title":"Černý synchronization conjecture","publication_version":2,"relationship":{"claim":{"id":"claim-cerny-one-cluster-sharp-family","version":1},"relation":"adjacent_result","relationship_note":"The example is sharp for the parameter-dependent theorem. 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Any copied graph data is CC0, as specified by its archive.","check_interval_days":90}]}}}