Introduction; main positive-level results; Section 10, Definition 10.1, Lemma 10.2 and Corollary 10.8.
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The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
The Černý conjecture asserts that every synchronizing automaton with n states has reset threshold at most (n−1)²; the classical examples of Černý (1964) show that this number would be best possible.
Passage ID: cerny-synchronization:entry:cerny-examples:v1
Source item: introduction-cerny-examples
Source record: cerny-synchronization:source:paper:v1
The level-zero circular case is already covered by older results. It is not the positive-level advance claimed by the 2026 proof.
Passage ID: cerny-synchronization:entry:circular:v1
Source item: circular
Source record: cerny-synchronization:source:paper:v1
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.
Passage ID: cerny-synchronization:entry:positive-level:v1
Source item: positive-level
Source record: cerny-synchronization:source:paper:v1
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.
Passage ID: cerny-synchronization:entry:question:v1
Source item: question
Source record: cerny-synchronization:source:paper:v1
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.
Passage ID: cerny-synchronization:entry:sharpness:v1
Source item: sharpness
Source record: cerny-synchronization:source:paper:v1
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v4
Source assessment
Proved
Černý’s examples attain (n−1)²
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Source assessment
Proved
Older circular case recalled
A letter acts as a cycle on all n states, so ℓ = 0.
Source date Jul 22, 2026 · Observed Oct 14, 2026
Source assessment
Proved
Positive-level one-cluster theorem reported
Synchronizing, unique a-cycle, least level ℓ ≥ 1.
Source date Jul 22, 2026 · Observed Oct 14, 2026
Source assessment
Proved
Positive-level one-cluster theorem reported
Synchronizing, unique a-cycle, least level ℓ ≥ 1.
Source date Jul 22, 2026 · Observed Oct 14, 2026
Proof explanation
Extend a subset until the cycle is covered
Source assessment
Open
General Černý conjecture remains open
All complete synchronizing deterministic finite automata.
Source date Jul 22, 2026 · Observed Oct 14, 2026
Scope
General conjecture
Source assessment
Proved
Exact reset threshold for the binary family
For every n ≥ 4, rt(F_n) = 3n−5.
Source date Jul 22, 2026 · Observed Oct 14, 2026
Bound
Reset threshold of F_n
Construction
A four-state member of the sharp family
Question
Černý synchronization conjecture
Connection
The positive-level one-cluster theorem reaches the conjectured bound within its restricted class.
Definition
One-cluster letter and level
History
Positive-level spectral proof is published
Further reading
Read the exact positive-level hypotheses and spectral proof.
Definition
Reset word
History
Černý’s reset-word question
Further reading
Understand reset words and the one-cluster restriction.
Further reading
Inspect the sharp family, its transitions and matching lower bound.
Recorded authorship
Yinfeng Zhu
Zhu reports obtaining the proof through interaction with OpenAI Codex using GPT-5.6 Sol in ultra mode and verifying the argument himself.
OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent
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