= Closure dominance for asynchronous threshold switching
Allow individual <vertices> to switch in either direction according to a fixed neighbour threshold, while the objective is to maximize the number in one designated state. Let $C$ be the <monotone threshold activation> closure of the initially designated <vertices>. At every time in any legal switching sequence, the designated <set> remains a <subset> of $C$: a loss preserves inclusion, while a gain outside $C$ is impossible because all currently designated neighbours lie in $C$ and the outside <vertex> has too few neighbours there. This <induction> proves an upper bound of $|C|$ on every attainable designated population. The monotone sequence attains $C$ itself and is therefore globally optimal. Temporary losses cannot improve this objective.
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