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.