Multiple finite closures (FCs) inevitably interact when combined into a larger system.In this paper we establish a \emph{Respect Principle} that characterises when such FCs cancoexist harmoniously. Under mild regularity assumptions ensuring Carathéodory/Filippovsolutions, we derive uniform \emph{boundary-size bounds} for the monitors associated with eachFC. These bounds lead to a simple budget inequality: a self–dissipation termmust dominate the interference influx from all other FCs. We prove a forward–invariancetheorem showing this inequality is sufficient for the product closure to remain in a saferegion. A converse shows the condition is also necessary: if it is violated, anadmissible coupling can force an FC to exit its safe region. A harmony–equivalencetheorem identifies respect as the minimal and indispensable condition for globalinvariance and yields a conservation/dissipation law on the boundary. Beyond the coreanalysis, we provide a Gershgorin-style interpretation, a quick–start checklist forpractitioners, and a $\SigOne$ verification procedure with explicit safety margins.The framework provides a verifiable, patent-independent certificate for finite-closure invariance, publishable for universal public use.