Residuation is the phenomenon of dual operations being connected by an adjunction. When two operations stand in a residuated relationship, knowing how one transforms a configuration tells you exactly how the other must respond.
The canonical instance in relationality is the law connecting Together and Implies:
Entails(Together(a, b), c) if and only if Entails(a, Implies(b, c))
This says: “a and b together entail c” is the same statement as “a alone entails that b implies c.” The meet operation and the implication operation are residuated partners — each is the adjoint of the other.
Residuation appears wherever dual containments arise. Close and Open are related by residuation. Flow and its resolution morphism form a residuated pair. At the meta level, systems of balanced operators exhibit residuation between their components.
The phenomenon matters because it means that dual aspects of relational structure are not independent. If you know the closure, the interior is determined. If you know how things combine, implication follows. Residuation is the formal expression of the relational claim that things exist through their relations — dual operations co-determine each other.
Derivational context
Residuation first appears implicitly in Movement I when the operations of combining and conditional relating (Together and Implies) are earned as adjoint partners. It becomes the primary connective structure in Movement IV: Geometric Cohesion, where the interplay between doing-then-stabilizing and stabilizing-then-doing governs how local and global structure translate into each other. At the physical level (Movement V), residuation governs the commutation of evolution and measurement.