Abstract

This paper develops a compositional theory for declared identity-persistence regimes. It asks when a certified output under one regime may become the identity-bearing input of another, which obligations must propagate backward through a chain or network of regimes, and which distinctions may be compressed into certificates that move forward. Its governing law is that compression is demand-indexed while soundness need not be: a demand-independent identity interface can always preserve everything, but canonical maximal compression is defined only relative to what downstream evaluation requires. For each certified interface, the paper defines a contravariant weakest-obligation transformer on Galois-closed demand families. These transformers compose laxly, which is sufficient to form a category of demand-indexed regimes: an interface from (R,Q_R) to (S,Q_S) is admissible when the upstream declared demand is strong enough to support the pullback of the downstream demand. Composition therefore closes without knowledge of future extensions to the network; a later stronger promise is represented by a new indexed object and a fresh interface obligation rather than by retroactively invalidating an earlier morphism. The corresponding normal forms form a coherent inverse system. If Q is refined to Q′, the finer certificate N_Q′(R) maps canonically onto N_Q(R). The obstruction to reversing that map is typed by the existing migration calculus: newly required distinctions, distinctions already lost by the old certificate, and the residual deficiency not repaired by an added channel. For fixed acyclic diagrams, backward obligation propagation computes the least admissible demand profile and the forward normal-form pass is maximally compressed among structurally admissible sound towers supporting the same ultimate promises. The network result is not limited to acyclic diagrams. On any fixed small directed network, the product of the closed-demand lattices is complete and the monotone network-obligation operator has a least fixed point. Ordinal iteration from the bottom converges abstractly to that least fixed point on any set-sized complete lattice. Cycles, infinite chains, and infinite branching therefore create no existence obstruction at this level; effective computation, closure-ordinal bounds, complexity, constructive implementation, and dynamic topology remain separate questions. The paper also gives exact laws for interface soundness, joint masking, locality, repair, and entitlement. At symmetric interfaces it imports the Orbit-Congruence Criterion and distinguishes three requirements: downstream demand must respect the declared symmetry, the symmetry must be compatible with the declared operations for the quotient to be operable, and the composite signature must be adequate for regime-level certification. Under these conditions, the semantic channel of any given sound symmetry-respecting interface descends through the orbit quotient, and every such interface canonically forgets down to the same terminal normal form. The resulting architecture is a dual flow: demands move backward; certificates move forward.