Dynamic sufficiency extends residuated sufficiency to account for temporal evolution.
Static residuated sufficiency asks whether two structures satisfy the residuation law at a single epoch. Dynamic sufficiency asks whether that satisfaction persists as the structures evolve. The question becomes: if I propagate a recognition forward through Flow, does it remain residuated-sufficient with respect to its counterpart?
Formally, the directed sufficiency relation under a temporal operator Flow is:
Includes(Together(Flow(a), b), c) if and only if Includes(b, Induces(Flow(a), c))
Flow here acts as a KZ comonad — it propagates recognitions forward across redshift while preserving the residuation structure. When this directed residuation holds, the inference system is dynamically sufficient: its claims remain internally consistent as the cosmological field evolves.
A dynamic imbalance function g(z, Δz) tracks whether sufficiency changes smoothly across redshift intervals. Smooth monotonic decline indicates stabilization. Persistent peaks indicate dynamic inconsistency — places where the temporal propagation of inference breaks down.
See Dynamic Sufficiency in Cosmological Inference and Reflexive Disequilibrium.