Observable Quotients and Exact Projected Dynamics: Closure, Memory, Minimal Dynamical Completion, and Effective Field Operators

Paper V of the seven-paper Shadow Theory sequence develops the observable and dynamical framework for source–readout models. It proves when a bounded observation map yields an exact observable quotient, distinguishes exact nonidentifiability from unstable inversion and controlled low-rank approximation, and gives a necessary-and-sufficient criterion for source dynamics to descend to an autonomous observable evolution. When autonomous descent fails, the paper derives the exact projected dynamics: deterministic dependence on unresolved initial data together with a memory term, without stochastic or Markov approximations. It then constructs the minimal dynamical completion needed to determine the resolved trajectory and gives a finite-dimensional observability criterion. The paper also treats nonlinear projectability, classical elimination of a field sector through Schur-complement effective operators, and variational effective stress-energy with total covariant conservation. All results are conditional mathematical and physical statements about models that specify a source–readout architecture; the paper makes no empirical claim that our universe has hidden source variables or is a projection of such a source.