Effective Hierarchy, Nonlinear Completion, and Regime-Stable Closure in the Finite-Capacity Latency–Erasure Field Theory From Low-Energy Source Closure to Nonlinear Branch Control, Effective-Field Ordering, and Regime-Stable Completion in a Finite-Capacity Gravity Program

We develop the effective-hierarchy and nonlinear-completion sector of the finite-capacity latency–erasure field theory by constructing a regime-stable ordering principle linking low-energy source closure, screened weak-field dynamics, cosmological overwrite activity, perturbative matter response, and strong-field branch control within one effective-field architecture. Earlier branches of the finite-capacity program established covariant source closure, dynamical wave consistency, benchmark-oriented numerical inference, and perturbative matter coupling. The present work provides the missing hierarchy principle that organizes these achievements into a controlled nonlinear theory rather than a collection of mutually compatible sectors. We define the latency–erasure theory as an ordered effective hierarchy in which source terms, kinetic terms, screening operators, overwrite-driven contributions, memory corrections, and fluctuation renormalizations appear with regime-dependent but systematically bounded importance. A nonlinear completion map is then introduced to distinguish admissible branches from merely formal ones and to identify the conditions under which weak-field, cosmological, nonequilibrium, stochastic, and strong-field regimes are embedded into a single regime-stable structure. We derive a branch-selection functional, a hierarchy-preservation criterion, and a nonlinear closure relation for screened and source-driven sectors. The theory is shown to possess a low-load perturbative ordering, an intermediate screened nonlinear ordering, and a saturation-adjacent completion regime, all governed by the same finite-capacity variables. We then formulate the conditions under which effective operators remain ordered, branch transitions remain controlled, and strong-field saturation does not destabilize the low-energy predictive sectors. The resulting framework upgrades the finite-capacity program from a source-closed and perturbatively operational theory to a regime-stable nonlinear effective architecture with explicit hierarchy logic. This paper therefore provides the nonlinear structural capstone of the finite-capacity program.