The Spectrum of a Compact Internal Space III: Finite-Window Dynamics of Cosmic Structure Formation

Cosmic matter forms concentrations whose external force can often be described by their mass and spatial moments. Encounters and material exchange can make finer structure relevant again. We determine when field and object descriptions preserve that force and the subsequent evolution of specified observables while retaining unresolved influence in residual variables and memory. The compact internal-space construction supplies the matter carriers and collective source. Their coupled variations determine the macroscopic evolution. The applications examine homogeneous response reuse and a constrained nonuniform Einstein-matter perturbation together with autonomous expansion from a local constitutive action. Reconstruction in a common state space combines object replacement with moving spectral boundaries and truncated memory for a finite reference evolution. Spatial moments bound the force error. Cloud-in-cell assignment admits an exact eight-moment cell representation. A positive lower bound on the source derivative of the stationary constraint bounds the error in reusing its response. Integrable stability and defect bounds on a certified state tube then control the accumulated reconstruction error. A common modulus of continuity transfers this bound to observables over the prescribed interval. The first accepted window is the least certified member of a finite ordered family. Finite Galerkin examples retain positive density and the prescribed accuracy with reduced arithmetic work including verification. A cosmological particle calculation encloses the continuous-time trajectory at fixed mesh resolution. Its weakly perturbed cold-sheet reconstruction also gives full-frequency force bounds and continuous position and momentum equation-defect bounds. These estimates select sufficient retained structure for the prescribed interaction and observables. They control the re-entry of finer coordinates.