Matter Coupling, Perturbation Closure, and Structure-Formation Consistency in the Finite-Capacity Latency–Erasure Field Theory From Unified Source Dynamics to Linear Response, Effective Growth Laws, and Regime-Consistent Matter Coupling in a Finite-Capacity Gravity Program

We develop the matter-coupling and perturbation sector of the finite-capacity latency–erasure field theory and derive a regime-consistent closure linking unified source dynamics to linear response, effective growth laws, and structure-sensitive observables. Earlier branches of the finite-capacity program established weak-field gravity, screened solar-system viability, moderated erasure cosmology, nonequilibrium latency memory, stochastic fluctuation effects, microphysical source genealogy, unified covariant source closure, dynamical wave consistency, and a benchmark-oriented numerical architecture. The present work closes the remaining sectoral gap between source closure and observational perturbation theory. Starting from the covariant latency action with unified source hierarchy, we formulate a canonical matter-coupling prescription, derive the linearized scalar–metric–matter perturbation system around homogeneous and weakly inhomogeneous backgrounds, and obtain effective equations for density growth, potential slip, clock-sensitive source activation, and fluctuation-dressed response. We show that the resulting perturbation theory admits a quasi-static weak-field limit, a horizon-weighted cosmological growth limit, a memory-dressed nonequilibrium response sector, and a stochastic renormalization sector, all as controlled reductions of a single branch-aware closure. We define an effective Poisson law, an effective growth factor, and a sector-sensitive response matrix for matter perturbations, and we identify the admissibility conditions under which matter clustering, scalar response, and background closure remain mutually consistent. The resulting framework upgrades the finite-capacity program from background- and source-level closure to perturbatively operational structure formation and cross-sector matter response. This paper therefore provides the perturbation capstone linking unified source architecture to observable matter-sector dynamics.