Covariant Source Closure and Effective Temperature Unification in the Finite-Capacity Latency–Erasure Theory From Patch Occupancy and Overwrite Dynamics to a Unified Matter–Latency Source Across Gravity, Cosmology, Memory, and Noise

We construct a unified covariant source closure for the finite-capacity latency–erasure theory. Previous branches of the program introduced a Landauer-based matter–information source for the latency field, a patch-occupancy microphysical substrate, overwrite-driven cosmological sourcing, history-dependent latency memory, and stochastic update fluctuations. Here these ingredients are reorganized as sectoral limits of a single source hierarchy rather than as parallel phenomenological additions. We begin from a finite-capacity patch medium with realized occupancy, occupancy-dependent latency, irreversible overwrite kinetics, internal memory retention, and stochastic update noise. Coarse-graining then yields a unified covariant source scalar for the latency potential. The total source is written as the sum of matter loading, irreversible overwrite activity, memory burden, and fluctuation-renormalized contributions, each weighted by an effective erasure-temperature prescription adapted to the local physical regime. A regime-consistent temperature unification is introduced that recovers the quasi-static laboratory limit, the weak-field gravitational limit, the horizon-defined cosmological limit, and the overwrite-active nonequilibrium limit within one common framework. The resulting closure provides a theory-wide parameter genealogy, removes source-level ambiguity across gravity and cosmology, and embeds microphysical patch dynamics directly into the covariant latency equation. Weak-field gravity, moderated erasure cosmology, history-dependent clock effects, and stochastic latency noise emerge as controlled reductions of the same source backbone. The result is a hardened multiscale architecture in which finite-capacity realization is sourced covariantly and interpreted microphysically.