Thermodynamic Consistency, Entropy-Production Closure, and Generalized Second-Law Architecture in the Finite-Capacity Latency–Erasure Theory From Unified Source Dynamics to Irreversible Accounting, Entropic Admissibility, and Regime-Stable Thermodynamic Closure in a Finite-Capacity Physical Program

We develop the thermodynamic sector of the finite-capacity latency–erasure theory by constructing an entropy-production closure, an entropic admissibility principle, and a generalized second-law architecture linking gravity, cosmology, nonequilibrium memory, stochastic latency noise, and saturation-adjacent strong-field structure within one irreversible framework. Earlier branches of the finite-capacity program established covariant source closure, dynamical wave consistency, benchmark-oriented numerical inference, perturbative matter coupling, and a regime-stable nonlinear hierarchy. The present work closes the thermodynamic sector by specifying how finite-capacity realization burden, overwrite activity, memory retention, fluctuation renormalization, and horizon-weighted sourcing are embedded into a common entropy balance. Starting from the patch-level realization substrate, we define realized-load entropy, overwrite entropy production, memory-burden free-energy storage, and fluctuation-induced coarse-grained entropy flow. These ingredients are then coarse-grained into a covariant entropy-current architecture for the latency field and its coupled sectors. We derive a generalized entropy-balance law, formulate a nonnegative entropy-production condition, and define thermodynamically admissible branches as those for which irreversible generation, horizon exchange, and delayed memory release remain globally ordered and locally nonpathological. Weak-field gravity, moderated erasure cosmology, history-dependent clock sectors, stochastic timing sectors, and saturation-adjacent strong-field branches are shown to arise as thermodynamic reductions of the same irreversible accounting system. We further construct a generalized second law for finite-capacity realization in which matter loading, overwrite erasure, memory release, and horizon-associated entropy exchange contribute to a total nondecreasing thermodynamic functional under admissible evolution. The resulting framework upgrades the finite-capacity program from source closure and nonlinear hierarchy to a thermodynamically closed physical theory with explicit entropy-production logic and branch-level entropic admissibility.