Mathematical Consistency, Stability, and Limit Structure of the Latency–Erasure Field Theory Well-Posedness, Linear Modes, Screening Stability, Relaxational Memory, Stochastic Dissipation, and Admissible Parameter Domains

A theoretical framework gains mathematical seriousness not only by generating phenomenology, but by demonstrating that its core dynamical structure is stable, internally consistent, and well-behaved across its relevant limiting regimes. In the finite-capacity latency–erasure program, later works have already established weak-field Yukawa sectors, moderated cosmological branches, nonequilibrium history-dependent latency, stochastic fluctuation sectors, and a microphysical closure architecture. What remains is to determine whether the underlying latency–erasure field theory possesses a mathematically admissible core: whether its perturbations are stable, whether its static branches are regular, whether its screened solutions behave consistently, and whether its parameter space contains nonpathological domains connecting the weak-field, cosmological, nonequilibrium, and fluctuation sectors. In this paper, we develop the first systematic mathematical audit of the latency–erasure field theory. We begin from a generic effective scalar-field formulation for the latency field , coupled to realized load and erasure-sensitive sources. The theory is then analyzed in several stages. First, we derive the linearized field equation around reference backgrounds and determine the conditions under which perturbative modes remain dynamically admissible. Second, we introduce an energy-like functional for the linearized latency sector and show that, on the admissible sign domain, the theory possesses a nonincreasing Lyapunov-type energy. Third, we study the static weak-field and screened branches, deriving regularity conditions, coercivity of the static functional, and asymptotic behavior of Yukawa-like solutions. Fourth, we analyze nonequilibrium relaxational sectors and stochastic fluctuation sectors as dynamical perturbation problems, identifying the conditions under which the theory avoids unstable growth and instead exhibits bounded, decaying, or mean-square stable behavior. Fifth, we classify the parameter domains into stable interior, marginal boundary, and pathological exterior regions and show that the principal weak-field, relaxational, and stochastic limits are mutually compatible on a nonempty admissible domain. The main results are as follows. The latency–erasure field theory admits a mathematically coherent effective structure provided the kinetic sign, screening mass scale, relaxation coefficients, and transport parameters satisfy a definite set of positivity and boundedness conditions. In these domains the weak-field branch is screened rather than runaway, the nonequilibrium sector is relaxational rather than explosive, the stochastic sector is dissipative rather than ill-posed, and the theory reduces continuously to its appropriate weak-latency and low-noise limits. Conversely, wrong-sign sectors or incompatible parameter combinations generate pathological branches, including unbounded growth, irregular screening behavior, anti-damped memory modes, and ill-posed stochastic transport. The resulting analysis shows that the finite-capacity latency–erasure program is not mathematically unconstrained. Its viable forms occupy a restricted but nonempty admissible domain. This provides an explicit consistency backbone for the wider program and marks an important transition from sectoral phenomenology toward a more disciplined field-theoretic foundation.