3+1 ADM Formulation and Numerical Evolution of Saturation Shells in the Latency–Erasure Field Theory

We develop the numerical-relativity branch of the finite-capacity latency–erasure program by constructing the full 3+1 decomposition of the latency–erasure field equations and deriving the dynamical evolution system relevant to compact collapse, strong-field time development, and horizon-scale structure formation. Starting from the covariant latency sector and its unified source hierarchy, we perform an ADM decomposition of the effective field equations, identify the Hamiltonian and momentum constraints, and obtain the corresponding evolution equations for the spatial metric, extrinsic curvature, lapse, shift, and latency field variables. The resulting system shows that classical horizon formation is replaced by a finite-capacity strong-field regime in which the latency field approaches saturation and generates a dynamical saturation shell rather than a featureless null surface. We derive the principal-part structure of the coupled gravity-latency evolution system, formulate the constraint-preserving initial-value problem, and obtain the numerical architecture needed for simulations of gravitational collapse and compact-object mergers. The theory predicts that the late-stage evolution of strongly loaded configurations is governed by shell formation, retarded overwrite dynamics, and bounded realization burden, producing a strong-field completion distinct from classical black-hole horizon formation. This establishes the numerical-relativity interface of the finite-capacity program and provides the mathematical infrastructure required for simulations of compact-object collapse and merger dynamics in a finite-capacity universe.