Weak-Field Yukawa Corrections in a Finite-Capacity Latency–Erasure Field Theory Perihelion Shift, Light Deflection, Shapiro Delay, Constraint Logic, PPN-like Embedding, and the Nonlinear Static Spherical Branch

We derive the weak-field test sector of a finite-capacity latency–erasure field theory whose central dynamical variable is a scalar latency field . Starting from the master equation of the theory, we show that the static weak-field branch reduces to a screened Poisson equation and therefore generates a Yukawa-type correction to the Newtonian potential. The resulting potential is used to derive, at first nontrivial order, the perihelion correction, the light-deflection correction, and the Shapiro time-delay correction. The perihelion shift is obtained in an exact first-order averaged form in terms of modified Bessel functions, while the lensing and time-delay corrections are written in closed integral form and reduced to compact expressions involving and . These results allow a direct constraint logic in the parameter plane, where is the Yukawa amplitude and is the latency correlation length. A PPN-like embedding is then constructed, showing that the weak-field metric structure remains formally GR-like while the effective source potential is Yukawa-deformed. We further derive a first observationally anchored weak-field constraint map using a Cassini-based effective propagation bound and a conservative Mercury perihelion proxy. Finally, we return to the full static spherical master equation in its nonlinear form, derive the dimensionless boundary-value problem, obtain its interior and exterior asymptotic branches, and show how the linear Yukawa law emerges as the outer weak-load limit of the nonlinear theory. The result is the first heavy-mathematical, directly testable phenomenological branch of the finite-capacity program.