Gravitational Shapiro Delay as Constitutive Retardation in an Effective Hyper-Elastic Vacuum Continuum

We derive the gravitational (Shapiro) radar time delay as a constitutive optical-retardation effect in an effective hyper-elastic continuum model of the vacuum. Using a weak-field refractive-index profile previously shown to reproduce the full Schwarzschild light-deflection angle via density scaling and anisotropic shear-modulus softening, we compute the excess light-travel time for a signal passing near a compact mass. The resulting delay contains the standard logarithmic term and matches the familiar weak-field Schwarzschild prediction at first order in GM/(rc²). This result closes the lensing-timing consistency loop in the static weak-field regime for the constitutive-optics approach: a single effective index profile accounts for both bending (lensing) and lag (time delay) within the regime of geometric optics and static spherical symmetry. Paper 4 in the Substrate Mechanics series.