Shell Reflectivity, Quasi-Normal Mode Deformation, and Echo Amplitude Hierarchy in the Finite-Capacity Latency–Erasure Theory

We develop the shell-reflectivity sector of the finite-capacity latency–erasure theory and derive the strong-field spectroscopic discriminator of the program. Earlier FCLET work established the local burden variables, completed the microphysical specification of the local information-capacity sector, derived the shell-onset condition , and obtained explicit Schwarzschild-shell benchmark estimates for the shell distance and echo-delay scale. What remained open was the boundary-response problem: once a finite-capacity saturation shell replaces the classical featureless horizon, how does that shell respond to incident perturbations? This paper answers that question. We introduce the shell reflectivity function as the frequency-dependent response coefficient of the FCLET saturation shell and derive its relation to local burden saturation, residual support margin, finite-capacity dissipation, and delayed overwrite dynamics. In the FCLET compact-object sector, the shell is neither perfectly absorbing nor perfectly reflecting. It is a partially transmissive, frequency-selective boundary layer whose response is controlled by the same finite-capacity microphysics that fixed the shell location. We therefore construct the shell response operator, derive the mixed near-shell boundary condition for perturbation modes, and obtain the leading quasi-normal mode deformation generated by shell reflection and delayed reprocessing. We show that the observationally decisive FCLET strong-field signature is not the logarithmic echo delay alone, since that timing scale is shared broadly by Planck-scale near-horizon shell models. The discriminator lies in the full spectroscopic package: shell reflectivity, echo amplitude hierarchy, cavity filtering, mode-dependent phase deformation, and damping modification. We derive the first general FCLET expressions for the reflected amplitude, echo-train suppression law, and complex frequency shift , and show how these observables inherit species dependence through the shell loading parameter and structural dependence through the shell-response scale . The result is decisive for the strong-field completion of the theory. FCLET now advances from shell-location kinematics to shell-response spectroscopy. The compact-object branch no longer predicts only where a finite-capacity shell forms. It predicts how that shell reprocesses strong-field perturbations, how the ringdown spectrum departs from classical general relativity, and where the genuinely theory-specific observational signatures of FCLET must appear.