Numerical Shell Spectroscopy and Ringdown Signatures in FCLET Compact Objects

We develop the numerical shell spectroscopy of finite-capacity latency–erasure compact objects by solving the finite-thickness coupled metric–latency shell transfer problem and extracting the physical shell reflectivity, echo hierarchy, and ringdown deformation of the strong-field sector. Earlier FCLET work established the shell-onset threshold , derived the shell location and echo-delay scale, constructed the minimal odd-parity thin-shell reflectivity law, derived the axial shell kernel coefficients from the quadratic action, obtained the shell coupling, derived the regular quenching branch of the saturated core, and then formulated the full finite-thickness shell matching problem in the coupled metric–latency sector. Those results completed the derivational architecture of the shell. What remained open was the observable problem: once the shell transfer system is fully specified, what physical reflectivity law does it generate, how strongly does it soften the rigid minimal branch, and what ringdown signatures does it imprint in the astrophysical band? This paper answers that question. We solve the finite-thickness shell transfer matrix numerically across the resolved shell layer, including shell-internal phase accumulation, gravitational-to-latency conversion, and core-side latency quenching. From the resulting transfer operator we extract the physical gravitational reflectivity amplitude, the effective shell phase, and the frequency-dependent shell return factor governing repeated shell-barrier cycling. We then compute the resulting echo spectrum, quantify the suppression and spectral reshaping of late-time echoes relative to the minimal branch, and derive the associated quasi-normal mode deformation of the compact-object response. The result is decisive. The physical shell sector is not maximally rigid. Finite shell width, coupled shell-layer dynamics, and quenching at the saturated core reduce the effective infrared reflectivity below the one-channel thin-shell limit and generate a resolved shell spectroscopy with nontrivial frequency structure. The strong-field FCLET program therefore advances from derivational shell dynamics to numerical shell phenomenology. The shell is no longer only a theoretical response layer. It becomes an observable transfer medium with a calculable spectroscopic signature.