Ringdown, Quasi-Normal Modes, and Strong-Field Spectroscopy of Saturation-Shell Compact Objects in the Finite-Capacity Latency–Erasure Theory Regge–Wheeler/Zerilli Completion, Effective Boundary Conditions, Small-Deformation Spectroscopy, and Observable Post-Merger Signatures

We develop the strong-field perturbation and ringdown framework of the finite-capacity latency–erasure theory (FCLET) by formulating the quasi-normal-mode problem of saturation-shell compact objects. Earlier FCLET work established a compact-object completion in which classical horizon formation may be replaced or dressed by a saturation-adjacent boundary layer, while the far-zone tensor sector remains effectively compatible with standard radiative propagation in the appropriate weak-background regime. The present manuscript performs the missing spectroscopy step. We define a canonical spherically symmetric saturation-shell background, derive Regge–Wheeler-type axial and Zerilli-type polar master systems dressed by the latency field, classify absorptive, reflective, and impedance-shell boundary conditions, and formulate the resulting complex mode problem. We further derive small-deformation expansions of the ringdown spectrum, shell-offset estimates for echo delays, and detector-facing observables suitable for comparison with present and future gravitational-wave data. The purpose of this paper is not to claim large deviations from general relativity in advance, but to remove ambiguity about what the FCLET strong-field branch must calculate if it is to be taken seriously in the era of black-hole spectroscopy. We therefore state the master equations, recovery limit, shell matching structure, perturbative mode shifts, benchmark branch taxonomy, and observational outputs required for referee-level assessment. The resulting framework is intentionally conservative: it allows branches that are nearly indistinguishable from general relativity, branches with small but measurable spectral deformations, and more aggressive partially reflective branches that may generate delayed echo-capable transfer functions. These branches are then classified by falsifiability, radiative discipline, and compatibility with the broader FCLET weak-field, thermodynamic, and survival-corridor sectors.