Kerr Extension of the FCLET Shell Theory

We extend the finite-capacity latency–erasure theory to rotating compact remnants by constructing the Kerr-sector shell theory in a mathematically controlled form. Earlier FCLET work completed the non-rotating strong-field branch: the saturation-shell onset was fixed internally, the shell location and echo-delay scale were derived, odd-parity perturbations were constructed, and the shell reflectivity spectrum was obtained through numerical spectroscopy. What remained open was the astrophysically decisive step. Gravitational-wave remnants are rotating compact objects, and any shell sector confined to Schwarzschild backgrounds remains structurally disconnected from the observational ringdown problem. In this paper we build the Kerr extension of the FCLET shell theory through four coupled tasks. First, the Schwarzschild shell radius is generalized to a covariant saturation surface, yielding an axisymmetric shell profile. The threshold condition is re-derived on the rotating background from the local burden-balance structure itself rather than imported by analogy, and its leading universality is shown to admit controlled spin-dependent corrections. Second, the latency field is extended from a purely radial Schwarzschild profile to a stationary axisymmetric rotating-background field through a lapse–shift–drag decomposition. In the Kerr sector, the FCLET burden couples not only to the lapse structure but also to the frame-dragging channel, so that the effective background deformation acts simultaneously on lapse, shift, and shell geometry. Third, the linearized metric–latency system is projected to the Teukolsky/Generalized Sasaki–Nakamura channel. The FCLET modification is decomposed into three distinct sectors—background deformation, effective source structure, and shell boundary data—and each sector is carried through separately in the perturbation problem. Exact separability is treated as a special admissible branch rather than assumed generically, while the non-separable regime is handled through a controlled slow-rotation branch with perturbative angular mixing. Fourth, the physical admissibility of the rotating shell sector is constrained by superradiant consistency and ergoregion stability. The paper identifies the stability corridor in which shell reflectivity, frame-dragging deformation, and cavity amplification remain compatible with a spectroscopically viable Kerr-sector solution. The main outputs are the detector-facing FCLET spectral observables in the rotating sector: the quasi-normal mode shift , the geometric and mode-projected echo-delay families, the excitation branching coefficients , and the ringdown damping correction derived from the full complex mode spectrum. These quantities supply the mathematical infrastructure required for the injection–recovery program of the next stage and for event-level confrontation with rotating compact-remnant data. The Schwarzschild shell sector is recovered in the limit, the standard Kerr–Teukolsky system is recovered in the limit, and the admissible Kerr FCLET branch is isolated explicitly from the non-separable or superradiantly unstable regime. The result is not a rotating extension by analogy, but the first controlled Kerr-sector perturbative foundation of the FCLET shell program. In addition, the paper reports the first stabilized numerical benchmark of the Schwarzschild FCLET shell spectrum: a two-segment continuation analysis in the plane produces explicit, publication-grade real-frequency shifts of approximately and for the and fundamental modes, establishing the numerical zero-spin anchor for the rotating spectral program.