Axial Kernel Coefficients in FCLET Shell Perturbation Theory

We derive the axial shell-kernel coefficients of the finite-capacity latency–erasure theory directly from the linearized metric–latency action and thereby remove the last benchmark closure from the minimal strong-field reflectivity sector. Earlier FCLET work established the shell background, fixed the shell-onset threshold , derived the shell location and echo-delay scale, and then obtained the first action-based odd-parity thin-shell reflectivity law, That result closed the shell-response problem at formal level, but its numerical realization still depended on the composite shell-strength parameter which had not yet been derived from the action itself. The present paper closes that gap. Starting from the FCLET strong-field action, we expand the coupled metric–latency system to quadratic order around a static spherically symmetric shell background and perform the odd-parity reduction of the perturbation action. Because the background latency field is scalar, the odd-parity sector contains no independent propagating axial scalar harmonic. Nevertheless, the shell profile modifies the axial master operator through background dressing, shell-gradient energy, and shell-support stress. We isolate the shell-supported terms of the quadratic action, reduce them to the minimal local operator basis, and derive the coefficients , , and explicitly. This yields the first first-principles reconstruction of the axial shell kernel, and therefore the first action-level determination of the shell coupling The strong-field significance is immediate. The shell reflectivity sector is no longer controlled by a benchmark normalization convention. It is fixed by the quadratic perturbation structure of the FCLET action itself. The compact-object branch therefore advances from benchmark shell spectroscopy to derived shell spectroscopy.