We develop the foundational scattering paper of the FCLET polar shell program by deriving, validating, and solving the corrected two-channel gravitational-scalar system that governs even-parity shell scattering on a Schwarzschild background. The problem is formulated in terms of a Zerilli-like gravitational master variable and a gauge-invariant scalar latency channel , coupled through shell-supported gradient and algebraic interaction terms. We show that the reduced action enforces an antisymmetric gradient-coupling structure: the term enters the two equations with opposite signs, and the derivative completion is mandatory for a physically consistent flux-conserving system. This corrected structure resolves the instability and flux pathology of incomplete benchmark formulations. We then construct the full-branch completion of the shell problem by activating the nontrivial sector and its induced coefficient functions , , and . The benchmark branch provides the minimal coupled scaffold and reproduces the vacuum limit exactly, while the full branch restores the missing shell-supported couplings and upgrades the system from a provisional numerical model to a physically closed scattering framework. Using a QR-renormalized transfer-matrix solver, we compute the gravitational reflection coefficient , the scalar leakage amplitude , and the associated flux balance. The vacuum limit satisfies exact unit flux conservation, and the full branch restores near-unit flux throughout the shell-active regime, improving the benchmark flux deviation by approximately two orders of magnitude. The resulting FCLET shell is not a passive barrier but an active conversion layer: over a finite frequency window, incident gravitational flux is partially redirected into the scalar channel, producing a structured response in , , and the scattering phase . This paper establishes the formal and numerical foundation on which the avoided-crossing analysis and the later parameter taxonomy rest. It provides the corrected equations of motion, the full-branch coefficient closure, the conserved-current structure, the transfer-matrix algorithm, and the first physically reliable FCLET shell-scattering observables.
Ali Caner Yücel (Tue,) studied this question.