Avoided-Crossing Resonance Structure in FCLET Shell Scattering

We show that the non-monotonic transparency structure of FCLET polar shell scattering is governed by a genuine avoided-crossing mechanism in the coupled transfer-matrix spectrum. Starting from the corrected two-channel formulation of the FCLET shell problem, we analyze the scalar leakage fraction and reflection response as functions of shell amplitude, multipole number, and shell geometry. The observed suppression windows are not produced by trivial profile-overlap effects, monotonic coupling growth, or boundary artifacts. They arise from a two-branch resonance structure in which the dominant leakage channel changes across parameter space while the minimum spectral gap remains strictly positive. A systematic two-branch tracking analysis identifies two distinct shell-amplitude windows in the sector, each exhibiting dominance exchange and nonzero minimum separation, thereby establishing true avoided crossing rather than exact mode crossing. We then extend the analysis across and show that the mechanism is universal: avoided crossing persists at every scanned multipole, while the spectral gap grows with . The shell radius and shell width remap the transparency dip and reshape the resonance morphology, while the scalar mass parameter suppresses the scalar channel through freeze-out and the quartic kinetic correction acts only on coupling amplitude. FCLET shell transparency is therefore not a one-channel transmission effect. It is the observable imprint of a universal two-branch resonance architecture.