Discovery of Finite-Order Resonance in Iterated Exchange Scattering: Identifying Sharp Peaks near Fine-Structure-Constant Inverse Values 137 and 128 with a Reproducible Wave-Packet Model

This preprint investigates sharp parameter-space peaks previously observed near values corresponding to inverse fine-structure constants 137 and 128 in two exchange-scattering wave-packet simulations. Full-range reanalysis, root-centered sweeps, eigenvalue analysis, and 50- and 80-digit calculations show that the peaks arise from exact finite-order recurrence of the antisymmetric eigenvalue of a two-channel unitary exchange operator. The roots are R_(n,m) = cos^2(pi m/n); the principal low-energy peak is R_(124,23), the former high-energy candidate is R_(122,23), and R_(620,117) lies within 0.1243 standard deviations of alpha^(-1)(M_Z^2) = 128.946 +/- 0.015 under the study's readout correspondence. Ideal even-order roots have zero gray error and hence formally infinite depth, denoting zero residual rather than divergent energy. The main contribution is a reproducible numerical model connecting exact operator recurrence to wave-packet localization transfer and metastable two-state readout. The study does not derive alpha: whether the proximity to the two physical alpha^(-1) values is an arithmetic coincidence or reflects an additional structural necessity remains open.