Coupled Gravitoelectromagnetic Response of a Magnetically Supported Generalized Hayward Black Hole in Nonlinear Electrodynamics

We construct the exterior coupled gravitational and electromagnetic response of a magnetically supported generalized Hayward black hole within a specified nonlinear electrodynamic completion. Both parity sectors reduce to canonical two channel wave systems with explicit potentials and the correct Schwarzschild and effective Reissner--Nordstr\"om limits. We compute the quasinormal spectrum, metric and optical characteristics, extremal throat weights, quadrupolar wave conversion, finite bandwidth transfer, and coherent absorption eigenchannels while verifying exterior hyperbolicity and flux conservation. An effective charged operator captures the leading weak core geometry and mode mixing; after its subtraction, the residual hierarchy is consistent with cubic spectral corrections in the primarily electromagnetic branch and quartic corrections in the primarily gravitational branch and principal scattering observables. The fundamental quadrupolar branches become intrinsically linewidth separated near $\chi\simeq0.70$, whereas parity splitting remains subleading. Metric and optical characteristic families separate monotonically toward extremality. At $\chi=0.9$, positive parity reflected conversion reaches $45.8\%$ and remains $40.8\%$ after moderate bandwidth averaging, while coherent incident combinations yield $98.4\%$ bright and $1.3\%$ dark absorption. These independently evaluated observables provide mutually consistent diagnostics of the same nonlinear electrodynamic operator.