Coupled Gravitoelectromagnetic Ringdown of a Maxwell Normalized Exponential de Sitter Core Black Hole

We formulate a two scale magnetic nonlinear electromagnetic completion of an exponential de Sitter core black hole. The additional sector recovers Maxwell electrodynamics at weak field, is exponentially suppressed at the regular center, and has an exact magnetic mass integral. The exterior domain is selected by the nonextremal outer horizon condition together with principal hyperbolicity, metric subluminality and the standard energy conditions. At an admissible benchmark, the coupled axial and polar fundamental modes for ℓ = 2, 3, 4 exhibit frequency splitting of 1.08–2.59 percent and damping splitting of 1.43–8.70 percent. Along the displayed quadrupolar source coupling sequence, the frequency splitting decreases monotonically in magnitude while the branch II damping splitting changes sign; the polar frequencies remain lower on both branches. The extraordinary optical frequency is lower throughout the admissible two parameter scan, while its Lyapunov shift changes sign. One way incident packets yield cross channel conversion of 29.8–43.6 percent, with extraction and domain systematics attached to the reported percentages. These results isolate the two scale source operator and its exterior admissibility conditions as the defining ingredients of the coupled response.