R-layer Mode Theory LXXXIV-C: Dynamical Black Holes, CUP Evolution, and Time-dependent Multi-band Observables

This Volume LXXXIV-C extends the R-layer Mode Theory (RLMT) to dynamical black holes, incorporating time-dependent mass, spin, and R-layer evolution. We introduce a time-dependent Kerr parametrization of the R-layer effective geometry and construct a dynamical CUP potential that governs the growth, saturation, and decay of the CUP region during black hole formation and accretion. The analysis includes dynamical Teukolsky and Sasaki-Nakamura equations with R-layer-induced mass and dissipation kernels, leading to dynamical quasi-normal modes (DQNM), dynamical echoes, and dynamical shadow deviations. We further extend the R-layer framework to an FRW cosmological background, derive cosmological CUP scaling laws, and discuss the impact of dynamical CUP potentials on the cosmic microwave background (CMB). Finally, we formulate a time-dependent Fisher matrix in the extended parameter space (M(t), a*(t), AR(t)) and connect the dynamical R-layer scenario to future multi-band, multi-epoch observations such as SKA, LISA, and CMB-S4. This Volume provides the dynamical counterpart to Volumes LXXXIV-A and LXXXIV-B and establishes a unified information-geometric description of evolving black holes and their observational signatures.