R-layer Mode Theory LXXXIV-A: Observational Predictions, Numerical Analysis, and Extensions of the Internal Structure

his Volume LXXXIV-A extends the R-layer Mode Theory description of black hole interiors developed in Volume LXXXIV. It constructs a mathematically rigorous framework for observational predictions and numerical analysis of the CUP-dominated core. A fully dimensionless formulation is introduced, together with the physically admissible parameter space for the background R-layer profile. Strict conditions for the existence of an inner potential well and an outer barrier in the R-layer effective potential are derived. The analysis includes the effective refractive index, photon sphere, and shadow radius, and quantifies their deviations from the Schwarzschild prediction. The quasi-normal mode spectrum is computed using a third-order WKB approximation, supported by numerical checks, and the parameter regions where R-layer corrections become potentially observable are identified. The inner-well and outer-barrier structure generically leads to echo-like signals, whose amplitudes are shown to be strongly suppressed for realistic CUP configurations. A Fisher-matrix framework is formulated to map the CUP parameters to observable quantities, including the shadow radius and low-lying quasi-normal mode frequencies. The resulting parameter degeneracies are quantified, providing a first estimate of how well the internal CUP structure can be constrained by combined electromagnetic and gravitational-wave observations. This Volume establishes the technical basis for future studies of R-layer phenomenology and its confrontation with high-precision observational data.