Shadow dependent phenomenology framework for rotating black hole metric

Abstract

We establish a formal thermodynamic-optical duality that bridges the semiclassical quantum evaporation of black holes with their classical macroscopic geometry. The physical viability of this framework is anchored by a stable multivariate coordinate transformation and a non-vanishing Jacobian determinant, which allows for a diffeomorphic inversion mapping that decouples intrinsic physical quantities such as bare mass from the unobservable spacetime interior. By re-parameterizing black hole properties entirely in terms of the analytical shadow radius ($R_{sh}$), we derive explicit, observable-based expressions for the weak deflection angle, Hawking temperature, and integrated semiclassical luminosity. We demonstrate the framework's predictive utility by applying it to standard Kerr, Kerr-MOG (Scalar-Tensor-Vector Gravity), and rotating Horndeski spacetimes. Our results provide a definitive mathematical solution to parameter degeneracy, revealing that distinct fundamental fields (vector vs. scalar) leave unique observational fingerprints on far-field astrometry and horizon-scale quantum thermodynamics. By confronting these models with Event Horizon Telescope (EHT) M87* data, we show that this formalism successfully breaks mass-parameter degeneracies, offering a robust and computationally efficient operational tool for testing the Kerr hypothesis and probing modified gravity theories with next-generation very-long-baseline interferometry (VLBI).