Extended Generalized Uncertainty Principle: Mathematical Framework for Astrophysical Applications

We present a comprehensive mathematical framework for extended Generalized Uncertainty Principle (GUP) formulations and their systematic application to astrophysical systems. Through rigorous dimensional analysis and comparative evaluation of competing theoretical models, we establish a unified approach for incorporating quantum gravity effects into macroscopic astronomical observations. Our analysis reveals fundamental scaling relationships between microscopic quantum gravity parameters and observable astrophysical phenomena, providing the theoretical foundation for observational constraints on minimal length scales. The framework developed here offers a systematic methodology for translating Planck-scale physics into testable predictions for neutron star structure, black hole thermodynamics, and gravitational wave signatures. We demonstrate that while individual quantum gravity corrections appear negligible, their cumulative effects in extreme astrophysical environments can potentially reach observational thresholds, particularly in gravitational wave observations of binary neutron star mergers, where tidal deformability measurements offer unprecedented sensitivity to the underlying equation of state modifications.