$q$-Deformed Quantum Mechanics and the Thermodynamics of Black Hole/White Hole Spectral pair

Abstract In this work, we investigate the thermodynamics of Schwarzschild black and white holes within a $q$-deformed Wheeler--DeWitt framework. By introducing a $q$-deformed Heisenberg--Weyl algebra at a root of unity, we derive a finite-dimensional Hilbert space, a bounded mass spectrum, and an adiabatic invariant leading to a bounded entropy-mass relation. The deformation results in a universal logarithmic correction, as well as a minimum temperature and a maximum entropy that matches the de Sitter bound. Also, we examine the interpretation of a cold remnant, which is dynamically stable because its radiation rate approaches zero, even though its heat capacity remains negative. We also explore the holographic implications of this limited entropy. 
Our results thus provide a consistent semiclassical picture, where quantum deformation naturally introduces an entropy bound, avoids divergences at the final evaporation stage, and suggests a smooth transition from quantum gravity to cosmology.