Black Hole Thermodynamics. Energy, Time and Structural Closure.

This article analyzes a structural consequence of black hole thermodynamics. Starting from the Schwarzschild--Bekenstein--Hawking relations among mass, energy, horizon radius, area, temperature, entropy and informational capacity, it argues that these magnitudes should not be read as isolated formulae, but as a linked physical configuration. In a finite-mass Schwarzschild black hole, mass determines geometry through G, geometry determines horizon area, and horizon area determines a finite entropic capacity through the Planck scale. The central claim is that a finite configuration causally delimited by a horizon cannot be interpreted as an ontological continuum of infinitely many physically distinguishable states. The mathematical continuum remains a powerful effective tool, but its formal continuity should not be identified without restriction with the ultimate physical support of black-hole configurations. Special attention is given to the structural role of ℏ. The quantum of action appears simultaneously in Bekenstein--Hawking entropy, in the Planck area scale, in Hawking temperature and in the relation E= ℏω. This shows that the configuration is not merely classically bounded, but quantum-normalized. The article also analyzes Hawking evaporation and the limit M → 0. If this limit is forced continuously, energy, horizon radius, area, entropy and entropic capacity tend to zero, while Hawking temperature diverges. This is interpreted not as a physically admissible final state, but as the breakdown of the continuous parametrization. Complete evaporation, if it occurs, must therefore involve a last admissible horizon state and a physically distinguishable final transition. Finally, the work formulates minimum admissibility premises for future microscopic theories of black holes. Any such theory must reproduce the full linked chain of mass, geometry, entropy, horizon accounting, quantum normalization, admissible transitions and exterior compatibility. The conclusion is not that the mathematical continuum is invalid, but that it survives only as an effective description, while physical admissibility reveals a finite and structurally restricted support.