Black Hole Thermodynamics and Finite Degrees of Freedom

This article analyzes a structural consequence of black hole thermodynamics. Starting from Bekenstein--Hawking entropy and Hawking temperature, it argues that an event horizon with finite entropy requires a finite capacity of physically accessible states. Therefore, a physical continuum understood naively as an infinitely divisible support with independent degrees of freedom at every point cannot be the ultimate description compatible with the finite entropy of black holes. The work distinguishes between the mathematical continuum as an effective approximation and the physical continuum as an ontological assumption. It shows that Hawking temperature belongs to the horizon and should not be interpreted as a volumetric temperature of an increasingly hot interior. The article also places this conclusion in relation to compatible frameworks such as the holographic principle, loop quantum gravity, causal sets, and other structural approaches in which the degrees of freedom of space are limited, quantized, or reorganized. The central thesis is that Bekenstein--Hawking thermodynamics points necessarily toward finite degrees of freedom and an effective minimum scale, even though the continuum remains a useful mathematical approximation at macroscopic scales.