Black Hole Thermodynamics, Quantum Closure of the Horizon, and Admissible RadiiStructural Closure.

This article analyzes a structural consequence of the thermodynamics of Schwarzschild black holes. Starting from the relations between mass, energy, horizon radius, area, temperature, entropy and entropic capacity, it argues that the event horizon should not be understood as an isolated magnitude or as a passive geometric surface, but as a closed causal boundary belonging to a physically linked configuration. The central thesis is that the chain formed by mass, radius, area, temperature, entropy and capacity does not describe independent variables, but a single functionally determined configuration. The global mass fixes the Schwarzschild radius; the radius fixes the area; the area fixes the entropy; and the entropy fixes the dimensionless capacity of the horizon. Therefore, any physical condition imposed on the horizon propagates to the entire configuration. The work argues that the capacity of the horizon, normalized by the Planck scale, should not be interpreted as a simple continuous measure of area, but as a quantum-normalized boundary capacity. If the horizon is a closed causal boundary, its admissibility must satisfy a closure condition. This condition is formulated through the integrality of the dimensionless capacity of the horizon, leading to a discrete family of admissible radii. The article distinguishes between the usual analogy with one-dimensional closed quantum systems — standing waves, cavities, particles on a ring or Bohr-type models — and the proper structure of the Schwarzschild horizon, whose topology is that of a two-dimensional sphere. For this reason, the closure of the horizon is interpreted not as a condition on a closed path, but as an integrality condition on a closed surface. From this condition it follows that the horizon radius cannot take any continuous value if it is to correspond to a physically admissible quantum boundary. The mathematical continuum may remain an effective tool in the macroscopic regime, but it should not be identified with an ontological continuum of physically distinguishable horizons. Once the exact closure radius is fixed, the mass appears as a consequence of the Schwarzschild geometry itself. Thus, the discreteness of the horizon propagates functionally to the entire chain: area, entropy, capacity, mass, energy and temperature. Quantization does not affect an isolated variable, but the complete configuration. The article also develops an interpretation of Hawking temperature as a boundary temperature. Under this reading, temperature does not measure internal heat or physical evaporation from the interior, but the thermal response of a causal boundary with an exactly determined maximum capacity. A larger black hole has more boundary degrees of freedom and a lower temperature; a smaller black hole has less capacity to absorb information and a stronger thermal response. Temperature is therefore interpreted as the resistance or congestion of a finite boundary faced with the flow attempting to enter it, not as evidence of real evaporation. Consequently, the work rejects the physical evaporation of a black hole understood as a real outward escape of radiation from within the horizon. Hawking evaporation may be understood, at most, as an effective semiclassical description associated with the boundary and the exterior. Moreover, even if an effective loss of energy were considered, it could not be ontologically continuous down to zero mass, but would have to be expressed as a succession of transitions between admissible levels. Finally, the article analyzes the closure action of the horizon. From the closure condition of the dimensionless capacity, it follows that the product of mass, limiting velocity and radius has a natural quantization as a characteristic closure action. This reading allows the horizon to be interpreted not only as a surface of discrete area, but as a complete configuration whose mass, energy, radius, entropy, temperature and exterior gravity are linked by the same integer of admissibility. The general conclusion is that any microscopic theory compatible with Bekenstein--Hawking thermodynamics must simultaneously reproduce the linked mass--geometry--entropy--capacity chain, explain the horizon as a closed causal boundary, justify the quantum normalization of its accounting and recover the admissible radii as closure solutions. The mathematical continuum retains its value as an effective tool, but the horizon belongs to the domain of quantum closure.