Abstract
Uniform black-hole area spectra are commonly postulated directly. Here, we show that an affine Schwarzschild area sequence can emerge within a local covariant squared-energy model. Within the positive isotropic homogeneous sector, an ultrarelativistic equal-weight virial condition selects a linear interaction whose independent squared contribution produces an isotropic oscillator in the squared-energy channel. A separate Schwarzschild projection maps the resulting spherical sequence to uniformly spaced geometric-area levels. This construction establishes a model-internal correspondence between ultrarelativistic scaling, linear interaction, exact spectral solvability, and a conditional black-hole projection. The statistical construction treats a stationary black hole as a fixed sector of distinguishable equal-energy ground-state units. The total statistical area of the completed ground-state distribution is assumed to limit the information capacity available at every relational position. Because the exterior description does not determine how formation information is assigned among these positions or among their admissible microscopic realizations, every position retains the full capacity permitted by the completed sector. Independent ordered assignments then generate a finite microscopic shell. Maximum ignorance on this previously specified shell yields a uniform microcanonical density matrix whose entropy equals Boltzmann’s constant times the logarithm of the number of admissible assignments. When the sector enlarges, the information capacity available at every pre-existing relational position also increases. This provides a density-matrix interpretation of entropy nondecrease in sector-enlarging processes, compatible with the second law of black-hole mechanics. The framework connects exterior no-hair simplicity, structural information conservation, Schwarzschild and Kerr sectors, fixed-sector additivity, and merger cross terms. At finite Planck constant, the assignment space and entropy remain finite. In the nonphysical statistical-singularity limit, the elementary spectral scales vanish while the number of effective positions, the assignment space, and the entropy diverge. The framework is restricted to stationary fixed-macrostate sectors and does not describe Hawking evaporation, the Page curve, or a dynamical information-recovery mechanism.