High-Frequency Suppression and Pseudo-Inversion of Weighted Frame Operators near Black Hole Horizons: A Rindler Model

We present a weighted frame operator framework for analyzing spectral stability near black hole horizons, using torus-compactified Rindler spacetime as a mathematically rigorous toy model. Employing horizon-adapted weights $w(x) = x^\alpha$ encoding surface gravity $\kappa$ (with $\alpha \sim \kappa / 2\pi$), we establish (1) uniform boundedness with explicit $\Gamma(\alpha + 1)$ estimates, (2) high-frequency suppression at the rate $|n|^{-\alpha}$, and (3) controlled pseudo-inversion with quantitative error bounds. These results provide a provable analytic foundation for horizon-modified quasi-normal modes and for the stabilization of local energy flux. Extensions to more general geometries and weights are outlined, with the core methods accessible to both mathematicians and physicists. We also discuss possible connections between these mathematical findings and open questions in black hole thermodynamics, quantum information, and the information paradox, with an emphasis on the interpretive nature of such links. Quantitatively, we find that the high-frequency suppression scales as $\|T_w f\| \sim |n|^{-\alpha}$ with $\alpha \approx \kappa/(2\pi)$, providing a rigorous spectral analogue of horizon thermal damping.