Near-Horizon Phase-Space Saturation in the Schwarzschild Geometry: Universal Shell Thickness from a Planck-Granular Effective Description

This paper investigates the near-horizon structure of the Schwarzschild geometry within an effective description in which the vacuum possesses Planck-scale granularity. The central result is a universal proper thickness of the near-horizon layer, derived independently by two methods and shown to be independent of black hole mass for M ≫ m_P. The refractive index n(r) = α(r)⁻² diverges at the horizon, where α(r) is the Schwarzschild lapse. This divergence is interpreted not as a geometric artefact but as the saturation of the propagating phase-space capacity of the vacuum degrees of freedom. A covariant phase-space analysis shows that the effective density of propagating modes scales as α(r)⁻³; this scaling is confirmed to be physically observable via Unruh–DeWitt detector response and energy flux arguments, and is grounded in the standard Bogoliubov transformation framework. A WKB spectral saturation condition — equating the local wavelength of dominant Hawking-scale modes to the Planck granularity floor — yields a maximum refractive index n_max ~ r_s/ℓ_P and a coordinate shell thickness δ ~ ℓ_P. Independently, the Israel junction conditions applied to a shell separating a Minkowski interior from the Schwarzschild exterior give a volumetric thickness δ_vol = ℓ_P²/(4πr_s). Integrating the exterior radial metric over this volumetric thickness yields the universal proper thickness L = ℓ_P / √π ≈ 0.5642 ℓ_P independent of black hole mass. The Schwarzschild radius r_s cancels exactly between the WKB spectral bound and the Israel junction result — universality is a consequence of the junction geometry, not an assumption. The jamming equation of state P_s ∝ (φ_c − φ)^(−β) is derived from an effective near-critical scalar field action rather than postulated. The stability bound β ≫ 1 follows explicitly from the linearised effective potential. The Minkowski void interior is motivated as the endpoint of a holographic flow in which the α⁻³ mode-density divergence forces projection of bulk degrees of freedom onto the boundary layer, consistent with the covariant entropy bound and brick-wall regularisation. This mechanism provides a phenomenological realisation of the AMPS firewall without invoking entanglement monogamy directly. Three independent calculations — WKB index saturation, Israel junction conditions, and semiclassical stress-tensor energy density — all resolve at the same physical scale, the Planck density ρ_P. The near-horizon reflectivity is R ~ exp(−2π√π r_s/ℓ_P). All corrections are parametrically suppressed by ℓ_P/r_s and consistent with all current observations. Universality is robust under alternative UV prescriptions, non-Minkowski interiors, and smooth shell profiles. This paper is part of a programme treating the gravitational vacuum as an effective optical medium. It connects directly to the companion ADM decomposition of the Plebański refractive index (doi:10.5281/zenodo.19510746), which establishes that all near-horizon divergences in the dielectric vacuum are lapse-sector effects — a result that underpins the mode-density scaling derived here.