Universal Proper Thickness of the Schwarzschild Quantum Hair: A Planck-Scale Grain Model and Israel Junction Derivation

This work derives a universal Planck-scale proper thickness for the Schwarzschild horizon within an effective model of the vacuum as a granular, polarisable medium. Using the optical-metric formulation of Jerzy Plebański, the Schwarzschild refractive index diverges at the horizon as a pure lapse effect. Imposing a local Planck-length cutoff on propagating modes, and identifying the relevant frequency scale for horizon transport with the curvature scale ω∼c/rsω ~ c/r_sω∼c/rs, yields a saturation refractive index nmax⁡∼rs/ℓPnₘₐₓ ~ r_s/_Pnmax∼rs/ℓP without introducing nonlocal microphysics. Independently, the Werner Israel junction conditions for a thin shell with Minkowski interior give a volumetric thickness δvol=ℓP2/(4πrs)δᵥₒₗ = _P^2/(4π r_s)δvol=ℓP2/(4πrs). Integrating the Schwarzschild proper distance across this layer produces a universal physical thickness L=ℓPπ≈0.564 ℓP,L = {_P}{√π} ≈ 0.564\,_P,L=πℓP≈0.564ℓP, independent of black hole mass for M≫mPM m_PM≫mP. The universality arises from an exact cancellation of rsr_srs between (i) the curvature-scale saturation of propagating modes and (ii) the near-horizon metric factor in the proper-distance integral. The result is therefore not imposed but emerges from the interplay of local Planck-scale physics and classical geometry. The horizon layer is interpreted as a gravitationally jammed Planck-grain monolayer (“quantum hair”). The factor-of-four discrepancy between naive grain counting and Bekenstein–Hawking entropy admits a natural explanation in terms of isostatic constraint counting in the Edwards–Oakeshott framework. Evaporation is modelled as WKB gradient-index mode mixing, reproducing the Hawking temperature scale. This construction is formulated within an effective description that retains the Schwarzschild metric and junction conditions exactly, augmented by Planck-scale granularity as a UV hypothesis. Several elements (mode selection, granular model, entropy interpretation) are physically motivated but not derived from a complete quantum gravity theory.