Particle Size from Trapped Vibration: Proton Radius, Baryon-Meson Pattern, and Fine Structure Constant from the Clausius-Mossotti Framework

We derive a base equation E = mc² = hf = (4/3) hc/ (2R) uniting mass-energy, energy-vibration, and D=3 sphere geometry. Combined with the unique density profile ρ (r) ∝ 1/r² (equal energy per shell, connected to the 3D Kepler law), this yields a zero-parameter prediction for the proton charge radius: Rₚ = (4/3) π × ℏ/ (mₚ c) = 0. 881 fm (0. 5% match with electron-scattering measurement). A baryon-meson size pattern is identified: baryons use (4/3) π (sphere volume), while the pion uses e^ (-3/4) (confinement wall transmission), giving R_π = 0. 668 fm (1. 3%). An observation α = (π²/OE + D/ (4π) ) × mₑ/mₚ = 1/137. 045 (0. 006%) eliminates α from the Bohr formula. Tested on 8 hadrons and 6 hydrogen-like ions. No free parameters. Paper 2026t in the Speed Gap (CM) Framework series.