What question did this study set out to answer?

The aim is to derive a unifying equation that predicts particle sizes based on energy and geometrical principles.

April 22, 2026Open Access

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

Key Points

The aim is to derive a unifying equation that predicts particle sizes based on energy and geometrical principles.
Developed a base equation integrating mass-energy, energy-vibration, and sphere geometry.
Examined density profiles and their relationships to baryon and meson sizes.
Tested predictions against measurements from hadrons and hydrogen-like ions.
Predicted proton charge radius aligns closely with experimental measurements at 0.5%.
Identified distinct size patterns for baryons and pions, with deviations of 1.3%.
Derived fine structure constant with 0.006% accuracy, independent of the Bohr formula.

Abstract

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.

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