Geometric Resonance and Binary Information Closure: A First-Principles Derivation of the Proton Charge Radius from Relational Projection Principles

The "Proton Radius Puzzle" has revealed a persistent discrepancy between proton charge radius measurements obtained via electron scattering and those from muonic hydrogen spectroscopy. In this work, we propose a first-principles, phenomenological model based on geometric resonance and relational projection to derive the proton radius analytically.

We postulate that baryonic stability arises from an orthogonal decomposition of the Compton frequency, governed by a binary bifurcation condition with quantum number n=2n = 2n=2. This leads directly to the expression rp=4λCr_p = 4 \lambda_Crp=4λC, yielding a theoretical radius of 0.8412 fm—in 99.98% agreement with the CODATA 2018 recommended value of  without any free parameters.

We further identify a dimensionless structural constant κ≈4\kappa \approx 4κ≈4, indicating that the proton radius is set by a fundamental relativistic confinement scale, not by empirical curve fitting. These results support the hypothesis that hadronic structure obeys universal principles of informational symmetry and geometric quantization, consistent with prior relational models applied to cosmology and atomic valence.