A Quantum Geometric Relation Between the Proton Charge Radius and Its Compton Frequency

This paper presents a remarkably precise numerical relation between the proton charge radius,measured experimentally, and its fundamental Compton frequency. Using the radius obtained frommuonic-hydrogen spectroscopy (2010) and a simple geometric relation involving the factor 𝜋/2, wealso demonstrate an equality between one quarter of the proton’s electromagnetic equatorialcircumference and the Compton wavelength. We extend this observation by showing that the sameequation reproduces the electron Compton frequency (i.e., the electron rest-frequency). We then showthat it also applies to the hydrogen atom and, in this framework, leads to the Hartree energy value,27.2 eV. These observations establish an intriguing bridge between a classical geometric quantity anda relativistic quantum property.