The Proton as a Phase Resonator: Quarter-Wave Origin of the Compton Wavelength

In a previous work [1], we highlighted a kinematic relation linking the measured charge radius of the proton 𝑅𝑝to a frequency 𝑓, whose numerical value coincides remarkably well with the Compton frequency of the proton: 𝑓=𝑐(𝜋/2) 𝑅𝑝This relation can also be rearranged into a compact geometric form: the non-reduced Compton wavelength of the proton is equal to one quarter of the equatorial perimeter associated with 𝑅𝑝, namely 𝜆𝐶,𝑝=𝜋2𝑅𝑝.The present article continues this analysis by proposing a wave interpretation of the appearance of the factor 𝜋/2. Relying on standard results from wave physics (standing waves, mixed boundary conditions, resonators), we show that in a wave-based reading this factor is neither arbitrary nor fitted, but constitutes the universal signature of a fundamental quarter-wave mode associated with a central phase constraint. Within this framework, the nucleus is no longer interpreted as a rigid boundary, but as a phase tuner. The same structure is immediately extended to the neutron by simple substitution, providing a coherent length scale associated with its Compton frequency.