Scale Relation Between Nuclear Radius and Atomic Radius via the Fine-Structure Constant

The fine-structure constant α occupies a central position in fundamental physics as a dimensionless parameter characterizing the strength of the electromagnetic coupling. Despite its structural role in quantum electrodynamics, its deep origin remains an open question. In this work, we highlight a scale relation connecting α to three independent experimental quantities: the proton–electron mass ratio (m_p/m_e), the proton charge radius (r_p), and the Bohr radius (a_0). We show that the constant can be written in the form: α = (1/4) (m_p/m_e) (r_p/a_0) where the factor 1/4 appears as a consequence of a minimal quarter-wave type kinematic relation applied to proton structure. Numerical substitution using current experimental values reproduces the measured value of α with remarkable precision. When adopting the ideal value derived from the proposed kinematic condition, the equality becomes exact. This approach does not constitute a dynamical derivation of α, but rather highlights a specific numerical organization suggesting a scale correlation between nuclear structure and atomic geometry.