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This work presents a Quantum Measurement Units (QMU) ledger-based decomposition of the hydrogen $2\mathrm{S}$--$6\mathrm{P}$ transition, using the Aether Physics Model (APM) as a geometric framework for interpreting atomic structure. The analysis is anchored to the 2026 high-precision spectroscopic measurement yielding a proton charge radius of $r_p = 0.8406(15)\,\mathrm{fm}$. The hydrogen spectrum is reformulated as a perturbative expansion in the fine-structure constant $\alpha$ on a single base frequency scale $F_q = m_e c^2 / h$. The conventional decomposition into Dirac, radiative (Lamb shift), and finite-size contributions is translated into QMU ledger form using the Compton wavelength $\lambda_C = h/(m_e c)$ and the invariant relation $F_q \lambda_C = c$. A central result is the derivation of the proton finite-size frequency shift in QMU form:\[\Delta \nu_{\mathrm{finite}}(2S)=-\frac{\pi^2}{3}\,\alpha^4\left(\frac{m_r}{m_e}\right)^3\left(\frac{r_p}{\lambda_C}\right)^2F_q,\]where $m_r$ is the reduced mass. This expression is obtained by direct substitution from the conventional bound-state QED formulation using $\hbar = h/(2\pi)$ and $\lambda_C = h/(m_e c)$, preserving dimensional and scaling consistency. Inversion of this relation provides a direct extraction of the proton charge radius from the measured frequency shift, yielding agreement with experiment at the $10^{-3}\,\mathrm{fm}$ level when recoil is included through the factor $(m_r/m_e)^3$. Within the QMU framework, the finite-size correction is interpreted as a geometric traversal mismatch between the electron’s bound-state path and the proton’s distributed Aether structure. The proton radius emerges as a dimensionless geometric ratio $r_p/\lambda_C$, linking nuclear structure directly to the electron Compton scale without introducing additional fundamental lengths. The paper also establishes a ledger identity flow connecting the Aether unit closure relation\[A_u \cdot \mathrm{curl} = {F_q}^2 {\lambda_C}^2\]to the observed hydrogen transition frequency, demonstrating that atomic structure can be expressed as successive geometric perturbations of a single invariant frequency scale. Predictions include stability of the ratio $r_p/\lambda_C$ across hydrogenic systems, sensitivity of hyperfine structure to distributed charge anisotropy, and consistency between electronic and muonic hydrogen when reduced-mass effects are treated as a coupled inertial ledger. This work provides a geometrically unified interpretation of the proton radius within the QMU/APM framework and identifies experimental pathways for testing traversal-based effects in precision spectroscopy.