  A long-distance effective theory of hydrogen-like atoms, dubbed the
relativistic Ritz approach was recently introduced and some its theoretical
consequences were explored. In this article, the relativistic Ritz approach is
used to fit extant measurements of atomic hydrogen and deuterium transitions
using information-theoretic analyses. As a result, the fine-structure constant
($\alpha$), a fundamental parameter of the Standard Model, may be determined
simultaneously with the ionization energies of hydrogen and deuterium,
$E_I^{\text{(H)}}$ and $E_I^{\text{(D)}}$. The best hydrogen analysis yields
$\alpha^{-1} = 137.035\,999\,185(25)$, in good agreement with the value
obtained by other methods and without relying on a separately determined
Rydberg constant. From the same analysis, I find that $E_I^{\text{(H)}} =
13.598\,434\,599\,684(25)\,\text{eV}$, an improvement of two orders of
magnitude in precision compared to previous determinations and in agreement
with the Standard Model prediction at 1.8 parts per trillion. The best
deuterium analysis yields $E_I^{\text{(D)}} =
13.602\,134\,636\,543(31)\,\text{eV}$, agreeing with the Standard Model at 2.3
parts per trillion. This study demonstrates how the relativistic Ritz approach
can be used for testing the Standard Model with the spectra of hydrogen-like
atoms.
