  We use effective field theory to compute the influence of nuclear structure
on precision calculations of atomic energy levels. As usual, the EFT's
effective couplings correspond to the various nuclear properties (such as the
charge radius, nuclear polarizabilities, Friar and Zemach moments {\it etc.})
that dominate its low-energy electromagnetic influence on its surroundings. By
extending to spinning nuclei the arguments developed for spinless ones in {\tt
arXiv:1708.09768}, we use the EFT to show -- to any fixed order in $Z\alpha$
(where $Z$ is the atomic number and $\alpha$ the fine-structure constant) and
the ratio of nuclear to atomic size -- that nuclear properties actually
contribute to electronic energies through fewer parameters than the number of
these effective nuclear couplings naively suggests. Our result is derived using
a position-space method for matching effective parameters to nuclear properties
in the EFT, that more efficiently exploits the simplicity of the small-nucleus
limit in atomic systems. By showing that precision calculations of atomic
spectra depend on fewer nuclear uncertainties than naively expected, this
observation allows the construction of many nucleus-independent combinations of
atomic energy differences whose measurement can be used to test fundamental
physics (such as the predictions of QED) because their theoretical
uncertainties are not limited by the accuracy of nuclear calculations. We
provide several simple examples of such nucleus-free predictions for
Hydrogen-like atoms.
