  Nuclear-structure effects often provide an irreducible theory error that
prevents using precision atomic measurements to test fundamental theory. We
apply newly developed effective field theory tools to Hydrogen atoms, and use
them to show that (to the accuracy of present measurements) all nuclear
finite-size effects (e.g. the charge radius, Friar moments, nuclear
polarizabilities, recoil corrections, Zemach moments {\it etc.}) only enter
into atomic energies through exactly two parameters, independent of any
nuclear-modelling uncertainties. Since precise measurements are available for
more than two atomic levels in Hydrogen, this observation allows the use of
precision atomic measurements to eliminate the theory error associated with
nuclear matrix elements. We apply this reasoning to the seven atomic
measurements whose experimental accuracy is smaller than 10 kHz to provide
predictions for nuclear-size effects whose theoretical accuracy is not subject
to nuclear-modelling uncertainties and so are much smaller than 1 kHz.
Furthermore, the accuracy of these predictions can improve as atomic
measurements improve, allowing precision fundamental tests to become possible
well below the 'irreducible' error floor of nuclear theory.
