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.
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