The inputs of the formulas for the hydrogen energy levels in the paper are: The Rydberg constant, the fine structure constant, the electron-to-proton mass ratio

The inputs of the formulas for the hydrogen energy levels in the paper are: The Rydberg constant, the fine structure constant, the electron-to-proton mass ratio, the electron-to-muon mass ratio, the Compton wavelength of the electron, and some nuclear properties (charge radius, Friar radius, and nuclear polarizability). All inputs except the nuclear properties are as fundamental as it gets according to our current understanding of physics (note that the Rydberg constant and Compton wavelength are simple combinations of other physical constants). Nuclear physics is dominated by quantum chromodynamics which is not nearly as well developed as QED. The constants are determined by fitting the theory to the best available measurements (not only in hydrogen). This is exactly what fundamental constants do: They convert unit-less theory expressions into measurable quantities.