  Proofs are developed to explicitly show that the ionization energy theory
(IET) is a renormalized theory, which mathematically exactly satisfies the
renormalization group formalisms developed by Gell-Mann-Low, Shankar and
Zinn-Justin. However, the cutoff parameter for IET relies on the energy-level
spacing, instead of lattice spacing in \textbf{k}-space. Subsequently, we apply
the earlier proofs to prove that the mathematical structure of the
ionization-energy dressed electron-electron screened Coulomb potential is
exactly the same as the ionization-energy dressed electron-phonon interaction
potential. The latter proof is proven by means of the second-order
time-independent perturbation theory with the heavier effective mass condition,
as required by the electron-electron screened Coulomb potential. The outcome of
this proof is that we can derive the heat capacity and the Debye frequency as a
function of ionization energy, which can be applied in strongly correlated
matter and nanostructures.
