  We systematically explore the exquisiteness of Bardeen-Cooper-Schriefer(BCS)
theory where the BCS-type electron-phonon interaction is unambiguously
reinforced as the only viable superglue in cuprate superconductors because
phonon-induced scattering is effectively nil for Cooper pairs (in its original
form), and also phonons are never required to Bose-condense. Here, we prove
that (i) the Cooper-pair binding energy can be strengthened to obtain high
superconductor transition temperature (Tsc) and (ii) the existence of a
generalized electron-phonon potential operator that can induce the
finite-temperature quantum phase transition between superconducting and strange
metallic phases. To lend support for this extended BCS theory, we derive the
Fermi-Dirac statistics for Cooper-pair electrons, which correctly captures the
physics of strongly bounded Cooper-pair break up with respect to changing
temperature or superconductor gap (gap-BCS). Finally, we further extend the BCS
formalism within the ionization energy theory to prove (iii) the existence of
optimal doping that has maximum Tsc(x_optimum) or gap-BCS (x_optimum), and (iv)
that the specific heat capacity jump at Tsc in cuprates is due to
finite-temperature quantum phase transition. Along the way, we expose the
precise microscopic reason why predicting (not guessing) a superconductor
material properly is in general a hard problem within any theory that require
pairing mechanism.
