  A Hamiltonian H(\Gamma) applicable to cuprate HTS, with a doping dependent
pairing interaction \Gamma(x) = V(x) + U(x), is linked to a Cu3d-O2p state
probability model(SPM). A consequence of doping induced electron hopping, the
SPM mandates that plaquettes with net charge and spin form in the CuO plane,
establishing an effective spin-singlet exchange interaction U(x). The U(x) is
determined from a set of probability functions that characterize the occupation
of the single particle states. An exact treatment of the average static
fluctuation part of H shows that diagonal matrix elements U_{k k} < 0 produce
very effective pairing, with significant deviation from the mean field
approximation, which also depends on a phonon-mediated interaction V. This
deviation is primarily responsible for the diverse set of HTS properties. The
SC phase transition boundary T_c(x), the SC gap \Delta(x), and the pseudogap
\Delta_{pg}(x) are fundamentally related. Predictions are in excellent
agreement with experiment, and a new class of HTS materials is proposed. Large
static fluctuation results in extreme HTS and quantum criticality.
