A Hamiltonian H(Γ) applicable to cuprate HTS, with a doping dependent pairing interaction Γ(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 kk < 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 ∆(x), and the pseudogap ∆ 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.