We present a strong-coupling approach to the theory of high-temperature superconductivity based on the observation of a quantum critical point in the plaquette within the $t,t′$ Hubbard model. The crossing of ground-state energies in the $N=2–4$ sectors occurs for parameters close to the optimal doping. The theory predicts the maximum of the $d_{x2}−y_{2}$-wave order parameter at the border between localized and itinerant electron behaviors and gives a natural explanation for the pseudogap formation via the soft-fermion mode related to local singlet states of the plaquette in the environment. Our approach follows the general line of resonating valence-bond theory stressing a crucial role of singlets in the physics of high-$T_{c}$ superconductors but focuses on the formation of local singlets, similar to phenomena observed in frustrated one-dimensional quantum spin models.