The study of multiple n-CuO \(_{2}\) -layer (multilayer) cuprate high-temperature superconductors in the singlet-bond (SB) superconductivity theory reveals that the large enhancements of superconducting transition temperatures \(T^{(n)}_{c}\) are the result of increased condensation energy which is added to the layer by the motion of interlayer-tunneled condensed SB-pairs from the adjacent layers. This mechanism is endorsed by the experimental observations of the large enhancements of muon-spin-relaxation rate \(\sigma ^{(n)}(0)\) proportional to inverse squared London penetration depth at \(T = 0\) , \(\sigma ^{(n)}(0) \propto 1/\lambda ^{(n)}(0)^{2}\) , where \(2 |\Psi ^{(n)}|^{2}\) in \(1/\lambda ^{(n)}(0)^{2} = 4 \pi 2 |\Psi ^{(n)}|^{2} e^{2}/mc^{2}\) ( \(\Psi ^{(n)}\) the order-parameter of Ginzburg-Landau free energy) is enhanced by the same factor as the condensation energy increase due to the kinetic energy increase. These enhancements are a very unique effect particular to multilayer superconductors where while SB-pairs in the normal-state pseudogap insulator phase are strictly two-dimensional and confined to one CuO \(_{2}\) -layer, SB-pairs in the superconductng phase can tunnel to adjacent layers where they increase both condensation and kinetic energies through the coherent motion. The calculations of \(T^{(n)}_{c}\) and \(1/\lambda ^{(n)}(0)^{2}\) give the results; (1) \(T^{(n)}_{c}\) and \(1/\lambda ^{(n)}(0)^{2}\) increase from \(n = 1\) to \(n = 3\) , decrease from \(n = 3\) to \(n = 5\) and then level off for \(n \ge 6\) , having the maxima \(T^{(3)}_{c}\) and \(1/\lambda ^{(3)}(0)^{2}\) always at \(n = 3\) , (2) the maximum superconducting transition temperature \(T^{max}_{c}\) that can be achieved at the ambient pressure by \(T^{(3)}_{c}\) in any multilayer cuprates is bounded by the maximum condensation onset-temperature \(T_{0}^{max} \sim 156 K\) at the optimal doping \(\delta _{m} \sim 0.15\) , (3) the multilayer enhancements \(T^{(n)}_{c} / T^{(1)}_{c}\) as a function of n of all the multilayer cuprate superconductors examined are the same and can be expressed by a singe curve (except with one correction to the low \(T^{(1)}_{c}\) values as explained in the text). This \(T^{(n)}_{c} / T^{(1)}_{c}\) - n curve is well reproduced by the multilayer SB superconductivity with the restricted entropy (RE) imposed on the pseudogap insulator phase.
