On the theory of supermodulation of the superconducting order parameter created by structural supermodulation of apex distance in optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$
Recently using Scanning Josephson Tunneling Microscopy (SJTM) in the group of Séamus Davis a super-modulation of the superconducting order parameter induced by super-modulation of the distance $δ$ between planar Cu and apical O was observed in [O'Mahony et. al., PNAS Vol. 119(37), e2207449119 (2022)]. The authors conclude that: "concurrence of prediction from strong correlation theory ... with these observations indicates that ... super-exchange is the electron pairing mechanism of Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$". In our current theoretical study we use the LCAO approximation, Hilbert space spanned on 5 atomic orbitals: Cu$4s$, Cu$3d_{x^2-y^2}$, O$2p_x$, O$2p_y$, O$2p_z$. For the only super-exchange amplitude $J_{sd}$ we use the Kondo double electron exchange between Cu$4s$ and Cu$3d_{x^2-y^2}$ orbitals and its anti-ferromagnetic sign is determined by adjacent to the Cu ion in-plane O orbitals. Within this approximations we have calculated: "Measured dependence of \dots electron-pair density $n_p$ on the displacement $δ$ of the apical O atoms from the planar Cu atoms" depicted in Fig. 5(C) of O'Mahony \textit{et. al.} and obtained an acceptable accuracy. As a whole, the analyzed SJTM experiment is one of the best confirmations of J. Röhler [J. Röhler, Physica B: Cond. Matter Vol. 284-288, 104 (2000)] idea that hybridization of Cu4$s$ with conduction band leads to increasing of $T_c$. The lack of an alternative explanations for SJTM data $n_p$ versus $δ$ and shape-$T_{c,max}$ correlations for the description of the critical temperature of optimally doped cuprates for several decades on the background of a simple view gives a hint that the long sought pairing mechanism has possibly been found and the Kondo exchange interaction as a property of strongly correlated quantum matter deserves further attention in the physics of layered cuprates.