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d-wave superconductivity in the virtual-electron pair quantum liquid

We find evidence that for zero spin density $m=0$, intermediate $U/4t$ values, and a range $x\in (x_c,x_*)$ of finite hole concentrations the ground state of the virtual-electron pair quantum liquid obtained from perturbing the square-lattice quantum liquid of Ref. \cite{companion2} by weak three-dimensional (3D) uniaxial anisotropy and intrinsic disorder has long-range d-wave superconducting order. Here $t$ is the effective nearest-neighbor transfer integral and $U$ the effective on-site repulsion. The long-range d-wave superconducting order emerges below a critical temperature $T_c$ for a hole concentration range centered at $x_{op}= (x_c+x_*)/2$. It results from the effects of the residual interactions of the charge $c$ fermions and spin-neutral two-spinon $s1$ fermions of Ref. \cite{companion2}, as a by-product of the short-range spin correlations. Rather than the U(1) symmetry contained in the $η$-spin SU(2) symmetry of the $SO(4) = [SU(2)\times SU(2)]/Z_2$ symmetry, the U(1) symmetry broken at $T_c$ is the $c$ fermion U(1) symmetry of Ref. \cite{bipartite}. It is contained in the extended global $SO(3)\times SO(3)\times U(1)=[SO(4)\times U(1)]/Z_2$ symmetry of the Hubbard model on the square lattice. Our preliminary results seem to indicate that combining the electronic correlations described by the square-lattice quantum liquid perturbed by 3D uniaxial anisotropy with the very weak effects of intrinsic disorder or superfluid-density anisotropy leads for the hole-concentration range $x\in (x_c,x_*)$ to a successful description of the universal properties of the hole-doped cuprate superconductors.