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The Hubbard model on a square lattice is one of the most studiedcondensed-matter quantum problems.Here we find evidence that for intermediate$U/4t$ values and a hole-concentration range $x\in (x_c,x_*)$ the ground stateof the Hubbard model on the square lattice perturbed by weak three-dimensional(3D) uniaxial anisotropy has long-range d-wave superconducting order. Here $t$is the effective nearest-neighbor transfer integral and $U$ the effectiveon-site repulsion. The lower critical concentration $x_c$ involves the Ginzburgnumber Gi and is approximately given by $x_c\approx {\rm Gi}+x_0\approx 0.05$.Here $x_0<{\rm Gi}$ is a small critical hole concentration that marks a sharpquantum phase transition from a Mott-Hubbard insulator with long-rangeantiferromagnetic order for $x<x_0$ to an Anderson insulator with short-rangeincommensurate spiral spin order for $x\in (x_0,x_c)$. The value of thecritical hole concentration $x_*$ depends on $U/4t$ and is given by $x_*\approx0.27$ for $U/4t\approx 1.525$. The long-range d-wave superconducting orderemerges below a critical temperature $T_c$ for a hole concentration rangecentered at $x_{op}= (x_c+x_*)/2\approx 0.16$. It results from the effects ofthe residual interactions of the charge $c$ and spin-neutral two-spinon $s1$fermions of Ref. \cite{companion2}, as a by-product of the short-range spincorrelations. The spin subsystem provides through such interactions the energyneeded for the effective pairing coupling between the $c$ fermions of thevirtual-electron pair configurations.

d-wave superconductivity in Hubbard model on the square lattice perturbed by weak 3D uniaxial anisotropy