  We present the $n$-orbital extension of the Hatsugai-Kohmoto (HK) model to
the orbital HK (OHK) model which obtains by covering the Brillouin zone with
$N/n$ Hubbard clusters each containing $n$-sites all connected via twisted
boundary conditions. We show that this defines a systematic computational
scheme to go from $n=1$ ``band'' HK to the full Hubbard model. Further, through
powerful scaling arguments, we show that the convergence to Hubbard goes as
$1/n^{2d}$ for $n^d$-orbital HK on a $d$-dimensional system implying that all
the fluctuations vanish in $d=\infty$. As evidence for the above, we employ
exact diagonalization and DMRG to show that the OHK model matches the exact
(from Bethe ansatz) ground state energy of the 1d Hubbard model within 1$\%$
with just $n=10$ orbitals. For a square lattice, we recover an insulating state
regardless of the strength of the interactions, double occupancy in agreement
with state of the art simulations, dynamical spectral weight transfer,
short-range antiferromagnetic correlations and charge neutral excitations
leading to the algebraic temperature dependence of the specific heat, all with
a fraction of the computational time of more advanced cluster methods and
making it possible to obtain analytical insights. The success of OHK and its
rapid convergence to the Hubbard model reiterates that a fixed point controls
the physics. Consequently, the $n$-orbital HK model offers a new tool for
strongly correlated quantum matter.
