  The physics model of a next-generation spallation-driven high-current
ultracold neutron (UCN) source capable of delivering an extracted UCN rate of
around an-order-of-magnitude higher than the strongest proposed sources, and
around three-orders-of-magnitude higher than existing sources, is presented.
This UCN-current-optimized source would dramatically improve cutting-edge UCN
measurements that are currently statistically limited. A novel "Inverse
Geometry" design is used with 40 L of superfluid $^4$He (He-II), which acts as
a converter of cold neutrons (CNs) to UCNs, cooled with state-of-the-art
sub-cooled cryogenic technology to $\sim$1.6 K. Our design is optimized for a
100 W maximum heat load constraint on the He-II and its vessel. In our
geometry, the spallation target is wrapped symmetrically around the UCN
converter to permit raster scanning the proton beam over a relatively large
volume of tungsten spallation target to reduce the demand on the cooling
requirements, which makes it reasonable to assume that water edge-cooling only
is sufficient. Our design is refined in several steps to reach
$P_{UCN}=2.1\times10^9\,/$s under our other restriction of 1 MW maximum
available proton beam power. We then study effects of the He-II scattering
kernel as well as reductions in $P_{UCN}$ due to pressurization to reach
$P_{UCN}=1.8\times10^9\,/$s. Finally, we provide a design for the UCN
extraction system that takes into account the required He-II heat transport
properties and implementation of a He-II containment foil that allows UCN
transmission. We estimate a total useful UCN current from our source of
$R_{use}=5\times10^8\,/$s from a 18 cm diameter guide 5 m from the source.
Under a conservative "no return" approximation, this rate can produce an
extracted density of $>1\times10^4\,/$cm$^3$ in $<$1000~L external experimental
volumes with a $^{58}$Ni (335 neV) cut-off potential.
