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 4He (He-II), which acts as the converter of cold neutrons to UCNs, cooled with state-of-the-art subcooled cryogenic technology to ∼ 1.6 K. Our source design is optimized for a 100 W maximum heat load constraint on the He-II and its vessel. In this paper, we first explore modifying the Lujan Center Mark-3 target for UCN production as a benchmark. In our Inverse Geometry, the spallation target is wrapped symmetrically around the cryogenic 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 a UCN production rate P UCN = 2.1 × 10 9 s - 1 under our other restriction of 1 MW maximum available proton beam power. We then study the effects of the He-II scattering kernel used as well as reductions in P UCN due to pressurization to reach P UCN = 1.8 × 10 9 s - 1. 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 × 10 8 s - 1 from an 18 cm diameter guide ∼ 5 m from the source. Under a conservative "no return" (or "single passage") approximation, this rate can produce an extracted density of > - > 1 × 10 4 UCN cm - 3 in < 1000 l external experimental volumes with a 58Ni (335 neV) cutoff potential.