Dichotomous electronic system in a bilayer Ni¹⁺ nickelate

``Infinite layer'' nickelates (ILNs) ${\cal R}$NiO$_2$ (${\cal R}$=rare earth elements), having empty apical O sites, become superconducting upon hole doping. They display a secondary electron Fermi surface (FS), giving hole doping, arising not from atomic orbitals but from a band based on interstitial density. Newly reported La$_3$Ni$_2$O$_5$F, formally Ni$^{1+}$, provides an unexpected example of ILN with essentially ideal two dimensional character. A partially occupied single band $E^*$, based on interstitial density, has distinct properties, as its strongly anisotropic shape extends over the three ``apical'' layers and leads to a cylindrical electron FS giving self-doping. This interstitial density is associated with a {\it network of valence bands}, including a Ni $d_{xz},d_{yz}$ pair that partners with $E^*$ to provide an incipient non-analytic Dirac point, leading to an unusual type of interstitial density--$d$ band coupling. The $E^*$ electron band and the conventional Ni $dp\sigma$ band will display a dichotomy of hole and electron quasiparticle behavior in normal state transport and far-IR properties, and likely resulting in unconventional superconducting state properties even for nickelates.