Kohn-Sham ensemble v-representability of Coulomb ground-state densities by potentials in L^{3/2}+L^infinity

Theorem 1.1: there is a finite positive integer $Z$ such that the three-electron Coulomb Hamiltonian with two nuclei of charge $Z$ at $\pm(D/Z)e_z$, $D=10^{200}$, has a normalizable absolute ground state $\Psi$ (sector $S_z=1/2$) whose spin-summed density is not the density of any ground-state ensemble of $H_s(v)$ for any real $v\in L^{3/2}+L^\infty$. The mechanism is the Gori-Giorgi-Gal-Baerends nodal-plane asymptotics, made rigorous, plus a kinetic certificate against all ensembles. A corollary rules out a local-potential first variation of the ensemble Hartree-exchange-correlation functional at that density. Not covered: larger potential classes, atoms, or physically realistic charges.

Theorem 1.1: there is a finite positive integer $Z$ such that the three-electron Coulomb Hamiltonian with two nuclei of charge $Z$ at $\pm(D/Z)e_z$, $D=10^{200}$, has a normalizable absolute ground state $\Psi$ (sector $S_z=1/2$) whose spin-summed density is not the density of any ground-state ensemble of $H_s(v)$ for any real $v\in L^{3/2}+L^\infty$. The mechanism is the Gori-Giorgi-Gal-Baerends nodal-plane asymptotics, made rigorous, plus a kinetic certificate against all ensembles. A corollary rules out a local-potential first variation of the ensemble Hartree-exchange-correlation functional at that density. Not covered: larger potential classes, atoms, or physically realistic charges.