A Pure-Spin-Singlet Obstruction to Single-Orbital Kohn–Sham Representability on Hubbard Rings

We provide an explicit pure correlated two-electron spin-singlet state on every real-hopping Hubbard ring with at least five sites whose initial site densities and first derivatives cannot be reproduced by a spin-restricted closed-shell Kohn–Sham state with one doubly occupied spatial orbital, the same real hopping and any scalar on-site potential. The obstruction follows from saturated phase-current edges and an incompatible global phase winding. Independent symbolic and finite-Hubbard-generator verifications corroborate the all-size analytical argument, and an associated nonconvexity result holds for the corresponding cycle phase-injection image. This is not a counterexample to unrestricted noninteracting or ensemble V-representability, and previously published lattice nonrepresentability results are acknowledged. Historical priority and external review remain outstanding.