Density functional theory (DFT) is usually presented as a single narrative: start with the Hohenberg-Kohn theorem, move to Kohn-Sham equations, approximate the exchange-correlation functional. This compressed story obscures the structure underneath.
The paper on exact DFT as parallel ensemble variational hierarchies (arXiv: 2603.23399) reconstructs the theory around a distinction that standard expositions blur: there are two parallel variational structures, not one. The interacting hierarchy is rooted in Lieb's ensemble formulation. The noninteracting hierarchy is rooted in exact ensemble noninteracting theory. Kohn-Sham theory is not either hierarchy — it's the auxiliary construction that links them on a common density class.
From this viewpoint, the Levy-Lieb constrained search, Hohenberg-Kohn picture, and ordinary pure-state Kohn-Sham formulations appear as specializations under additional restrictions. The exchange-correlation functional — usually treated as “the unknown remainder” — becomes the interface quantity between the two hierarchies, not a leftover.
The reorganization also places fractional particle number, derivative discontinuity, and fractional orbital occupations within a single variational picture, as natural consequences of the ensemble framework rather than separate complications.
The through-claim: DFT's conceptual difficulties arise not from the physics but from compressing two distinct variational structures into one narrative. The exchange-correlation functional is mysterious only when viewed as a correction to Kohn-Sham. Viewed as the interface between two parallel hierarchies — interacting and noninteracting — it becomes structurally necessary rather than merely empirically fitted.
2603.23399. Quantum chemistry / density functional theory / Kohn-Sham / variational methods / exchange-correlation.