Exact density-functional theory as parallel ensemble variational hierarchies: from Lieb's formulation to Kohn-Sham theory

Exact density-functional theory is reconstructed here from its convex variational structure as two parallel exact ensemble hierarchies: an interacting hierarchy rooted in Lieb's ensemble formulation and a noninteracting hierarchy rooted in the exact noninteracting ensemble theory. The Kohn-Sham construction links the two on a common admissible density class. In this organization, Levy-Lieb, Hohenberg-Kohn, and ordinary determinant-based Kohn-Sham formulations appear as constrained specializations of broader ensemble variational structures. Fractional particle number and fractional occupations enter naturally in the same ensemble variational setting, while piecewise linearity, one-sided chemical potentials, derivative discontinuity, and Janak-type relations emerge as consequences of the associated variational geometry. We also clarify several distinctions that are often compressed together in standard expositions, including functional domain versus representability class, state-space realization versus density-level supporting-potential structure, and density reproduction versus spectral interpretation.