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

Exact densit y-functional theo ry as pa rallel ensemble va riational hiera rchies: from Lieb’s fo rmulation to K ohn–Sham theo ry Nan Sheng 1 Institute for Computational and Mathematical Engineering (ICME), Stanfor d University , Stanford, CA 94305, USA. (Dated: 25 March 2026) Exact ground-state density-functional theory contains two parallel variational structures that are often compressed into a single narrati ve: an interacting hierarchy rooted in Lieb’ s ensemble formulation and a noninteracting hierarchy rooted in exact ensemble noninteracting theory . W e reconstruct e xact DFT around this parallel structure and distinguish both exact frame works from the Kohn–Sham auxiliary density-functional construction that links them on a common admissible density class. From this viewpoint, the Levy–Lieb constrained search, the Hohenberg–K ohn picture, and ordinary pure-state noninteracting or Kohn–Sham formulations appear as narrower specializations under additional restrictions. The same organization also places fractional particle number, piecewise linearity , one-sided chemical potentials, deri vati ve discontinuity , fractional orbital occupations, and Janak-type relations within a single v ariational picture. Exchange-correlation structure is reconsidered from the same standpoint, where it appears as the interface quantity between the interacting and noninteracting hierarchies rather than merely as the unknown remainder of the K ohn–Sham decomposition. The result is a formal reorganization of exact DFT that clarifies distinctions often blurred in compressed expositions, including functional domain v ersus representability class, noninteracting supporting-potential structure versus K ohn–Sham auxiliary construction, and density reproduction versus spectral interpretation. I. INTRODUCTION AND HISTORICAL PERSPECTIVE Ground-state density-functional theory is usually introduced through a historical sequence beginning with the Hohenber g– K ohn theorem and leading to the Kohn–Sham construction. 1,2 That narrativ e is elegant and indispensable, b ut it is also highly compressed. It tends to fold sev eral formally distinct layers of the exact theory into a single storyline and thereby to blur distinctions between objects that do not have the same logical status. On the interacting side, the original Hohenberg–K ohn theo- rem, the Levy constrained search, and Lieb’ s ensemble formu- lation are often presented as successi ve variants of one basic idea. 1,3,4 Formally , howe ver , they reorganize the theory in dif ferent ways. The Hohenberg–K ohn setting is tied to ground- state densities and potential uniqueness under nondegenerac y assumptions. The Le vy constrained search enlarges the density domain to the N -representable class and defines a univ ersal functional through minimization ov er states. Lieb’ s formula- tion passes to ensembles, places the theory in a Banach-space setting, and identifies the exact density functional as the conv ex dual of the ground-state ener gy viewed as a functional of the external potential. These are not merely dif ferent presentations of the same starting point; they correspond to different formal lev els. A parallel compression occurs on the noninteracting side. In many standard presentations, noninteracting structure en- ters only when one introduces the Kohn–Sham equations. But before one constructs the K ohn–Sham auxiliary density func- tional for the interacting problem, one already has an exact noninteracting ensemble variational theory of its own, with its own constrained-search functional, dual ener gy functional, and representability questions. 5,6 The K ohn–Sham construc- tion in the stricter sense begins only when this noninteracting framew ork is coupled back to the interacting one through the decomposition of the exact uni versal functional. The central claim of the present paper is therefore organiza- tional. For the structural questions emphasized here, e xact DFT is most clearly read as two parallel ensemble v ariational hier- archies, namely an interacting hierarchy and a noninteracting hierarchy , together with the K ohn–Sham auxiliary construction that links them on a common admissible density class. In this reconstruction, Lieb’ s ensemble theory is the natural broad starting point for the interacting side, while exact ensemble noninteracting theory is the corresponding broad starting point for the noninteracting side. Levy–Lieb, Hohenberg–K ohn, and ordinary pure-state noninteracting or K ohn–Sham formulations then appear as narro wer specializations obtained under addi- tional restrictions or assumptions. The ensemble vie wpoint matters on both sides, b ut for dis- tinct reasons. On the interacting side, it is the natural setting in which con vexity , duality , subgradients, representability , and fractional particle number can be read as parts of one exact variational picture. 4,7 On the noninteracting side, it is the nat- ural setting in which supporting potentials for T s , fractional occupations, and the distinction between ensemble and pure- state noninteracting representability become exact variational questions. 5,6 These two ensemble structures are parallel but not identical. The role of exact Kohn–Sham theory is not to erase that distinction, but to bridge the two exact frame works while retaining the density of the interacting problem. This reorganization helps clarify several distinctions that are often blurred in compressed expositions of exact DFT : functional domain versus representability class, definition of a univ ersal functional versus existence of a supporting potential, fractional particle number versus fractional orbital occupation, density reproduction versus spectral interpretation, and exact noninteracting ensemble theory versus the K ohn–Sham auxil- iary density functional built from it. Read from the ensemble lev el downw ard, these do not appear as later cav eats attached to an otherwise simple story; they appear as nati ve features of the exact theory . 2 The present article is not a theorem-driven contrib ution in the usual sense. It does not claim a ne w isolated result compara- ble to the classical papers it discusses. Rather , it offers a formal reorganization of material that already e xists in the literature: the HK and KS papers, 1,2 the Le vy and Lieb formulations, 3,4 the exact-conditions literature on fractional char ge and deriv a- tiv e discontinuity , 8–10 modern work on representability and density-to-potential mappings, 7,11 and ongoing discussions of orbital energies and Janak-type interpretations. 12,13 The goal is to place these ingredients in a single formal hierarchy and to make clear which statements belong to the interacting theory , which belong to the noninteracting theory , and which belong specifically to their K ohn–Sham coupling. The remainder of the paper is organized as follo ws. Section II dev elops the interacting ensemble theory , emphasizing Lieb duality , supporting potentials, and the v ariational origin of frac- tional particle number , piecewise linearity , and deriv ativ e dis- continuity before turning to Levy–Lieb and Hohenber g–K ohn as more specialized settings. Section III dev elops the exact noninteracting ensemble framew ork in parallel, with emphasis on T s , its dual energy functional, supporting potentials, and the relation between ensemble and pure-state noninteracting repre- sentability . Section IV then develops e xact Kohn–Sham theory as the auxiliary construction that couples the two hierarchies through the exchange-correlation decomposition, and re visits Janak-type relations, exchange-correlation structure, and the gap decomposition from that vie wpoint. I I. EXA CT INTERACTING DFT IN THE ENSEMBLE FRAMEW ORK A. Lieb’s formulation, duality , and representabilit y For the structural questions emphasized in this paper , Lieb’ s density-matrix formulation provides the broadest natural start- ing point for the interacting side of exact DFT . The point is not only that density matrices enlarge the admissible state class beyond pure states. More importantly , they place the theory directly in a con vex v ariational setting adapted to duality , sub- gradients, closure, and supporting potentials. 4,11 Three features are central here. First, at the density-matrix lev el, the map from state to density is linear . Second, the expectation value of the Hamiltonian is affine in the state. Third, while the set of pure states is not conv ex, the set of density matrices is. Passing from pure states to density matrices therefore con vexifies the state space without changing either the linearity of the density map or the affinity of expectation values. This is precisely the setting in which con vex analysis becomes natural. In that sense, the mov e to ensembles is not merely a more permissive notation; it is the relaxed v ariational formulation naturally selected by the structure of the problem. W e consider the standard nonrelativistic electronic Hamilto- nian H [ v ] = T + W + V [ v ] , (1) with T = − 1 2 N ∑ i = 1 ∆ i , W = ∑ 1 ≤ i < j ≤ N 1 | r i − r j | , V [ v ] = N ∑ i = 1 v ( r i ) . (2) The exact interacting uni versal functional is defined by F [ ρ ] = inf Γ 7→ ρ tr  Γ ( T + W )  , (3) where Γ ranges over positiv e trace-class operators with tr Γ = 1 , together with the fixed-particle-number condition tr  Γ ˆ N  = N in the ordinary N -electron setting. 4 Equation (3) should be read as more than a generalized con- strained search. The admissible fibers cut out by the condition Γ 7→ ρ are now con vex subsets of a con vex state space, and the objectiv e tr [ Γ ( T + W )] is affine. The resulting functional is therefore naturally compatible with con vex-analytic opera- tions. This is the sense in which Lieb’ s formulation is formally broader than the pure-state Le vy picture: it is not simply less restrictiv e, but better adapted to the primiti ve geometry of the density-constrained variational problem. Lieb’ s framework places the density in the Banach space X = L 1 ( R 3 ) ∩ L 3 ( R 3 ) , (4) and the external potential in the dual space X ∗ = L ∞ ( R 3 ) + L 3 / 2 ( R 3 ) . (5) This matters because it turns the density–potential relation into a genuine dual pairing rather than a merely heuristic corre- spondence. Once density and potential are recognized as dual variables, the natural potential-side object is the ground-state energy functional E [ v ] = inf Γ ≥ 0 , tr Γ = 1 tr  Γ ( T + W + V [ v ])  . (6) Since Eq. (6) is an infimum of affine functionals of v , the map v 7→ E [ v ] is concave, and the interacting universal functional may be recov ered by Legendre–Fenchel duality: F [ ρ ] = sup v  E [ v ] − ⟨ v , ρ ⟩  , (7) E [ v ] = inf ρ  F [ ρ ] + ⟨ v , ρ ⟩  . (8) W e use a unified pairing notation throughout. For densities ρ ∈ X and external potentials v ∈ X ∗ , ⟨ v , ρ ⟩ = Z v ( r ) ρ ( r ) d r . For density matrices and observ ables, we retain the standard trace notation tr ( Γ A ) , viewed as the corresponding expectation pairing on the state space. This duality is not merely a reformulation of the variational principle. It also clarifies what an exact external potential represents in the theory: namely , a supporting functional of the exact density functional. In the conv ex-analytic frame work, − v ∈ ∂ F ( ρ ) ⇐ ⇒ ρ ∈ ∂ E ( v ) , (9) 3 so exact representability is naturally expressed as a subdif- ferential question. 7,11 At diff erentiable points this reduces to an Euler equation; at nondifferentiable points the subgradient formulation remains meaningful in exact form. One of the main advantages of the ensemble setting is precisely that it distinguishes cleanly between the existence of the functional and the existence of a supporting potential. That distinction should be made explicit. The exact inter- acting functional is defined on the full N -representable density domain, dom F = I N . (10) By contrast, the densities admitting an exact supporting poten- tial form a smaller class, dom ∂ F ⊆ dom F . (11) W ithin the present con vex-analytic perspecti ve, dom ∂ F is the relev ant interacting ensemble v -representable class. 7,11 Thus one must distinguish between the statement that F [ ρ ] is well defined and the statement that ρ is realized as an exact ensem- ble ground-state density of some external potential. The corresponding closure statement, cl  dom ∂ F  = dom F , (12) again in the appropriate density topology , should be interpreted with the same care. 4,7 It says that densities with exact support- ing potentials are dense in the full N -representable domain; it does not say that e very N -representable density is itself e x- actly interacting ensemble v -representable. The distinction between closure and pointwise membership will reappear on the noninteracting side. Seen from this perspective, the familiar Hohenberg–K ohn density-to-potential mapping is no longer the broadest formal starting point of the interacting theory . Rather, it emerges on the subdomain of densities that admit supporting potentials and under the additional uniqueness assumptions appropriate to the Hohenberg–K ohn setting. The mapping language remains central, but here it is understood as a downstream specialization of a broader ensemble variational theory . B. F ractional particle numb er, one-sided slop es, and derivative discontinuity Once the interacting theory is formulated in ensemble form, fractional particle number enters naturally . One may impose particle number only in expectation, tr Γ = 1 , tr  Γ ˆ N  = N , (13) with N not necessarily an integer . 8 This remains a zero- temperature exact v ariational problem and should not be con- fused with thermal smearing. In this sense, fractional particle number is not an external correction appended to exact DFT ; it is already present once the exact state space has been con ve xi- fied. Let N = M + ω , with M ∈ N and 0 < ω < 1 . If Γ M and Γ M + 1 are minimizing ensembles in the adjacent inte ger sectors M and M + 1, then Γ M + ω = ( 1 − ω ) Γ M + ω Γ M + 1 (14) is admissible at expected particle number M + ω . Since the expectation value of the Hamiltonian is af fine in the density matrix, tr  Γ M + ω H [ v ]  = ( 1 − ω ) E [ v , M ] + ω E [ v , M + 1 ] . (15) The ensemble construction therefore always provides an affine upper bound between adjacent inte ger sectors. Under the usual con vexity or no-phase-separation assumption, that upper bound is exact and one obtains the Perdew–Parr –Levy–Balduz (PPLB) piecewise-linearity relation 8 E [ v , M + ω ] = ( 1 − ω ) E [ v , M ] + ω E [ v , M + 1 ] . (16) The role of the adjacent sectors is conceptually important. Once the exact problem is relax ed to a con vex ensemble state space, the segment connecting neighboring integer energies is the natural variational candidate at intermediate particle number . The particle-number constraint is local in N , so the relaxed problem points first to mixtures of the two adjacent sectors; more complicated ensembles inv olving nonadjacent particle numbers do not improve upon this affine segment in the absence of phase separation. The same point may also be vie wed from an optimization perspectiv e: when the admissible ensemble class is conv ex and the energy is affine on it, mini- mization at fixed av erage particle number has the flav or of a linear-programming problem, with the optimum supported on an extreme decomposition, here realized by adjacent particle- number sectors. The optimization and geometric readings are thus complementary . This also clarifies the role of con vexity . The affine mix- ture of adjacent sectors always gi ves an admissible competitor and hence an upper bound. What the conv exity or no-phase- separation hypothesis excludes is the possibility that a more complicated decomposition at the same av erage particle num- ber could do better . Exact piecewise linearity therefore says that the af fine candidate supplied by the relaxed ensemble ge- ometry is already optimal. In this sense, the PPLB line is not best viewed as an externally imposed interpolation rule, but as the energetic signature of the relaxed v ariational geometry itself. The physical interpretation is equally important. Fractional N does not describe a fraction of an electron attached to a single pure-state molecule; it is the language of an open-system or av erage-particle-number viewpoint. In that language, the exact energy between integers encodes electron addition and remov al energetics through its one-sided slopes. Once piecewise linearity holds, the relation to ionization energy , electron af finity , and chemical potential becomes im- mediate. The left and right deriv ativ es at integer M are µ − M = d E d N     M − , µ + M = d E d N     M + . (17) 4 Because the energy is affine on each adjacent interval, these one-sided deri vati ves are exactly the corresponding finite dif- ferences: µ − M = E [ v , M ] − E [ v , M − 1 ] = − I M , µ + M = E [ v , M + 1 ] − E [ v , M ] = − A M . (18) Thus ionization energy and electron af finity arise internally in exact DFT as one-sided slope data of the interacting ensemble energy . The same structure may be read in three equi valent lan- guages. In energetic language, one has the finite differences I and A . In thermodynamic language, one has the one-sided chemical potentials µ ± M . In con vex-analytic language, one has the left and right supporting slopes of the exact energy with respect to particle number . These are not distinct facts but distinct descriptions of the same geometric structure. In the same language, the familiar inequality I M ≥ A M is equi valent to µ − M ≤ µ + M , that is, to a nonnegati ve jump between the one- sided slopes at integer particle number . It should therefore be read as the physical expression of the slope ordering at the in- teger , rather than as the primary origin of the piecewise-linear structure itself. This is why curv ature between integers in approximate func- tionals is not merely a numerical imperfection but a structural failure: it signals that the approximation has lost the affine geometry implied by the exact ensemble theory and therefore corrupts the exact slope data from which electron addition and remov al energetics are read. 14–16 The many-body fundamental gap is therefore E true g = I M − A M = µ + M − µ − M . (19) This prepares the ground for the deriv ativ e discontinuity . At integer particle number , the exact energy has distinct one-sided slopes, so the supporting hyperplanes of the e xact interacting functional need not coincide from the left and from the right. Because adding a constant to a potential does not change the associated density class, the mismatch between the left and right supporting potentials appears as a spatially constant jump. That jump is the deriv ati ve discontinuity . 9,10,17 It is useful to state this relation in both density-space and potential-space terms. In density-space language, integer parti- cle number is special because the exact energy as a function of N is not differentiable there: the left and right slopes dif fer . In potential-space language, the same fact appears as the f ailure of a single supporting potential to represent both one-sided variational directions. Since potentials differing by an additi ve constant generate the same density class, the mismatch appears as a constant jump. The deriv ativ e discontinuity is therefore the potential-side image of the same nondif ferentiable structure already visible in the particle-number dependence of the exact energy . This is why piecewise linearity and deri vati ve discontinuity should be read together . The former is the ener gy-side manifes- tation of the relaxed ensemble geometry; the latter is the corre- sponding potential-side manifestation. Approximate curvature between integers and the absence of the corresponding disconti- nuity are therefore not merely two unrelated numerical defects. Both indicate that the approximation has failed to preserve the exact geometry of fractional particle number . 14,15,18,19 C. Pure-state sp ecializations: Levy–Lieb and Hohenb erg–K ohn Having established the interacting ensemble framew ork, one may now ask how the more familiar pure-state formulations arise within it. The key point is not simply chronological. It is to identify what is being restricted, and what formal structure is correspondingly lost, when one passes from the ensemble theory to narrower pure-state settings. The Levy constrained-search functional is F LL [ ρ ] = inf Ψ 7→ ρ ⟨ Ψ | T + W | Ψ ⟩ . (20) This is an exact pure-state v ariational problem defined on the N -representable density domain. 3,4 Its relation to the ensemble theory is transparent at the level of admissible classes. The Lieb functional minimizes tr [ Γ ( T + W )] ov er all admissible interacting density matrices reproducing ρ , whereas the Le vy functional minimizes the same quantity over the restricted subclass of pure states only . Since the pure-state admissible class is contained in the ensemble class, one immediately has F [ ρ ] ≤ F LL [ ρ ] . (21) Equality holds whenev er the ensemble infimum is attained by a pure state. This is the precise sense in which Le vy–Lieb is a narrower specialization of Lieb’ s formulation. It is not inexact, nor does it define a different physical problem. Rather, it imposes an additional admissibility restriction. Whenev er that restriction does not exclude the exact minimizer, the two formulations coincide on the density in question. On densities for which the ensemble minimizer is attained by a pure state, one therefore has F [ ρ ] = F LL [ ρ ] . (22) This observation helps locate the Hohenber g–K ohn picture more precisely . In the nondegenerate pure-state setting, one may write F HK [ ρ ] =  Ψ ρ   T + W   Ψ ρ  , ρ ∈ A N , (23) where A N denotes the set of interacting ground-state densities generated by external potentials in the ordinary Hohenberg– K ohn setting. 1 This domain is narrower than the full N - representable domain and should not be identified with the broader interacting ensemble v -representable class that appears naturally in Lieb’ s formulation. On densities lying in this fur- ther specialized nondegenerate pure-state v -representable class, the three functionals coincide: F HK [ ρ ] = F LL [ ρ ] = F [ ρ ] . (24) Thus the difference among these formulations lies not in their exact v alue on such densities, but in their admissible classes, 5 domains of definition, and the hypotheses under which they are introduced. Stated this way , the relation among the three formulations is more informati ve than the looser claim that they are merely different “exact formulations” of DFT . The dif ference is v ari- ational as well as notational. In the pure-state setting, the N -representable density domain remains explicit, but the prim- iti ve con vex geometry of the state space is no longer present at the same le vel. As a result, sev eral distinctions native to Lieb’ s formulation, such as the clean separation between functional domain and supporting-potential existence and the direct subd- ifferential reading of representability , become less transparent, ev en when the underlying exact content remains a vailable in restricted form. More broadly , Lieb’ s ensemble functional may be vie wed as the con vex, lower -semicontinuous relaxation of the pure-state constrained-search functional in the appropriate density-space setting. In that sense, the passage from Levy–Lieb to Lieb is not only a change in the admissible state class, but also a genuine con vexification of the underlying v ariational problem. This is precisely why the ensemble formulation is the natural broad starting point for the present reconstruction: it places the interacting theory directly at the lev el where con vexity , duality , and supporting-potential structure are nativ e rather than recov ered only after further reformulation. The hierarchy Lieb/ensemble exact DFT → Levy–Lieb → Hohenberg–K ohn (25) should thus be read as a sequence of increasingly restrictiv e admissible classes and hypotheses. The point is not to dis- place the familiar pure-state formulations, which remain exact where applicable, but to place them more precisely within the broader interacting theory . In the present reconstruction, the ensemble formulation is the broad variational starting point; Levy–Lieb is its pure-state constrained-search specialization; and the Hohenberg–K ohn framework is the further special- ization obtained when one restricts attention to the densities generated by external potentials under the usual pure-state uniqueness assumptions. I I I. EXA CT NONINTERACTING DFT IN THE ENSEMBLE FRAMEW ORK A. The noninteracting constrained-search problem and its dual energy functional The noninteracting side should be dev eloped in deliberate parallel with the interacting one. The aim at this stage is not yet to construct the Kohn–Sham auxiliary density functional for the interacting problem, but to isolate the e xact noninteracting en- semble variational frame work in its own right. This distinction is essential to the present paper , because the noninteracting theory already possesses a constrained-search functional, a dual energy functional, and its o wn representability structure before it is coupled back to the interacting hierarchy . The exact noninteracting kinetic-ener gy functional is T s [ ρ ] = inf Γ s 7→ ρ tr ( Γ s T ) , (26) where Γ s ranges over admissible noninteracting fermionic ensembles. 2,5,6 This is the noninteracting analogue of Lieb’ s interacting ensemble functional. It is not yet the K ohn–Sham total density functional, b ut the exact density-constrained varia- tional object associated with the auxiliary noninteracting prob- lem itself. The corresponding noninteracting ground-state energy func- tional is E s [ v s ] = inf Γ s ≥ 0 , tr Γ s = 1 tr  Γ s ( T + V [ v s ])  . (27) Since Eq. (27) is an infimum of affi ne functionals of v s , the map v s 7→ E s [ v s ] is concav e. The same Legendre–Fenchel logic used on the interacting side then yields the dual relations T s [ ρ ] = sup v s  E s [ v s ] − ⟨ v s , ρ ⟩  , (28) E s [ v s ] = inf ρ  T s [ ρ ] + ⟨ v s , ρ ⟩  . (29) Thus T s is not only a constrained-search functional; it is also the con vex dual of the e xact noninteracting ground-state energy . This establishes a sharp formal parallel between the interact- ing and noninteracting ensemble theories. On the interacting side, the pair ( E , F ) defines the exact many-body v ariational frame work. On the noninteracting side, the pair ( E s , T s ) defines an exact auxiliary variational framew ork of the same general type. The two pairs are parallel, b ut they should not yet be iden- tified with the K ohn–Sham construction proper . K ohn–Sham theory in the stricter sense be gins only when these two e xact framew orks are linked on a common admissible density class through the exchange-correlation decomposition. The noninteracting duality also clarifies the status of the aux- iliary potential v s . In the same conv ex-analytic sense in which the interacting external potential is a supporting functional of F , the noninteracting potential is a supporting functional of T s . One has − v s ∈ ∂ T s ( ρ ) ⇐ ⇒ ρ ∈ ∂ E s ( v s ) . (30) Noninteracting representability is therefore again most natu- rally read as a subdifferential question. That distinction matters here for the same reason as on the interacting side. The functional T s may be well defined on a broad density domain, whereas the existence of an exact supporting noninteracting potential v s may hold only on a smaller class. The noninteracting theory thus already contains its o wn density–potential relation, but it is properly understood as a supporting-potential or subgradient structure rather than as a nai ve one-to-one map. This point is conceptually important because the later K ohn–Sham construction must operate inside this pre-existing noninteracting supporting-potential geometry rather than create it from nothing. 6 B. Rep resentability , supp orting p otentials, and orbital realization Suppose now that ρ is noninteracting ensemble v - representable. Then there exists a one-body potential v s such that − v s ∈ ∂ T s ( ρ ) . (31) Relati ve to the constrained-search problem Eq. (26) , this means that v s is the supporting potential associated with the density constraint. Whenev er a classical multiplier picture is valid, one may regard v s as the Lagrange multiplier for the density constraint; more generally , it is a subgradient representativ e of T s at ρ . The multiplier and supporting-potential vie wpoints are therefore two readings of the same v ariational structure rather than two independent constructions. Once such a supporting potential e xists, the corresponding noninteracting reference system is generated by the one-body Hamiltonian − 1 2 ∆ + v s ( r ) , (32) with orbitals satisfying  − 1 2 ∆ + v s ( r )  φ i ( r ) = ε i φ i ( r ) . (33) The density is then represented as ρ ( r ) = ∑ i n i | φ i ( r ) | 2 , 0 ≤ n i ≤ 1 , ∑ i n i = N . (34) This is the exact noninteracting ensemble realization of the density . 5,6 It is important to emphasize that, in the ensemble setting, the occupation numbers { n i } are not merely a passive bookkeeping de vice for the orbital representation. They belong to the admis- sible noninteracting v ariational structure itself, subject to the constraints 0 ≤ n i ≤ 1 and ∑ i n i = N . This is one of the main formal dif ferences between ensemble and pure-state noninter- acting theory . In the ensemble theory , fractional occupations are part of the exact admissible class and therefore enter the variational problem on the same footing as the orbitals. For fix ed supporting potential v s and stationary orbitals, the noninteracting energy is the occupation-weighted sum of one- particle eigen values, so that ∂ E s ∂ n i = ε i . (35) In this limited but exact sense, the appearance of orbital en- ergies as occupation deriv ativ es already belongs to the non- interacting ensemble theory itself. The more consequential statement comes later , when the same occupation-space struc- ture is carried into the Kohn–Sham construction and gi ves rise to the corresponding Janak-type stationarity relation for the full auxiliary total-energy functional. T wo lev els should therefore be kept distinct. The exact noninteracting framework already determines the admissi- ble ( ρ , v s ) pairs through the subdifferential condition − v s ∈ ∂ T s ( ρ ) . The later Kohn–Sham construction does not replace this structure, but works within it. Its outer density variation must ultimately produce an effecti ve potential compatible with the same supporting-potential set already encoded by the non- interacting ensemble theory . This is also the point at which density reproduction and spec- tral interpretation begin to diver ge. The noninteracting sup- porting potential is introduced to realize a giv en density within the exact auxiliary frame work. That does not, by itself, justify reading the resulting one-particle spectrum as a literal many- body addition/remov al spectrum of the interacting problem. The exact role of the noninteracting framework is therefore v ariational and density-reproducti ve first; any stronger spectral interpretation belongs to a separate question and cannot be assumed simply from the existence of a supporting potential. C. Pure-state sp ecialization and noninteracting rep resentability classes The next question is how the more familiar determinant- based formulation arises from the broader noninteracting en- semble theory . As on the interacting side, the answer is not merely historical. It is that one passes to a narrower admissible class, and this restriction may or may not leav e the minimizing state unchanged. Let E s ( ρ ) = { Γ s | Γ s ≥ 0 , tr Γ s = 1 , Γ s noninteracting , Γ s 7→ ρ } (36) denote the admissible noninteracting ensemble class, and let P s ( ρ ) ⊆ E s ( ρ ) denote the subclass of pure noninteracting states, equiv alently Slater determinants reproducing ρ . Then T s [ ρ ] = inf Γ s ∈ E s ( ρ ) tr ( Γ s T ) , (37) whereas the corresponding pure-state noninteracting functional is T ps s [ ρ ] = inf Γ s ∈ P s ( ρ ) tr ( Γ s T ) . (38) Since P s ( ρ ) ⊆ E s ( ρ ) , (39) one has T s [ ρ ] ≤ T ps s [ ρ ] . (40) This inequality should be read in the same way as its interact- ing analogue. It does not merely say that the pure-state problem is formally smaller . Rather, it sho ws that exact equiv alence between the ensemble and pure-state noninteracting theories requires the ensemble minimizer to lie in the idempotent, or pure-state, subclass. When that happens, the determinant pic- ture is exact on the density in question. When it does not, the difference between the tw o formulations is genuinely vari- ational: one has changed the admissible class and thereby potentially changed the minimizing state. 7 There is also a useful one-body way to phrase the same dis- tinction. At the lev el of the noninteracting one-body density matrix, the determinant restriction corresponds to idempotency , whereas the broader ensemble theory permits non-idempotent occupation structure associated with fractional occupations. The present paper does not require the full reduced-density- matrix formalism, b ut this observation helps make clear wh y the pure-state restriction is substanti ve rather than merely cos- metic. This is why the passage from ensemble to pure-state nonin- teracting theory must be discussed together with representabil- ity . At the density lev el, the broad N -representable domain is not the main source of distinction between the two formula- tions. The substanti ve differences arise instead at the le vel of noninteracting v -representability and of the admissible min- imizing states, rather than at the level of the broad density domain itself. 5,6 The determinant picture is therefore best un- derstood not as an automatically equi valent reformulation of the noninteracting ensemble theory , but as a more restrictiv e specialization of it. The conceptual outcome of Section III may now be stated plainly . The exact noninteracting ensemble framework already contains its own constrained search, dual energy functional, supporting-potential structure, and representability hierarch y . Fractional occupations, Fermi-lev el degeneracy , and the dis- tinction between ensemble and determinant realizations are therefore not peripheral technicalities appended to K ohn–Sham theory after the fact. They belong to the e xact noninteracting variational geometry itself. This is precisely why the Kohn– Sham construction, to be discussed ne xt, must be understood as an auxiliary theory built from this noninteracting frame work rather than identified with it from the outset. IV. EXA CT KOHN–SHAM THEORY AS AN AUXILIARY DENSITY-FUNCTIONAL CONSTRUCTION A. The Kohn–Sham auxiliary density functional The interacting and noninteracting ensemble theories dev el- oped in the preceding sections are formally parallel, but they remain distinct exact frame works. On the interacting side, one has the exact pair ( E , F ) ; on the noninteracting side, one has the exact pair ( E s , T s ) . Up to this point, howev er , these two theories hav e only been de veloped side by side. Kohn–Sham theory in the stricter sense begins only when one asks whether the minimizing density of the interacting problem can also be represented within the noninteracting framew ork on a common admissible density class. This is the essential new step. The existence of the non- interacting dual pair ( E s , T s ) by itself does not yet constitute K ohn–Sham theory . The Kohn–Sham construction begins only when the interacting and noninteracting frameworks are linked at the same density . In that sense, Kohn–Sham theory is not identical with the exact noninteracting ensemble theory de- veloped in Section III. Rather , it is the auxiliary construction obtained when that noninteracting theory is coupled back to the interacting one. The key bridge is the e xact decomposition F [ ρ ] = T s [ ρ ] + J [ ρ ] + E xc [ ρ ] , (41) with J [ ρ ] = 1 2 ZZ ρ ( r ) ρ ( r ′ ) | r − r ′ | d r d r ′ , (42) and E xc [ ρ ] = F [ ρ ] − T s [ ρ ] − J [ ρ ] . (43) This decomposition does not merely rewrite the interacting problem in a new notation. Its content is precisely that the exact noninteracting kinetic functional T s is inserted into the interacting density-functional problem, while the exchange- correlation functional records the part of the interacting theory not contained in T s and J . It is worth stating explicitly what this does and does not mean. If one were to define T s by minimizing over all inter- acting states yielding the density , then the distinction between the interacting and noninteracting constrained-search problems would already have been erased. But that distinction is the heart of Kohn–Sham theory . The K ohn–Sham construction is therefore not simply the noninteracting theory itself, nor is it merely a formal rearrangement of the interacting theory . It is the auxiliary density-functional scheme obtained when the exact noninteracting frame work is used to represent the same density on which the interacting functional is ev aluated. The resulting K ohn–Sham total density functional is E KS [ ρ ] = T s [ ρ ] + ⟨ v ext , ρ ⟩ + J [ ρ ] + E xc [ ρ ] , (44) to be minimized over the appropriate common admissible den- sity class. W e write E KS [ ρ ] in order to distinguish this auxiliary total density functional from the potential-side ground-state energy functionals E [ v ] and E s [ v s ] . By Eq. (41), E KS [ ρ ] = F [ ρ ] + ⟨ v ext , ρ ⟩ , (45) so the exact Kohn–Sham and exact interacting total-energy functionals agree numerically as functionals of the density . What differs is not their value on density space, but their de- composition and therefore their interpretation. This distinction is essential to the exact status of the K ohn– Sham construction. Exact K ohn–Sham theory is exact not because it identifies the interacting problem with the bare non- interacting one-body problem, b ut because the auxiliary decom- position is arranged so as to reproduce the same minimizing density as the interacting density-functional problem, pro vided the rele vant common admissible class and supporting-potential conditions are satisfied. The exactness lies at the le vel of den- sity reproduction and density-space total-energy minimization, not in a literal identification of the interacting and noninteract- ing theories. This also shows wh y the Kohn–Sham construction must re- main compatible with the noninteracting supporting-potential geometry already encoded by T s . The outer variation of Eq. (44) , understood in the appropriate dif ferentiable or subd- ifferential sense, yields an eff ective potential candidate. But 8 the same density must also satisfy the inner noninteracting supporting-potential relation associated with T s . K ohn–Sham theory therefore closes only when the outer auxiliary variation and the inner noninteracting realization are consistent with one another at the same density . In the exact ensemble setting, this means that the Kohn– Sham construction closes on a density ρ only insofar as ρ lies in a common admissible class on which both the interacting and noninteracting supporting-potential structures are nonempty . Only on such a class can the decomposition F = T s + J + E xc function as the bridge between the two exact v ariational hierar- chies. Pure-state formulations impose stricter conditions still, since one must further require the relev ant minimizers to lie in the corresponding pure-state subclasses. B. Stationarit y , effective p otentials, and Janak-type relations Suppose that ρ belongs to a common admissible density class on which both the interacting and noninteracting ensem- ble framew orks can be realized, and suppose further that the relev ant stationarity conditions hold. Then the outer variation of Eq. (44) , understood in the appropriate dif ferentiable or sub- differential sense, requires that the ef fectiv e potential entering the noninteracting realization of the density be v s ( r ) = v ext ( r ) + v H ( r ) + v xc ( r ) , (46) where v H ( r ) = δ J [ ρ ] δ ρ ( r ) , v xc ( r ) = δ E xc [ ρ ] δ ρ ( r ) . Strictly speaking, Eq. (46) holds only up to an additiv e con- stant, since the noninteracting supporting potential is defined only modulo constant shifts. This constant is immaterial for the associated density class and may therefore be fixed by a con venient choice of gauge. The content of the exact Kohn–Sham construction is that the same v s determined by the outer stationarity of the aux- iliary density functional must at the same time belong to the supporting-potential set of T s at the same density . In that case, the noninteracting realization described in Section III applies: the density is generated by the orbitals and occupations associ- ated with the one-body problem defined by v s , as in Eq. (34) and the corresponding eigenv alue equation given there. The exact K ohn–Sham equations therefore arise when the outer stationarity condition and the inner noninteracting supporting- potential relation are satisfied consistently at the same density . This vie wpoint is important because it makes clear that the ef fective Kohn–Sham potential is not introduced independently of the underlying noninteracting v ariational structure. Rather, it is the potential at which the outer density variation of the auxiliary functional closes compatibly with the inner exact non- interacting realization of that density . The effecti ve potential therefore belongs to the same supporting-potential geometry already encoded by T s , e ven though it is obtained through the full K ohn–Sham decomposition. The role of fractional occupations is best understood in this same framew ork. In the ensemble setting, the occupation num- bers { n i } are not merely a representational device attached to orbitals after the fact. They belong to the admissible vari- ational structure itself. Let the exact minimizing density be represented by orbitals and occupations as in Eq. (34) , and consider the total energy as a functional of { φ i } and { n i } , E = E [ { φ i } , { n i } ] . (47) If the orbitals are stationary for fixed occupations, then one has the Janak relation 20 ∂ E ∂ n i = ε i . (48) In the present reconstruction, Eq. (48) should be interpreted as an occupation-space stationarity statement internal to ex- act ensemble K ohn–Sham theory . That is the sense in which it is structurally important here. Once the occupations them- selves belong to the exact admissible class, dif ferentiation with respect to n i is no longer an ad hoc maneuver; it is the infinites- imal expression of moving within the noninteracting ensemble sector while maintaining the orbital stationarity conditions of the auxiliary problem. This interpretation is deliberately narrower than stronger readings often attached to Kohn–Sham orbital energies. By itself, Eq. (48) does not imply that ev ery orbital energy is to be read as a literal many-body ionization or affinity ener gy in a general finite system. 12,13 Such a spectral interpretation re- quires additional assumptions and is entangled with asymptotic structure, piecewise linearity , deri vati ve discontinuity , and the distinction between the exact K ohn–Sham reference system and the interacting many-body problem. For the purposes of the present paper, the cleanest reading is therefore the nar- rower one: Janak’ s relation is an occupation-space differential statement belonging to the exact auxiliary variational structure. This distinction is not merely interpreti ve caution. It matches the general logic of the paper . On the interacting side, frac- tional particle number already required one to distinguish ordi- nary differentiability from one-sided or subdif ferential struc- ture. On the noninteracting side, fractional occupations lead to an analogous need for care. The exact theory need not be smooth in the naiv e sense, especially at integer particle number or in the presence of degeneracy . The v alue of the ensemble formulation is precisely that it allo ws one to replace ov ersim- plified deriv ativ e statements by one-sided or subdifferential information without losing formal control. 7,11 The conceptual outcome of this subsection is therefore twofold. First, the effecti ve Kohn–Sham potential is fixed, up to an irrele vant additi ve constant, by the consistent closure of the outer auxiliary variation with the inner noninteracting supporting-potential structure. Second, Janak’ s relation is best understood not as a blanket spectral identification, b ut as the occupation-space stationarity statement naturally associated with exact ensemble K ohn–Sham theory . 9 C. Structural properties of the exchange-correlation functional The preceding discussion has sho wn how e xact Kohn–Sham theory closes only when the interacting and noninteracting framew orks are linked consistently at the same density . The exchange-correlation functional is the quantity through which that link is completed. Defined by Eq. (43) , E xc records the part of the interacting theory that is not contained in the exact noninteracting kinetic functional T s and the Hartree term. This point is important conceptually . The exchange- correlation functional is not merely a residual Coulombic cor - rection term. It contains both the genuinely nonclassical inter- action contribution and the kinetic remainder that appears when the interacting and noninteracting hierarchies are compared at fixed density . In that sense, E xc is the interface quantity left after separating out the Hartree term from two exact but nonidentical v ariational theories. If E xc is introduced only as “whatev er is left over , ” then the exact K ohn–Sham decompo- sition appears formally thin. Read instead through the inter- acting/noninteracting hierarchy dev eloped in this paper , E xc is already a structured object before approximation enters. It is useful here to separate e xchange and correlation more explicitly . In exact K ohn–Sham theory one may write E xc [ ρ ] = E x [ ρ ] + E c [ ρ ] . (49) Exchange is the contribution already generated by antisymme- try within the noninteracting reference system, whereas corre- lation is the further remainder beyond exchange once the e xact interacting and noninteracting problems have been matched at fixed density . This is not merely bookkeeping. Different exact constraints attach naturally to exchange, correlation, or the full exchange-correlation functional in different w ays, and this difference reflects the distinct roles these quantities play in comparing the two hierarchies. One important family of e xact properties concerns uniform coordinate scaling. 21–23 If the density is scaled as ρ γ ( r ) = γ 3 ρ ( γ r ) , γ > 0 , (50) then the corresponding exact functionals obe y T s [ ρ γ ] = γ 2 T s [ ρ ] , J [ ρ γ ] = γ J [ ρ ] . (51) Exchange scales linearly , E x [ ρ γ ] = γ E x [ ρ ] , (52) whereas correlation obeys the more subtle coupling-constant- related scaling structure familiar from e xact DFT . 21,22 These relations matter here not as isolated formal curiosities, but because they show that exchange and correlation transform differently precisely because they encode dif ferent aspects of the comparison between the interacting and noninteracting theories. Another exact structure concerns the one-electron limit and self-interaction cancellation. For every one-electron density , the exact correlation energy vanishes and exchange exactly cancels the Hartree self-repulsion: E c [ ρ 1 e ] = 0 , E x [ ρ 1 e ] = − J [ ρ 1 e ] . (53) Hence E xc [ ρ 1 e ] = − J [ ρ 1 e ] . (54) This is the exact one-electron self-interaction cancellation. 24,25 Its significance is both formal and practical. Formally , it sho ws that the interface quantity E xc is already highly constrained in the simplest possible density sector . Practically , it explains why self-interaction errors in approximate functionals are structural failures rather than minor empirical defects. Global energetic bounds and local hole structure provide another family of exact constraints. The best known example is the Lieb–Oxford bound, which places a univ ersal lower bound on the exchange-correlation energy . 26–28 In one common form, E xc [ ρ ] ≥ − C LO Z ρ ( r ) 4 / 3 d r , (55) with a universal constant C LO . The precise optimal v alue of the constant is not needed here. What matters for the present dis- cussion is that the existence of such a bound sho ws that the ex- act remainder term is subject to global many-body restrictions that are independent of any specific approximation. Closely re- lated are exchange-correlation hole constraints, which translate exact energetic structure into local information through nor- malization and sum rules. 25,29 T aken together , these constraints reinforce the same point: E xc is not an arbitrary leftover , but a tightly restricted interface quantity . The adiabatic connection is especially important in the present framew ork. 30–32 The exchange-correlation energy may be represented as an integral o ver interaction strength along a path that keeps the density fixed: E xc [ ρ ] = Z 1 0 W λ xc [ ρ ] d λ . (56) Here W λ xc [ ρ ] = tr  Γ λ ρ W  − J [ ρ ] , (57) where Γ λ ρ denotes a minimizing state of the coupling-constant- resolved constrained-search problem F λ [ ρ ] = inf Γ 7→ ρ tr  Γ ( T + λ W )  . (58) This representation is especially valuable here because it makes explicit that E xc is not only defined by subtraction in the K ohn– Sham decomposition. It is also the accumulated ener getic trace of moving from the exact noninteracting hierarchy at λ = 0 to the exact interacting hierarchy at λ = 1 along a density- preserving path. The adiabatic connection may be placed in the same varia- tional language used throughout this paper . For each fixed λ , the corresponding density-side functional F λ [ ρ ] and potential- side ground-state energy E λ [ v ] form a dual pair , with the two endpoint theories recov ered as F 0 [ ρ ] = T s [ ρ ] , F 1 [ ρ ] = F [ ρ ] . (59) In this sense, the adiabatic connection is the coupling-constant- resolved family linking the noninteracting and interacting 10 density-side hierarchies already isolated in the present recon- struction. This viewpoint also clarifies what kind of re gularity is actu- ally needed. At fixed density ρ , the map λ 7→ F λ [ ρ ] (60) is the infimum, ov er all admissible states yielding ρ , of af fine functions tr [ Γ ( T + λ W )] of λ , and is therefore monotone non- decreasing and concave. The adiabatic-connection formula should thus not be read as requiring naive ev erywhere dif- ferentiability in λ . Rather , one-sided deriv ati ves exist every- where, and differentiability almost ev erywhere is the natural generic situation. Under appropriate minimizer -selection and Hellmann–Feynman conditions, these slopes recov er the usual adiabatic-connection integrand. The same distinction emphasized throughout the paper also reappears here. It is one question for F λ [ ρ ] to be well defined at fixed density , and another for there to exist, for ev ery λ , a supporting potential v λ that realizes the same density on the potential side. The former is a density-functional statement; the latter is a stronger representability requirement along the entire coupling-constant path. Seen from this angle, se veral exact properties line up nat- urally . Scaling relations sho w that exchange and correlation transform dif ferently because they encode different parts of the interacting/noninteracting comparison. The one-electron limit shows ho w self-interaction must cancel exactly . The Lieb– Oxford bound and hole constraints express global and local restrictions on the same interface quantity . The adiabatic con- nection shows that E xc is the density-fixed interaction-strength integral linking the two endpoint theories. T aken together , these properties support the central claim of this section: in exact DFT , the exchange-correlation functional is the struc- tured interface between two exact v ariational hierarchies, not a miscellaneous repository of whatev er remains after everything else has been defined. This is also the methodological reason exact constraints mat- ter for approximation. 25,33 Their importance does not come merely from their usefulness as design principles for density- functional approximations. It comes from the fact that they arise from the exact theory itself and therefore reflect genuine structural information about how the interacting and noninter - acting hierarchies are related. D. F undamental gaps and the derivative discontinuity The decomposition of the fundamental gap is the natural point at which the two e xact hierarchies come back together . It is also the point at which the difference between an interacting slope quantity and an auxiliary spectral quantity must be kept especially clear . On the interacting side, the exact ensemble theory yields the many-body fundamental g ap through the one-sided slopes of the ground-state energy as a function of particle number . At integer particle number M , one has µ − M = E [ v , M ] − E [ v , M − 1 ] = − I M , µ + M = E [ v , M + 1 ] − E [ v , M ] = − A M , (61) and hence E true g = I M − A M = µ + M − µ − M . (62) The exact interacting gap is therefore fundamentally a state- ment about many-body addition and remov al energetics en- coded in one-sided slope data. Before any auxiliary one-body reference problem is introduced, the gap already e xists as an interacting ensemble quantity . On the noninteracting side, the K ohn–Sham reference prob- lem yields a dif ferent object, namely the exact one-body spec- tral gap E KS g = ε L − ε H , (63) where ε H and ε L are the highest occupied and lo west unoccu- pied Kohn–Sham lev els. This quantity belongs to the nonin- teracting hierarchy . It is exact auxiliary spectral data of the density-reproducing reference system, but it is not, merely by construction, the full man y-body addition/removal gap of the interacting problem. This distinction is structural rather than accidental. The true gap is defined by comparing neighboring particle-number sec- tors and is therefore intrinsically a man y-body slope quantity . The Kohn–Sham gap, by contrast, is defined within a fixed noninteracting one-body problem and is therefore intrinsically an auxiliary spectral quantity . These two objects are related, but they are not the same kind of object. Their mismatch is thus not best understood as a defect in an otherwise exact one-to- one identification. It is the natural consequence of comparing two dif ferent, though linked, exact v ariational hierarchies. This is precisely why the present paper has insisted on keep- ing the interacting and noninteracting sides distinct before coupling them through the K ohn–Sham construction. If one compresses the two hierarchies into a single storyline, the failure of the bare K ohn–Sham gap to reproduce the true many- body gap can look like an unexpected shortcoming. Once the two hierarchies are kept separate, the situation appears more natural. The interacting theory provides exact slope data of the many-body ener gy; the noninteracting theory provides e xact one-particle spectral data of the auxiliary density-reproducing system. The point of the gap decomposition is to explain ho w these two exact b ut nonidentical structures meet. The standard relation E true g = E KS g + ∆ xc (64) should therefore be read as a structural statement about the interface between the two theories rather than merely as a numerical correction formula. 8–10,17 The exchange-correlation deriv ati ve discontinuity ∆ xc is the term that records the part of the true many-body gap not carried by the bare auxiliary spectrum. From the viewpoint adopted here, ∆ xc is best understood as an interfacial quantity . It is not purely interacting in the same 11 sense as I M − A M , since it is defined relative to the K ohn–Sham reference system. But neither is it purely spectral, since its origin lies in the interacting ensemble structure and in the jump of the exact supporting potentials at integer particle number . It is the quantity through which many-body slope informa- tion survives when the theory is reexpressed in Kohn–Sham language. One may say the same thing from the perspectiv e of ex- act Kohn–Sham theory itself. The Kohn–Sham construction succeeds exactly at the lev el for which it is designed, namely density reproduction and total-energy minimization on density space. The true fundamental gap, howe ver , is not merely a density-reproduction datum. It is a statement about how the interacting energy changes when one crosses an integer particle number . The deriv ativ e discontinuity is precisely the term that remembers this interacting slope-side information when the exact theory is re written in terms of the auxiliary K ohn–Sham system. This is why the gap decomposition is such a useful closing example for the present reconstruction. It displays, in one place, both the exactness and the limitation of the K ohn–Sham representation. The Kohn–Sham system is exact as an auxiliary density-reproducing reference problem. But bare eigenv alue dif ferences alone do not constitute a complete man y-body addi- tion/remov al theory . The deri vati ve discontinuity is the missing interfacial term required to connect the auxiliary spectrum to the true many-body gap. CONCLUDING REMARKS The contribution of the present work is not a ne w theorem but a formal reor ganization of exact DFT . The central proposal is that exact DFT is read most clearly as two parallel ensemble variational hierarchies, namely an interacting hierarchy and a noninteracting hierarchy , together with the K ohn–Sham aux- iliary construction that links them on a common admissible density class. From this viewpoint, Lieb’ s ensemble formulation is the broad natural starting point for the interacting side, while the exact ensemble noninteracting framew ork built from T s and E s is the corresponding starting point for the noninteracting side. Le vy–Lieb, Hohenberg–K ohn, and ordinary pure-state noninteracting or Kohn–Sham formulations then appear as narrower specializations that remain exact on the domains where the relev ant admissible-class restrictions do not exclude the minimizing state. This reorganization clarifies se veral distinctions that are of- ten blurred in compressed presentations of exact DFT . In partic- ular , it helps separate functional domain from representability class, closure from exact pointwise membership, fractional particle number from fractional orbital occupation, and density reproduction from spectral interpretation. It also clarifies why fractional particle number , piece wise linearity , deri vati ve dis- continuity , fractional occupations, and Janak-type relations are best understood not as isolated exact facts b ut as consequences of the variational geometry of the interacting and noninteract- ing ensemble theories and of their K ohn–Sham coupling. The same perspectiv e sharpens the status of the exchange- correlation functional. In exact theory , E xc is not merely an unknown remainder introduced for practical approxima- tion. It is the interface quantity that records the difference between the interacting and noninteracting hierarchies once the Hartree contribution has been separated out. This is why so many e xact properties of E xc are structural rather than merely approximation-oriented. The aim of the paper has therefore been conceptual clarifica- tion rather than formal no velty . 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