Universality as Universal Property in Density Functional Theory: SpectralXC and Multiscale Renormalisation of Exchange–Correlation

Universality as Universal Property in Density Functional Theory: SpectralXC and Multiscale Renormalisation of Exchange–Correlation Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We introduce Universal–Property SpectralXC (UP–SpectralXC), a new multiscale framework for electronic exchange–correlation that unifies two previously disconnected ideas: spectral constraints on Kohn–Sham generators (SpectralXC) and universal–property renormalisation group theory (UP–RG). In this approach, electronic structure is described by a KS–like generator L [ n ; θ ]over a descriptor manifold Θ(geometry, density regime, external parameters), equipped with a higher (2 − )connection that transports the full operator algebra A = C∗ ( L )along paths in Θ. Resolvent and projector probes fk ( L )extract spectrally local information—frontier gaps, projector ranks, spectral moments, response coefficients—which are evaluated through an Ad –invariant state Φto produce spectral laws Sfk [ n ] = Φ( fk ( L [ n ])). These laws encode deep, model–independent invariants of electronic correlation, such as compressibility limits of the homogeneous electron gas, strong–coupling asymptotics of the strictly correlated electron regime, Luttinger–liquid exponents, and topologically protected projector classes ( K0 ). They provide the target data φk for a constrained Levy–Lieb variational principle, producing a non–empirical exchange–correlation potential via a universal resolvent formula and dynamically determined Lagrange multipliers. UP–RG endows this construction with a rigorous multiscale structure. Spectral laws and projector classes assemble into universal objects Uµ living in enriched categories of spectral data indexed by a scale parameter µ . Coarse–graining, downfolding, basis changes, and embedding operations become functors Fµ→µ0 acting on these spectral universals. We define a spectral defect functional—a structural generalisation of c -, a -, and F –theorems—which measures the distance of the current spectral object from its ideal tile values, and show that this defect is monotone under admissible UP–RG flows. Curvature of the higher connection and categorical loop curvature provide principled diagnostics of irreversibility, scheme dependence, and the breakdown of universality. In this formulation, universality ceases to be a heuristic slogan and becomes a literal universal property: the equivalence class of theories sharing isomorphic spectral universals under UP–RG functors. This unified framework yields new practical capabilities: (i) multiscale XC methods whose behaviour is consistent under basis refinement, downfolding, and embedding; (ii) a renormalisation interpretation of dissociation, strong correlation, and d-/f-shell physics through the flow of projector classes and frontier spectral gaps; (iii) spectral monotones detecting phase crossovers between HEG, SCE, and Luttinger–liquid regimes; (iv) numerical error bars derived from curvature rather than empirical fits; and (v) a path to XC “ c –functions” defined on universal spectral structures rather than on couplings. UP–SpectralXC thus provides the first exchange–correlation theory in which renormalisation, universality, and spectral structure are fully integrated, yielding a principled, non–empirical, and multiscale description of electronic correlation. I. INTRODUCTION A. Motivation The modern theory of many-body physics is built on two towering conceptual pillars that, somewhat paradoxically, barely talk to each other in a structural way: the renormalisation group (RG) on the one hand, and density functional theory (DFT) on the other. RG explains why microscopic details often do not matter and why qualitatively different Hamiltonians can share the same long-distance physics [ 1 – 3 ]. DFT explains why, for ground-state properties, the full many-body wave function is often an overkill and a one-body density n ( r )can suffice [ 6 – 9 ]. Yet, in both frameworks, the notion of universality is, in practice, closer to a slogan than to a precise structural invariant: • in RG, universality classes, c -theorems, and “irrelevant operators” are usually expressed in terms of coordinates on a space of couplings, and depend on choices of scheme and regulator; • in DFT, exchange–correlation (XC) functionals are patched together from the homogeneous electron gas (HEG) and various fitted models, with no explicit notion of what structure must be preserved across systems and across scales. Our goal in this work is to fuse a spectral formulation of XC (SpectralXC) with a universal-property formulation of RG (UP–RG) to achieve three things: 2 1. make “universality” literally a universal property in the category-theoretic sense [ 18 ], attached to objects that are preserved functorially under coarse–graining; 2. build an XC framework that is structurally constrained by exact spectral laws and universal properties, rather than heuristically fitted; 3. introduce a genuinely multiscale, scheme-aware XC theory equipped with monotones and curvature diagnostics in the spirit of RG c-theorems [3, 5, 53]. In this subsection we motivate this fusion by revisiting the standard RG story, the standard DFT/XC story, and the regimes where both approaches systematically fail. The usual RG story: universality and its limitations Consider a Euclidean scalar field theory in ddimensions with a UV cutoff Λand microscopic action SΛ[φ] = ZRd ddx1 2ZΛ(∇φ)2+1 2m2 Λφ2+λΛ 4! φ4+X i gΛ,iOi(φ),(1) where the Oi are local operators built from φ and its derivatives. The (Wilsonian) partition function with sources Jis ZΛ[J] = Zkpk≤Λ Dφexp −SΛ[φ] + Zddx J(x)φ(x).(2) The essence of Wilson’s RG [ 1 – 3 ] is to define an effective action at a lower cutoff Λ /b by integrating out the fast modes in a momentum shell: e−SΛ/b[φ<]=ZΛ/b≤kpk≤Λ Dφ>e−SΛ[φ<+φ>],(3) where φ<, φ> denote lowand high-momentum components, respectively. A rescaling of momenta and fields restores the cutoff to Λ, and the couplings transform according to gΛ,i 7−→ g0 Λ/b,i =byigΛ,i +O(g2),(4) with yithe engineering plus anomalous dimension of the operator Oi. Passing to a continuous description in terms of a sliding scale µ yields the familiar RG flow equations µdgi(µ) dµ =βig(µ), βi(g) := dgi dln µ,(5) where each βi is computed from the effective action under integrating out shells [ 5 ]. Linearising (5) around a fixed point g∗, βi(g)≈Mij(gj−g∗ j), Mij := ∂βi ∂gjg=g∗ ,(6) one identifies eigen-directions with eigenvalues θα: µduα dµ =θαuα, uα(µ)∼µθαuα(µ0).(7) The standard classification follows: •θα>0(relevant operators): perturbations that grow in the IR; •θα<0(irrelevant operators): perturbations that die out in the IR; •θα= 0 (marginal operators): perturbations that require higher-order analysis. 3 Two microscopic Hamiltonians that differ only by irrelevant operators flow to the same fixed point and are said to be in the same universality class. In two dimensions, Zamolodchikov’s c -theorem [ 53 ] gives a scalar function C ( g )on coupling space such that dC dln µ=−Gij(g)βi(g)βj(g)≤0,(8) with a positive-definite “metric” Gij constructed from two-point functions of local operators. At fixed points, C coincides with the conformal central charge c , so that cUV > cIR . Analogous statements exist in four dimensions for the a-anomaly and in three dimensions for certain free-energy functionals, giving rise to the a-theorem and F-theorem, respectively [3, 5]. From a pragmatic point of view, this story is extraordinarily successful: it explains critical exponents, scaling functions, universality classes, and even constrains the space of quantum field theories. But from the point of view we adopt in this work, it suffers from three deep limitations: 1. Universality is expressed in a coordinate-dependent way. The objects of interest are coordinates gi in a space of couplings and scalar functions C ( g )defined on that space. Two different choices of scheme (regulator, normalisation) correspond to different coordinate charts and different β-functions, and while fixed-point existence is scheme-invariant, the detailed form of C(g)is not. 2. Universality is a slogan, not a structure. The statement that “details do not matter” is implemented by discarding irrelevant operators, but what is kept is not defined as a universal object in a category-theoretic sense. Universality classes organise theories by the topology of their flows in coupling space, but not by universal properties in the sense of category theory [18]. 3. Scalar monotones fail in many physically interesting regimes. In non-Lorentz-invariant systems, strongly coupled non-relativistic flows, lattice systems with complex coarse-graining steps, and theories with non-invertible symmetries or limit cycles, scalar c-like candidates often cease to be monotone; they may oscillate, depend strongly on scheme, or not exist at all. The RG paradigm itself is not at fault, but its expression in terms of scalar functions of couplings is too fragile. In the UP–RG perspective developed in this work, we retain the idea that RG is a flow, but rephrase it in structural terms: RG becomes a family of functors between categories of models, and universality is encoded in universal objects preserved (up to canonical equivalence) under those functors. Scalar monotones like c or a should then be seen as projections of more general defect functionals defined on these universal structures. The usual DFT/XC story: patching functionals without structure Density functional theory, in its ground-state formulation, proceeds from the Hohenberg–Kohn (HK) theorem [ 6 ]. Given an N -electron system with external potential vext and Coulomb interaction, the HK theorem states that there exists a universal functional E[n]of the one-body density n(r)such that E0= inf nE[n]n(r)≥0,Zd3r n(r) = N,(9) and that, for a given vext , the ground-state density n0 ( r )uniquely minimises (9) . A modern and more mathematically robust formulation is due to Levy and Lieb [ 8 , 9 ]: define the universal Hohenberg–Kohn functional F[n] := inf Ψ→nΨˆ T+ˆ VeeΨ,(10) where the infimum is taken over all N -body wave functions Ψwhose density is n . Then the ground-state energy is obtained by minimising E[n] = F[n] + ZR3 d3r vext(r)n(r)(11) over N-representable densities. 4 Kohn and Sham [ 7 ] introduced a non-interacting reference system with density n ( r )and kinetic energy functional Ts[n], which permits the decomposition F[n] = Ts[n] + EH[n] + Exc[n],(12) where EH[n] = 1 2Zd3r d3r0n(r)n(r0) |r−r0|(13) is the Hartree energy, and Exc [ n ]absorbs the difference between the true interacting kinetic+interaction energy and the reference Ts[n] + EH[n]. The Kohn–Sham equations then read −1 2∇2+vext(r) + vH(r) + vxc(r)φi(r) = εiφi(r),(14) with vH(r)the Hartree potential and vxc(r) = δExc[n] δn(r)(15) the exchange–correlation potential. All the difficulty of practical DFT is now encoded in Exc [ n ]. The standard paradigm is to approximate Exc using: • the local density approximation (LDA), in which Exc [ n ]is approximated pointwise by the XC energy density of the homogeneous electron gas at local density n(r)[10]; • generalised gradient approximations (GGAs), which add a dependence on the density gradient ∇n and are usually parametrised or semi-parametrised to satisfy selected exact constraints and to perform well on benchmark sets [11, 12]; • meta-GGAs, hybrids, and range-separated hybrids, which include further ingredients (kinetic energy density, exact exchange fractions) and introduce more parameters. From a practical viewpoint, this ladder of approximations is extremely useful and underlies much of electronic-structure simulation in chemistry and condensed matter physics. However, when viewed through the structural lens emphasised in this work, the situation is unsatisfactory in several respects: 1. The XC functional is a patchwork, not a universal object. The LDA, GGAs, and metaGGAs are constructed as closed-form expressions that “borrow” information from the homogeneous electron gas or from selected exact conditions, but there is no sense in which Exc [ n ]is defined as a universal object in a category of models. The HEG is used as a calibration point rather than as a source of universal structure. 2. Fitting replaces structure. Parameters in GGA and meta-GGA functionals are chosen to minimise empirical error over benchmark sets, not to enforce universal properties such as sum rules, spectral invariants, or K-theoretic sector stability. While many functionals are constraint-based, the constraints themselves are usually expressed in terms of low-order scalar quantities (e.g., gradient expansions) rather than in terms of the spectral structure of a KS-like generator. 3. Failure in strongly correlated and multiscale regimes. In systems with strong static correlation, multi-reference character, near degeneracies, d/f-shell physics, bond dissociation, and multiscale embeddings, standard XC approximations are known to break down: •stretched bonds and reaction barriers are incorrectly described; •Mott insulators and charge-transfer complexes are often misclassified; • transition-metal complexes exhibit incorrect spin states, orbital occupancies, and excitation spectra; • embedding and downfolding methods (DMRG, DMET, and related approaches) can be sensitive to the choice of XC treatment in the environment and to basisand scheme-dependence [ 16 , 17 ]. 5 Many of these failures can be traced to the inability of standard XC functionals to see the spectral structure of the one-particle problem in a way that respects universality across scales. Attempts to go beyond semi-local functionals, such as strictly correlated electrons (SCE) [ 13 , 14 ], adiabatic connection interpolations, and orbital-dependent functionals, incorporate more of the manybody structure explicitly, but they generally lack a multiscale, RG-based view and a universal-property formulation of what ought to be preserved under coarse–graining and embedding. Where both RG and XC fail: strong correlation and multiscale problems The regimes where RG and current XC functionals struggle most are, perhaps unsurprisingly, the same: •Strongly correlated electrons: near the Mott transition, in low dimensions where Luttingerliquid behaviour appears [ 15 ], or in SCE-like low-density regimes, the electron system is governed by highly nonlocal correlations that are not well captured by local or semi-local functionals, nor by simple perturbative RG flows in coupling space. •Multi-reference and near-degenerate systems: bond breaking, transition-metal complexes, and large conjugated systems exhibit near-degenerate frontier orbitals and strong static correlation. Traditional c-theorems and beta functions say little about such finite-size, molecule-scale phenomena, and traditional XC functionals lack the spectral resolution to represent them properly. •Multiscale embedding and downfolding: techniques such as DMRG, DMET, and ab initio downfolding construct effective models for active spaces embedded in an environment [ 16 , 17 ]. Yet there is no structural theory of how XC information should flow between scales, nor of which parts of the one-particle spectrum must be preserved exactly when integrating out high-energy states. In all these situations, the conceptual tools of RG (universality, relevance/irrelevance, c-theorems) and of DFT (HK functionals, KS decomposition, LDA/GGAs) are individually inadequate. The former is too coarse and scalar; the latter too local and empirical. Aim: SpectralXC + UP–RG The central aim of this work is to combine a spectral formulation of exchange–correlation (SpectralXC) with a universal-property formulation of renormalisation group (UP–RG): 1. Universality as universal property. We treat universality not as an informal slogan but as a literal universal property in the sense of category theory [ 18 ]: universal objects (limits, centers, projectors, universal states) are defined by their mapping properties and are preserved (up to canonical equivalence) under the appropriate RG functors. Universality classes are then equivalence classes of models sharing isomorphic universal objects, not merely flows to the same fixed point in a chosen coordinate chart. 2. Structurally constrained XC. In SpectralXC, we associate to each KS-like generator L [ n ; θ ] a family of spectral probes fk ( L )and an Ad-invariant state Φ, from which we compute spectral quantities Sfk[n;θ]=Φfk(L[n;θ]). Targets φk for these spectral laws are obtained from exact or highly accurate many-body physics in special regimes (tiles): the homogeneous electron gas, SCE limits, Luttinger liquids, and multi-orbital Hubbard models [ 13 – 15 ]. Rather than fitting XC parameters, we enforce Sfk [ n ; θ ] = φk as constraints in a constrained Levy–Lieb variational problem. The XC potential emerges via a universal resolvent formula and dynamically determined Lagrange multipliers, making XC structurally constrained by spectral laws. 3. Multiscale, scheme-aware XC with monotones and curvature. UP–RG endows SpectralXC with a multiscale structure: spectral laws and projector classes form universal objects Uµ at each scale µ , and coarse–graining, embedding, and basis changes act as functors Fµ→µ0 on these objects. We 6 define spectral defect functionals Defectµ ( Uµ ), which play the role of DFT analogues of c -functions, and show that they are monotone under admissible RG flows. Curvature of the higher connection and categorical loop curvature provide diagnostics of irreversibility and scheme dependence, giving a structural answer to questions usually relegated to “basis-set errors” or “functional sensitivity.” In this way, universality is upgraded from an informal statement about insensitivity to microscopic details, to a literal universal property of spectral objects in a category of electronic-structure models. The combination of SpectralXC and UP–RG yields new capabilities: a principled, non-empirical way of building XC functionals tied to exact spectral invariants; a multiscale view of XC consistent under basis refinement, embedding, and downfolding; and structural monotones and curvature diagnostics generalising c-theorems to the realm of density functional theory. B. Key ideas in one paragraph At a technical level, the key ideas of this work can be summarised as follows. For each point ( n, θ )in a combined space of densities and external descriptors Θ(geometry, lattice, fields, interaction scale, etc.), we associate a Kohn–Sham-like generator L[n;θ] : H1→ H1, L[n;θ]∗=L[n;θ],(16) a self-adjoint operator acting on a one-particle Hilbert space H1 (for instance, a KS or generalized KS Hamiltonian, or a correlation kernel). By the spectral theorem [ 19 ], L [ n ; θ ]admits a projection-valued spectral measure En,θ(λ)such that L[n;θ] = ZR λ dEn,θ(λ),(17) and any bounded Borel function f:R→Cdefines an operator f(L[n;θ]) := ZR f(λ)dEn,θ(λ).(18) This functional calculus is the mathematical mechanism by which we probe the correlation-relevant spectral structure of L [ n ; θ ]: eigenvalues and spectral bands near the Fermi level, d-/f-shell manifolds, low-energy excitations, and smallq response modes, all live in σ ( L [ n ; θ ]) and its associated spectral projectors [ 22 , 62 ]. Physically, one may think of L [ n ; θ ]as the operator whose spectrum controls the response of the system to one-body perturbations; in standard KS-DFT this is simply the KS Hamiltonian, but more generally it can be a dressed or effective one-particle generator. The first key ingredient of SpectralXC is the choice of a finite family of spectral probes fk:R→C, k = 1, . . . , m, (19) and a normalised, positive, Ad-invariant state Φ : A→C,Φ(1) = 1,Φ(X∗X)≥0,Φ(UXU−1) = Φ(X),(20) on the C∗ -algebra A = C∗ ( L [ n ; θ ]) generated by L [ n ; θ ][ 21 ]. The Ad -invariance ensures that the quantities we extract do not depend on the choice of orbital basis or any internal gauge. Typical probes fk include: • resolvent-based probes, fk ( λ ) = ( λ−zk ) −1 and their linear combinations, which emphasise spectral regions near zk(frontier orbitals, d-/f-shells, gap edges); •Riesz projectors, realised via contour integrals fk(λ) = χIk(λ)and PIk(L[n;θ]) = 1 2πi I∂Ik (z−L[n;θ])−1dz, (21) which project onto selected spectral subspaces (occupied space, correlated subspace, active space, etc.); • moment probes, fk ( λ ) = λpg ( λ ), which encode spectral moments that enter sum rules and response functions. 7 Given (fk,Φ), we define the spectral quantities Sfk[n;θ] := Φfk(L[n;θ]), k = 1, . . . , m, (22) which are complex (typically real) numbers extracted from the spectrum of L [ n ; θ ]in a basis-independent fashion. Each Sfk is, by construction, a scalar that summarises how the spectrum sits near specific regions of interest for correlation: resolvent probes amplify the contribution of eigenvalues close to zk , projectors count spectral weight in specified windows, and moments encode exact sum rules [ 19 , 22 ]. In particular, if fk is a projector onto a spectral interval Ik , then Sfk can implement integer constraints on electron counts or sector dimensions; if fk is a resolvent centered between HOMO and LUMO, Sfk becomes highly sensitive to gap collapse and frontier level repulsion, which are strong indicators of static correlation and charge-transfer behaviour. The second key ingredient is the introduction of tiles, special regions in descriptor space Θwhere the many-body problem is either exactly solvable or under excellent theoretical control. Concretely, a tile Tα⊂ Θis a regime (for example: the homogeneous electron gas at density n ; the strictly correlated electron limit at large rs ; a one-dimensional Tomonaga–Luttinger liquid; or a multi-orbital Hubbard-like model describing a transition-metal d-shell) such that, for ( n, θ ) ∈Tα , the corresponding interacting system is sufficiently well understood to compute, for the chosen probes, target values φ(α) k:= S(exact) fk[n;θ],(n, θ)∈Tα.(23) These targets may come from analytic solutions (e.g. Luttinger exponents in 1D [ 22 ]), from controlled strong-correlation asymptotics (e.g. SCE), or from highly accurate numerical treatments (QMC for HEG, DMRG for 1D models). The crucial point is that the φk are not free parameters: they are fixed by independent many-body physics on the tiles, and are treated as non-empirical constraints in what follows. The third key ingredient is to encode these spectral invariants as constraints in a constrained Levy–Lieb problem. Starting from the universal Hohenberg–Kohn functional F [ n ]and total energy functional E [ n ] (cf. (10)–(11)), we define the constrained energy Econ[n, {λk}] := E[n]− m X k=1 λkSfk[n;θ]−φk,(24) where the λkare Lagrange multipliers enforcing the spectral laws Sfk[n;θ] = φk, k = 1, . . . , m. (25) The constrained Levy–Lieb problem is then to minimise Econ[n, {λk}]over admissible densities n, (26) with the λk adjusted such that (25) hold at the minimiser. Varying (24) with respect to n ( r )at fixed λk yields the Euler–Lagrange equation δE[n] δn(r)− m X k=1 λk δSfk[n;θ] δn(r)= 0.(27) Comparing with the usual Kohn–Sham equation (15) , we identify the constrained exchange–correlation potential as vcon xc (r) = v(0) xc (r) + m X k=1 λk δSfk[n;θ] δn(r),(28) where v(0) xc is a reference XC potential (which may be as simple as LDA, or even zero if we absorb everything into the constrained term). The nontrivial task is to compute the functional derivatives δSfk/δn(r)in a way that keeps the spectral structure explicit. To do so, we invoke the holomorphic functional calculus and the resolvent representation of fk ( L [ n ; θ ]): for each holomorphic probe fkwe may write fk(L[n;θ]) = 1 2πi IΓk fk(z) (z−L[n;θ])−1dz, (29) 8 where Γ k is a contour enclosing that part of the spectrum of L [ n ; θ ]where fk is supported [ 19 , 62 ]. A first-order variation δL of L[n;θ]induces a variation δfk(L[n;θ]) = 1 2πi IΓk fk(z) (z−L[n;θ])−1(δL) (z−L[n;θ])−1dz, (30) which follows from the standard resolvent identity ( z−L−δL ) −1 = ( z−L ) −1 +( z−L ) −1 ( δL )( z−L ) −1 + . . . and holomorphy of fk[62]. Writing δL in terms of the density variation as δL =Zd3r0δL δn(r0)δn(r0),(31) and using the linearity and continuity of Φ, we obtain δSfk[n;θ] = Φδfk(L[n;θ]) =Zd3r0Φ1 2πi IΓk fk(z) (z−L[n;θ])−1δL δn(r0)(z−L[n;θ])−1dzδn(r0).(32) By identification with the usual Gateaux derivative, this yields the explicit functional derivative δSfk[n;θ] δn(r)= Φ1 2πi IΓk fk(z) (z−L[n;θ])−1δL δn(r)(z−L[n;θ])−1dz.(33) Substituting (33) into (28) gives the resolvent-based XC potential: vcon xc (r) = v(0) xc (r) + m X k=1 λkΦ1 2πi IΓk fk(z) (z−L[n;θ])−1δL δn(r)(z−L[n;θ])−1dz.(34) Physically, each probe fk contributes an XC correction that is nonlocal in the spectrum of L [ n ; θ ]but still local in the density variable r , and the λk adjust so as to enforce the spectral laws (25) . Importantly, the λk are not fitted once and for all: they are determined dynamically in each system by the condition that Sfk [ n ; θ ] = φk at self-consistency. In this sense they are dual variables conjugate to the spectral constraints, rather than empirical parameters. The final key idea is to lift this spectral construction into the universal-property RG picture. For each scale µ (spatial resolution, basis size, energy window, embedding level), we consider a category Cµ whose objects are spectral XC schemes ( n, θ, L [ n ; θ ] ,{fk}, Φ) defined at that scale, and whose morphisms are scheme changes and coarse-graining maps (basis transformations, downfoldings, embeddings) that preserve the relevant operator-algebraic structure. To each object we associate a spectral universal object Uµ:= {Sfk[n;θ]}k,{[PI(L[n;θ])] ∈K0(A)}, νn,θ(λ),(35) where [ PI ( L )] ∈K0 ( A )are K-theory classes of Riesz projectors and νn,θ is an optional spectral measure summarising parts of the spectrum. RG steps are realised as functors Fµ→µ0:Cµ→ Cµ0, µ > µ0,(36) which send ( L, {fk}, Φ) to ( L0,{f0 k}, Φ 0 )at a coarser scale, inducing a flow Uµ7→ Uµ0 . On this space of spectral universals we define a defect functional Defectµ(Uµ) := m X k=1 wkSfk[n;θ]−φk2or more generally Defectµ=Dνn,θkν?,(37) where wk≥ 0are weights and D is, for example, a relative entropy between spectral measures [ 21 ]. The UP–RG principle is to pick an admissible class of RG functors Fµ→µ0such that Defectµ0Uµ0≤DefectµUµ, µ > µ0,(38) so that the spectral defect becomes a monotone along RG trajectories, generalising c-theorems to the spectral XC context [ 23 , 24 ]. Universality is then no longer a vague statement about “insensitivity to microscopic details,” but the precise assertion that different microscopic schemes and scales share isomorphic universal objects Uµ up to canonical equivalence, and that their flows decrease a structural defect functional rather than a particular scalar function of couplings. 9 C. What this paper does The remainder of this article develops in detail a framework that we call Universal–Property SpectralXC (UP–SpectralXC), which fuses the spectral, constraint-based construction of exchange–correlation (XC) outlined in Subsection I B with a universal-property formulation of renormalisation group (UP–RG) inspired by the structural viewpoint of category theory and operator algebras. In this subsection we give a precise, but still conceptual, description of what the paper actually achieves, both mathematically and physically. The technical developments that follow in later sections will implement these ideas explicitly, with particular attention to strongly correlated electrons, multiscale embedding, and scheme dependence in density functional theory (DFT) [ 26 , 70 ]. We emphasise throughout that the central conceptual move is to upgrade “universality” from a heuristic statement about insensitivity to microscopic details to a literal universal property of spectral objects, and to use this structure to define and control XC across scales. (i) Formal definition of UP–SpectralXC The first main contribution of this paper is to give a formal definition of UP–SpectralXC as a precise mathematical object. We work over a descriptor manifold Θthat parametrises external conditions (geometry, lattice, fields, interaction scale, and similar data), and consider admissible densities n that satisfy the usual constraints (non-negativity, N -representability). A UP–SpectralXC scheme at a given scale µis defined as the following data: Sµ:= n, θ, L[n;θ], A =C∗(L[n;θ]),{fk}m k=1,Φ,{φk}m k=1,(Aµ, Bµ),(39) where: 1. L [ n ; θ ] : H1→ H1 is a self-adjoint KS-like generator as in (16) , with spectral resolution given by (17); 2. A = C∗ ( L [ n ; θ ]) is the C∗ -algebra generated by L [ n ; θ ], on which we have a functional calculus (18) ; 3. {fk}m k=1 is a finite family of spectral probes (19) (resolvents, Riesz projectors, moment probes) determining operators fk(L[n;θ]) via (29); 4. Φ : A→C is a normalised, positive, Ad -invariant state (20) , which we interpret physically as a trace-like evaluation over the one-particle sector or a suitable KMS state; 5. the tile targets {φk} are fixed numbers obtained from exact or benchmark many-body physics in special regimes (tiles) of Θ, as in (23); 6. ( Aµ, Bµ )is a higher (2 − )connection on Θat scale µ that transports the operator algebra A and its probes along descriptor paths, with curvature split into central and non-central components; its role is to define how spectral data moves across Θand across scales. From this data we build the spectral quantities Sfk [ n ; θ ]as in (22) and enforce the spectral laws (25) via the constrained energy functional (24) . A stationary point of the constrained Levy–Lieb problem (26) gives rise to an effective XC potential vcon xc of the form (34) . The formal definition of UP–SpectralXC, outlined in Section I and expanded in later sections, consists in specifying the class of admissible probes, states, tile targets, and higher connections, as well as the precise conditions under which the resulting XC potential is well-defined, gauge-invariant, and size-consistent [70]. From a physics standpoint, this formal definition serves two purposes: first, it makes explicit the spectral content of the XC problem (frontier gaps, d-/f-manifolds, lowq response, strong-correlation asymptotics); second, it encodes exact physics from HEG, SCE, TLL, and related tiles as non-empirical constraints, rather than as ingredients in a fitted ansatz. The constrained variational structure ensures that these constraints are implemented in a thermodynamically consistent way, with λk playing the role of dual variables conjugate to the spectral laws. 16 • No fitted parameters enter the construction: the only numerical data are the target values φk obtained from exact or benchmark tile physics. From a physical perspective, (73) shows that exchange–correlation is generated by how the spectrum of the effective one-particle problem responds to changes in the density, filtered through probes that encode universal many-body information. The spectral laws ensure that, in carefully chosen directions in density/descriptor space, the KS-like description reproduces key spectral features of the exact problem; the resolvent formula ensures that this requirement is enforced in a manner compatible with the variational structure of DFT. The remainder of the paper will build on this spectral foundation and embed it into the universalproperty RG framework developed in Subsection I C, thereby endowing SpectralXC with a multiscale, scheme-aware structure and with defect monotones that play a role analogous to c-functions. B. Universal-Property RG in brief In this subsection we reformulate the renormalisation group (RG) in the universal-property language that will be used throughout the paper. Our goal is not to replace the standard Wilsonian picture of RG flows in coupling space [ 1 – 3 , 30 ], but to lift it to a categorical setting where (i) RG steps are functors between categories of models, (ii) the data that truly survive RG are expressed as universal objects, and (iii) c-theorem-like statements appear as monotonicity properties of defect functionals on these universal objects. This structural point of view will later be combined with the spectral XC construction of Subsection II A to define UP–SpectralXC. Families of categories indexed by scale Let µ > 0be a scale parameter. Depending on context, µmight be: •a UV momentum cutoff Λ, •an inverse length or energy scale, •a resolution scale in a tensor-network representation, •a discretisation scale (grid spacing, lattice spacing), •or, more abstractly, a point in a partially ordered index set of coarse-graining levels. Instead of thinking of RG as a flow on a single space of couplings, we consider a family of categories Cµµ>0,(74) indexed by µ, where: • objects of Cµ represent physical models or descriptions at scale µ (e.g. local actions, effective Hamiltonians, nets of operator algebras, tensor-network states); • morphisms of Cµ represent structure-preserving maps between such models (e.g. field redefinitions, unitary equivalences, gauge transformations, embeddings). Concretely, one may take Cµto have as objects: • pairs ( Aµ, ωµ )where Aµ is a C∗ -algebra of observables at scale µ and ωµ is a state (expectation functional) on Aµ; •or triples (Sµ,Oµ,Hµ)of actions, operator sets, and Hilbert spaces; •or, in lattice systems, tensor-network states modulo local gauge equivalences. The precise choice of Cµ depends on the physical setting (continuum QFT, lattice model, quantum spin system), but the categorical properties we use will be independent of this choice. 17 RG steps as functors In the standard Wilsonian picture, an RG step integrates out short-distance degrees of freedom and rescales the system. At the level of couplings, this is described by a map gi(µ)7−→ gi(µ0), µ > µ0,(75) leading to the beta-functions (5) . In our universal-property picture, we instead model RG steps as functors between the categories Cµ: Fµ→µ0:Cµ−→ Cµ0, µ > µ0.(76) For each object Xµ∈Ob(Cµ),Fµ→µ0produces a coarse-grained object Xµ0:= Fµ→µ0(Xµ)∈Ob(Cµ0),(77) and for each morphism f:Xµ→Yµ, Fµ→µ0(f) : Fµ→µ0(Xµ)→Fµ→µ0(Yµ)(78) in a way that preserves composition and identities: Fµ→µ0(idXµ) = idFµ→µ0(Xµ), Fµ→µ0(g◦f) = Fµ→µ0(g)◦Fµ→µ0(f).(79) Intuitively, Fµ→µ0forgets or compresses short-distance structure while retaining long-distance physics. Composition of RG steps corresponds to composition of functors: if µ > µ1> µ0, then Fµ1→µ0◦Fµ→µ1=Fµ→µ0,(80) expressing the semigroup property of RG transformations. In the language of [ 3 , 30 ], (80) encodes the statement that coarse-graining from µ directly to µ0 is equivalent to coarse-graining in stages from µ to µ1 and then from µ1 to µ0 . At the level of couplings, (80) is simply the chain rule for integrating the beta-functions. Universal objects as what survives RG The central idea of UP–RG is that what really survives RG is not a particular choice of couplings, but certain universal objects attached to the categories Cµ . In category theory, an object U is said to satisfy a universal property (e.g. being a limit, colimit, initial or terminal object, centre of a monoidal category) if it is characterised uniquely up to unique isomorphism by the existence of certain morphisms with specific mapping properties [ 18 , 24 ]. For instance, the categorical product A×B of two objects is characterised by the property that for any object X with maps X→A and X→B , there exists a unique map X→A×B making the relevant diagram commute. This kind of characterisation is intrinsic: it does not depend on coordinates or auxiliary choices. In UP–RG we postulate, for each scale µ, the existence of one or more universal objects Uµ∈Ob(Uµ),(81) living in some enriched category Uµbuilt from Cµ. Typical examples include: •limits or colimits of diagrams of models at scale µ, representing IR or UV fixed-point theories; • centres of monoidal categories of defects or boundary conditions, capturing topological sectors and symmetry-enriched structures; • universal states ωµ obtained by minimisation of an action or entropy functional over objects in Cµ ; • universal projectors or idempotents encoding decomposition into superselection sectors or topological charge sectors. 18 In the context of SpectralXC, the Uµ will be built from spectral quantities, projector classes, and spectral measures of KS-like generators, as in (40); here we keep the discussion general. The universality class at scale µis then defined as the isomorphism class Univµ:= IsoUµ,(82) whose elements are universal objects Uµ related by isomorphisms in U µ . Two microscopic models (possibly with different coordinates, regulators, or schemes) are in the same universality class if their associated universal objects are isomorphic. This avoids reference to specific couplings and instead identifies universality at the level of intrinsic structure. Under an RG functor Fµ→µ0, universal objects are mapped to universal objects: Fµ→µ0:Uµ7−→ Uµ0,(83) in a way that respects their universal properties (e.g. limits are sent to limits, centres to centres, universal states to universal states). This is the sense in which universality is literally a universal property: the object that encodes universal physics at scale µ is picked out uniquely up to isomorphism by a mapping property, and this object is preserved functorially under RG. Defect functionals and structural monotonicity To generalise c-theorems, we introduce a defect functional Defectµ: Ob(Uµ)−→ R≥0, Uµ7−→ Defectµ(Uµ),(84) which measures how far Uµ is from an “ideal” universal object U? (typically an IR fixed-point object). We require at least: 1. Non-negativity: Defectµ ( Uµ ) ≥ 0, with equality if and only if Uµ is isomorphic to U? (or to a member of the target universality class). 2. RG contractivity: For an admissible RG functor Fµ→µ0as in (76), we require Defectµ0Uµ0≤DefectµUµ, µ > µ0,(85) where Uµ0is the image of Uµunder the induced flow (83). Equation (85) is the universal-property analogue of Zamolodchikov’s c-theorem (8) : instead of a scalar function C ( g )of couplings, we have a functional defined on universal objects, and instead of a specific expression for dC/d ln µ , we have the abstract contractivity property (85) . In applications, Defectµ will often be a relative entropy or distance between states or spectral measures, for which (85) follows from a data-processing inequality [27]. Example: defect from relative entropy. To make (85) concrete, consider the case where Uµ is a state ωµ on an algebra Aµ , and U? is a reference state ω? representing an IR fixed point. Suppose that an RG step Fµ→µ0induces a completely positive, unital map Φµ→µ0:Aµ→ Aµ0,(86) and that ωµ0is the push-forward of ωµ: ωµ0(X) = ωµΦ† µ→µ0(X), X ∈ Aµ0.(87) Define the Umegaki relative entropy between two states ω, η on the same algebra Aby S(ωkη) := (Trρω(log ρω−log ρη),if ω, η are normal with densities ρω, ρη, +∞,otherwise.(88) Then the data-processing inequality states that for a completely positive, trace-preserving map Φ†, S(ωkη)≥S(ω◦Φ†kη◦Φ†),(89) see e.g. [27]. Taking η=ω?and ω=ωµ, and defining Defectµ(Uµ) := S(ωµkω?),Defectµ0(Uµ0) := S(ωµ0kω?),(90) we see that (89) implies (85) . This is an operator-algebraic realisation of a c-theorem: the defect, measured as relative entropy from a fixed-point state, decreases along coarse-graining steps. In UP–SpectralXC, an analogous construction will be used with spectral measures and spectral laws in place of full states, leading to defect functionals of the form (44). 19 Curvature: irreversibility and scheme dependence The last ingredient of UP–RG is the notion of curvature associated with the family of categories {Cµ} and the RG functors Fµ→µ0 . Intuitively, curvature measures the failure of RG evolution to be path-independent in a combined space of scales and schemes, and decomposes into two physically distinct contributions: •adjoint curvature, associated with irreversibility and information loss; •loop curvature, associated with scheme dependence and anomalies in RG space. To make this precise, it is useful to think of µ as a continuous parameter and to consider infinitesimal RG steps. Formally, we may write the functor implementing an infinitesimal change µ→µ+dµ as Fµ+dµ→µ= IdCµ+dµ Γµ+O(dµ2),(91) where Γ µ is an endofunctor-valued one-form (a “connection” on the bundle of categories over scale space). The composition law (80) implies that the Γ µ satisfy a flatness condition if and only if RG is path-independent in µ alone. However, when one also allows changes in scheme (e.g. different regulators, different normalisations, different coarse-graining kernels), the base space becomes multi-dimensional, and Γbecomes a connection on this higher-dimensional base. The curvature of this connection is defined formally as R:= dΓ+Γ∧Γ,(92) and measures the failure of parallel transport (RG evolution) to be path-independent in the extended space of scales and schemes. Adjoint curvature. Adjoint curvature is related to the failure of RG functors to admit exact adjoints. Suppose that for an RG functor Fµ→µ0 , we seek a “reconstruction” functor Gµ0→µ such that Gµ0→µa Fµ→µ0 (adjoint pair). If F were an equivalence, one could find quasi-inverses with F◦G≃Id and G◦F≃Id . For genuine coarse-graining steps, however, F is typically far from invertible: information is lost about microscopic degrees of freedom, and any adjoint G can at best reconstruct a family of preimages consistent with the coarse description. Operationally, adjoint curvature can be defined as a measure of the failure of the unit and counit maps of an adjunction to be isomorphisms. If η : IdCµ⇒Gµ0→µ◦Fµ→µ0 and ε : Fµ→µ0◦Gµ0→µ⇒IdCµ0 are the unit and counit natural transformations, then Radj µ→µ0(X) := IdX−ηX⊕IdFµ→µ0(X)−εFµ→µ0(X),(93) evaluated on objects X∈ Cµ , encodes the deviation from perfect invertibility. A suitable norm or invariant of Radj µ→µ0 (e.g. a trace norm, a categorical dimension) then quantifies the irreversibility of the RG step: the larger the adjoint curvature, the more information has been irretrievably lost. Loop curvature. Loop curvature captures scheme dependence. Consider two different paths in the space of scales and schemes connecting the same endpoints: Xµ Fµ→µ1 −−−−→ Xµ1 Gµ1→µ0 −−−−−→ Xµ0, Xµ Hµ→µ1 −−−−−→ X0 µ1 Kµ1→µ0 −−−−−→ X0 µ0,(94) where F, G, H, K are RG or scheme-change functors in an admissible class. If the RG plus scheme dependence were completely flat, we would have Xµ0≃X0 µ0 canonically for all Xµ . In practice, this is not the case: different choices of regulator, blocking kernel, or basis lead to different coarse-grained descriptions. Loop curvature quantifies this discrepancy. Formally, let γ be a closed loop in the parameter space of scales and schemes, and let Pγ denote the corresponding composition of functors around γ. Then the loop holonomy Hγ(X) := Pγ(X)∈ Cµ(95) measures how much X changes when transported around the loop. Loop curvature is then some invariant of Pγ (or of Hγ ) measuring the failure of Pγ to be isomorphic to the identity. In UP–SpectralXC, we will apply Pγ not to full models X but to spectral universal objects Uµ , and interpret the resulting curvature in terms of scheme dependence of spectral laws and XC functionals. Physically, adjoint curvature and loop curvature together answer two key questions: 20 • How irreversible is RG? Adjoint curvature quantifies the degree to which coarse-graining loses information irretrievably. • How scheme-dependent is RG? Loop curvature quantifies how much universal structures change under different choices of regulator, basis, and embedding path. By integrating these notions into the spectral XC framework, we obtain a multiscale, scheme-aware XC theory with built-in structural diagnostics. III. UNIVERSALITY AS A UNIVERSAL PROPERTY FOR XC A. Descriptor manifold, KS generator, and higher connection In order to make “universality” into a literal universal property for exchange–correlation (XC), we must first formalise the parameter space over which universality is to be understood and the way in which electronic-structure data are transported over this space. In the present framework, this parameter space is a descriptor manifold Θ, and the objects being transported are operator algebras generated by Kohn–Sham-like (KS-like) generators L [ n ; θ ]and their spectral probes. The appropriate mathematical language is that of connections and curvature on bundles over Θ, extended to a higher (2-) connection that can separately track isospectral and non-isospectral transport [ 32 , 33 , 36 ]. In this subsection we construct this structure in detail and relate it to the SpectralXC ingredients introduced in Subsection II A. The descriptor manifold Θ We begin by specifying the descriptor manifold Θ. Conceptually, Θis the space of all “external” parameters that characterise the one-particle problem, at fixed particle number and species. Typical components of Θinclude: •nuclear positions, e.g. R= (R1, . . . , RM)∈R3Mfor a molecule with Mnuclei; • lattice vectors and cell shapes, e.g. a point in the space of Bravais lattices modulo rigid motions for a periodic solid; •external fields (electric, magnetic, spin-orbit couplings), parametrised by strengths and directions; • interaction scaling parameters, such as the adiabatic connection coupling constant λ∈ [0 , 1] or an effective screening parameter; •dimensional reduction parameters, tuning between quasi-1D, quasi-2D, and 3D regimes. We model Θas a smooth, finite-dimensional manifold (possibly with boundary and/or corners to accommodate physically meaningful limiting regimes). Locally, in a patch U⊂ Θ, we may introduce coordinates θ= (θ1, . . . , θd)∈U⊂Rd,(96) where d = dim Θis the total number of independent descriptor parameters. We shall write ∂i := ∂ ∂θi for the coordinate vector fields, and dθifor the corresponding cotangent basis. In what follows we will regard the density n as a field that is dependent on θ through the solution of the KS equations at ( n, θ ); for notational simplicity we suppress this dependence and focus on the explicit dependence on θ , keeping in mind that n itself is determined by a constrained minimisation problem at each point in Θ. KS generator as a field over Θ For each θ∈ Θ, we associate a KS-like generator L [ n ; θ ]as in (50) , a self-adjoint operator on a one-particle Hilbert space H1: L[n;θ] : H1→ H1, L[n;θ]∗=L[n;θ].(97) 21 For fixed θ,L[n;θ]could be, for instance, L[n;θ] = −1 2∇2+vext(·;θ) + vH[n](·) + v(0) xc [n;θ](·),(98) or a generalised KS operator including non-local or orbital-dependent terms. More generally, L [ n ; θ ]may incorporate effective correlation kernels or quasi-particle corrections, as long as it remains self-adjoint and admits a well-defined spectral calculus (52). To organise the dependence on θ , it is convenient to think of L [ n ; θ ]as a section of a trivial Hilbert bundle over Θ: π: Θ ×H1−→ Θ, π(θ, ψ) = θ, (99) with fibres H1over each θ. On this bundle, Lis a map Θ−→ L(H1), θ 7−→ L[n;θ],(100) which we assume to be smooth in θ in an appropriate operator topology. Likewise, the C∗ -algebra generated by L[n;θ], Aθ:= C∗(L[n;θ]),(101) forms the fibre of an algebra bundle A:= G θ∈Θ Aθ−→ Θ,(102) whose sections encode the spectral data of the KS problem across Θ. In this language, SpectralXC probes (53) and states (56) are to be understood as fields over Θas well: the function fk is kept fixed, but the operator fk(L[n;θ]) and its evaluation by Φθvary with θ. Higher connection (A, B)on Θ We now introduce the central geometric structure: a higher connection ( A, B )on Θthat governs how the algebras Aθ and their generators L [ n ; θ ]are transported along paths in Θ. The guiding idea is borrowed from gauge theory and Berry-phase physics [ 32 , 34 , 35 ]: when a Hamiltonian H ( θ )depends smoothly on parameters, the eigenstates undergo parallel transport governed by a connection one-form (the Berry connection), and its curvature encodes geometric phases. Here, however, we need a richer structure because we want to separate isospectral changes (unitary basis transformations) from non-isospectral changes (genuine deformation of the spectrum, e.g. gap closing). To formalise this, fix a coordinate patch U⊂ Θwith local coordinates θi . We define two families of operator-valued one-forms on U: A= d X i=1 Ai(θ)dθi, B = d X i=1 Bi(θ)dθi,(103) where, for each θand each index i: •Ai ( θ )is an element of (or more precisely generates) the Lie algebra of inner derivations of Aθ , i.e. acts on operators X∈Aθas [Ai(θ), X]; •Bi ( θ )is an element of Aθ (or a derivation in a suitable extension) representing a “source” of non-isospectral deformation. The pair ( A, B )plays the role of a 2-connection on a 2-bundle in the sense of higher gauge theory [ 36 ]: A is a connection one-form valued in a Lie algebra of gauge transformations, and B is a higher-form field encoding additional parallel transport data. We postulate that the infinitesimal variation of L[n;θ]with respect to θiis given by ∂iL[n;θ] := ∂L[n;θ] ∂θi= [Ai(θ), L[n;θ]] + Bi(θ), i = 1, . . . , d. (104) This is the central structural equation of the higher connection. It says: 22 • The commutator term [ Ai, L ]generates an isospectral flow: if Bi = 0, then L [ n ; θ ]at different points along a path is related by unitary conjugation, and the spectrum is preserved. • The Bi term generates a (potentially) non-isospectral flow: it can change the eigenvalues of L [ n ; θ ], opening or closing gaps, shifting levels, and modifying spectral weights. To see the isospectral effect of the A part more explicitly, consider a smooth path γ : [0 , 1] → Θ, t7→ θ(t), with tangent vector ˙ θi(t)∂i. Along this path, define At:= X i ˙ θi(t)Ai(θ(t)), Bt:= X i ˙ θi(t)Bi(θ(t)).(105) Then (104) implies d dtL[n;θ(t)] = [At, L[n;θ(t)]] + Bt.(106) If Bt≡0, (106) reduces to d dtL[n;θ(t)] = [At, L[n;θ(t)]],(107) whose solution is L[n;θ(t)] = U(t)L[n;θ(0)] U(t)−1, U(t) := Pexp Zt 0 At0dt0,(108) where P denotes path-ordering. Thus the A component can be interpreted as a Berry-like connection generating unitary parallel transport of the eigenspaces of L [ n ; θ ][ 34 , 35 ]. The inclusion of Bt in (106) then permits controlled departures from isospectrality. 2-curvature (F, G)and integrability The presence of the higher connection ( A, B )implies a corresponding 2-curvature ( F, G ), which measures the failure of the evolution of L [ n ; θ ]to be path-independent in Θ. To derive it, we compute mixed second derivatives of (104) and use the fact that coordinate derivatives commute: ∂i∂jL=∂i[Aj, L] + Bj= [∂iAj, L]+[Aj, ∂iL] + ∂iBj = [∂iAj, L]+[Aj,[Ai, L] + Bi] + ∂iBj = [∂iAj+ [Aj, Ai], L]+[Aj, Bi] + ∂iBj,(109) and similarly with i and j interchanged. Subtracting the two equations and using [ ∂i, ∂j ] L = 0, we obtain 0 = ∂i∂jL−∂j∂iL= [Fij, L] + Gij,(110) where we have defined Fij := ∂iAj−∂jAi+ [Ai, Aj],(111) Gij := ∂iBj−∂jBi+ [Ai, Bj]−[Aj, Bi].(112) Equations (111) – (112) define the components of the 2-curvature ( F, G )associated with ( A, B ); they are the natural analogues of the curvature of an ordinary gauge connection and its higher counterpart in 2-gauge theory [36]. Equation (110) is an integrability condition for the family L [ n ; θ ]with respect to the higher connection: it ensures that the dependence of L[n;θ]on θis well-defined globally. It can be interpreted as follows: • The piece [ Fij, L ]measures the obstruction to finding a local gauge in which A is flat (i.e. pure gauge) and the evolution of Lis accounted for solely by B. • The piece Gij measures the obstruction to commuting the Bi -generated deformations in different coordinate directions. To connect this to universality, we need to examine how ( F, G )affects spectral quantities Sf [ n ; θ ]and, in particular, the spectral laws Sfk[n;θ] = φkintroduced in (61). 23 Central vs non-central curvature and spectral transport The crucial observation is that spectral quantities Sf [ n ; θ ]only depend on L [ n ; θ ]through its spectrum and, more precisely, through the evaluation of f(L[n;θ]) by an Ad-invariant state Φθ(57): Sf[n;θ]=Φθf(L[n;θ]).(113) The Ad-invariance of Φ θ implies that purely isospectral transformations generated by A do not change Sf: if Lis replaced by ULU−1, then Sf[n;θ]7→ Φθf(ULU−1)= ΦθUf(L)U−1= Φθf(L)=Sf[n;θ],(114) so only the B-driven part of the evolution can affect Sf. This suggests that we should decompose Bi ( θ )into parts that act centrally (i.e. commute with all operators in Aθ) and parts that do not. Let Z(Aθ)denote the centre of Aθ, and let πc:Aθ→Z(Aθ), πnc := Id −πc,(115) be the projections onto central and non-central parts, respectively. Define Bc i:= πc(Bi), Bnc i:= πnc(Bi),(116) and similarly for Gij . Because central elements commute with L and with all spectral projectors, they shift eigenvalues without mixing eigenspaces; non-central elements can also mix spectral subspaces. To see how this affects spectral quantities, recall from (72) that the variation of Sf with respect to a change in L is given by a resolvent sandwich. For a variation generated by δL = δB with δB central, one finds δSf[n;θ] = Φθ1 2πi IΓ f(z) (z−L[n;θ])−1δB (z−L[n;θ])−1dz.(117) If δB lies in the centre, it commutes with (z−L)−1, so we can simplify: δSf[n;θ] = Φθ1 2πi IΓ f(z) (z−L)−2dz δB = ΦθδB gf(L),(118) for some function gf derived from f . Thus central deformations of L induce spectral changes that depend only on central components of B. We can now define central and non-central curvature fields: Fc ij := πc(Fij), F nc ij := πnc(Fij), Gc ij := πc(Gij), Gnc ij := πnc(Gij).(119) Because Fij is defined via commutators and derivatives of Ai , it is naturally valued in derivations of Aθ ; after projection, the central part Fc ij acts trivially on L (in the sense that [ Fc ij, L ] = 0), whereas Fnc ij acts by nontrivial commutators. Similarly, Gc ij and Gnc ij encode central and non-central obstructions to commuting deformations. Central curvature and exact spectral transport. The central claim we make, and use in later sections, is that the central curvature governs the exactness of spectral transport for SpectralXC. More precisely: Proposition. Let f be a fixed probe and let Sf [ n ; θ ]=Φ θ ( f ( L [ n ; θ ])) be the corresponding spectral quantity. Suppose that along a simply connected domain U⊂ Θ, the central part of the 2-curvature vanishes, Fc ij(θ)≡0, Gc ij(θ)≡0,∀θ∈U. (120) Then for any two points θ0, θ1∈U , the value of Sf [ n ; θ1 ]is uniquely determined by Sf [ n ; θ0 ]and the (central) line integral of Bc along any path connecting θ0 to θ1 , and this value is independent of the path. In particular, if Bcis also exact (or zero), then Sfis constant on U. Sketch of proof. Consider the one-form on Udefined by ωf:= d X i=1 ωf,i(θ)dθi, ωf,i(θ) := Φθ1 2πi IΓ f(z) (z−L[n;θ])−1Bc i(θ)(z−L[n;θ])−1dz,(121) 24 so that, by the same reasoning as in (72), the differential of Sfis dSf[n;θ] = ωf.(122) Now compute the exterior derivative of ωf . A straightforward but somewhat lengthy calculation, using (111), (112), and the fact that Bc iand Gc ij lie in the centre, shows that dωf=X i<j ΦθΞf,ij(θ)dθi∧dθj,(123) for some operator-valued coefficients Ξ f,ij that are linear combinations of Fc ij and Gc ij . Under the central flatness condition (120) , all these coefficients vanish, so dωf = 0. In a simply connected domain, closed one-forms are exact, so there exists a scalar function Sf such that ωf = dSf , and path integrals of ωf between θ0 and θ1 are independent of the path. The additional statement about Bc being exact reduces to the special case where ωf= 0, hence Sfis constant.  This proposition captures the idea that central flatness of the higher connection ( A, B )is precisely the condition under which spectral quantities Sf (and hence spectral laws Sfk = φk ) can be transported exactly along Θwithout ambiguity. We will call paths in Θalong which (120) holds exact legs. On such legs, once the spectral laws have been fixed at a single point (say, on a tile where the many-body problem is exactly known), they can be propagated without further adjustment. Non-central curvature and controlled drift. In contrast, when Fc and/or Gc are nonzero, the one-forms ωf need not be closed, and Sf becomes path-dependent. In this case, the central curvature components measure the geometric drift of spectral laws under transport: different paths between the same descriptors give slightly different values of Sf . In the UP–SpectralXC framework, this drift is controlled by the defect functionals introduced in (37) : non-zero central curvature contributes to the defect and signals the need for additional tiles or modified constraints. The non-central parts Fnc and Gnc are equally important physically: they capture true mixing between spectral sectors and the possibility of topology-changing events such as gap closing and re-opening. For instance, changes in band ordering, level crossing, or topological phase transitions (quantised changes in projector K0 classes) require non-central curvature. In this sense, non-central curvature controls the allowed corrections to spectral laws and the limits of their applicability: as long as non-central curvature is small in a given regime, we can treat spectral laws as approximately preserved, with quantitative error bars derived from the curvature. Physical interpretation and relation to Berry connections It is instructive to compare the higher connection ( A, B )with the standard Berry connection in parameter-dependent quantum systems [ 34 , 35 ]. In the usual setting, one has a Hamiltonian H ( θ ) depending smoothly on θ∈ Θ, and for each non-degenerate eigenstate |n ( θ ) i one defines the Berry connection A(n) i(θ) := ihn(θ)|∂in(θ)i,(124) with Berry curvature F(n) ij =∂iA(n) j−∂jA(n) i.(125) The holonomy of this connection around loops in Θgives Berry phases; in the non-Abelian generalisation, degenerate subspaces carry a matrix-valued connection and curvature, leading to non-Abelian geometric phases [ 32 , 35 ]. In all these constructions, however, the Hamiltonian spectrum is typically assumed to vary slowly and without topological changes; the connection is defined on an eigenbundle. In our setting, A plays a role analogous to the Berry connection, but at the level of the full operator L [ n ; θ ]rather than individual eigenspaces. The A -driven part of the evolution (107) is purely isospectral and can be thought of as implementing sector-wise Berry parallel transport of eigenprojectors. The B -driven part allows for changes in eigenvalues and even in spectral projectors; its curvature G controls how these deformations fail to commute. The split into central and non-central curvature separates scalar deformations of eigenvalues (which are invisible to unitary basis changes) from mixing deformations of eigenspaces. 25 From the XC perspective, this geometry is what allows us to talk about universality in a structural way: tiles (HEG, SCE, TLL, etc.) provide anchor points where spectral laws are known exactly; the higher connection ( A, B )tells us how to propagate these laws across Θ; and the central vs non-central curvature decomposition quantifies when this propagation is exact, approximate, or invalid. In this sense, the descriptor manifold Θ, the KS generator field L [ n ; θ ], and the higher connection ( A, B )together form the geometric backbone on which UP–SpectralXC is built. In the next subsection we will combine these structures with the spectral quantities Sfk [ n ; θ ]and universal objects Uµto make precise the notion of universality as a universal property for XC. B. Spectral probes, spectral laws, and universal objects We now combine the geometric structure of the descriptor manifold Θand higher connection ( A, B ) from Subsection III A with the spectral construction of SpectralXC from Subsection II A. The goal is to define, at each descriptor point ( θ, µ ), a universal object Uµ,θ built from spectral probes and their laws, and to make precise in what sense universality for XC means that these objects are preserved (up to controlled defects) under changes in θ, in scale µ, and in scheme (basis/embedding). Spectral probes revisited Let L [ n ; θ ]be the KS-like generator at descriptor θ , as in (97) , and let Aθ := C∗ ( L [ n ; θ ]) be the associated C∗ -algebra (101) . A spectral probe is a bounded Borel function f : R→C (often holomorphic in a region containing the spectrum of L [ n ; θ ]) which we apply to L [ n ; θ ]via the functional calculus (52) : f(L[n;θ]) = ZR f(λ)dEn,θ(λ),(126) where En,θ(λ)is the spectral measure of L[n;θ]. In UP–SpectralXC we will use a finite, but conceptually expandable, family of probes F:= {fk}m k=1,(127) chosen so that each fkisolates a physically meaningful part of the spectrum: •Resolvent-type probes. For zk∈C\σ(L[n;θ]), define fk(λ) = (λ−zk)−1, fk(L[n;θ]) = (L[n;θ]−zk)−1.(128) If zk is chosen near the HOMO–LUMO gap or a specific band edge, fk ( L )strongly amplifies the contribution of eigenvalues near zk , making these probes highly sensitive to frontier gaps, level splittings, and low-energy excitations. • Riesz projectors onto spectral windows. Fix a bounded interval Ik⊂R and let Γ k be a contour in the complex plane encircling σ ( L [ n ; θ ]) ∩Ik and no other part of the spectrum. The corresponding Riesz projector is PIk(L[n;θ]) = 1 2πi IΓk (z−L[n;θ])−1dz, (129) which can be thought of as fk(L)for a probe fkapproximating the indicator function of Ik. Such projectors select spectral subspaces: –the occupied subspace (below the Fermi level), –the unoccupied subspace, –a correlated d-/f-shell manifold in transition metals, –an active space used in embedding. 32 functors on universal objects send Uµ,θ to an object isomorphic to Uµ0,θ0 in the relevant category U µ0 . That is, there exists an isomorphism Uµ0,θ0≃b T(Uµ,θ),(155) for some composite b Tof induced functors. In particular, within a universal leaf Lα at fixed scale µ , all points share isomorphic universal objects Uµ,θ (up to curvature-controlled defects), and thus belong to the same XC universality class at that scale. Changes in scale and scheme then extend this notion of universality across the full space of UP–SpectralXC schemes. This formulation turns the informal RG slogan “systems with the same critical exponents are universal” into the categorical statement: Universality is the property of sharing isomorphic universal spectral objects Uµ,θ under functorial transport along integrable directions and RG transformations. Critical exponents and other scalar invariants become derived quantities from the structure of Uµ,θ , while XC functionals are constrained to respect these universal structures by construction. Physical examples of exact legs and integrable directions To build intuition, let us briefly mention three physically relevant situations where exact legs or integrable directions are expected to exist: • Uniform density scaling near HEG tiles. In regions of Θcorresponding to weakly inhomogeneous, nearly uniform densities (HEG tiles), small variations in the average density and weak modulations can be viewed as integrable directions: central curvature associated with HEG spectral laws is small, and spectral quantities related to compressibility and structure factor remain nearly constant. In the limit of exact HEG, these directions become exact legs along which HEG spectral laws are transported rigidly. • Adiabatic bond stretching before strong correlation sets in. For a diatomic molecule, nuclear separation R is a coordinate in Θ. Near equilibrium and up to the onset of strong static correlation, adiabatic changes in R may form approximate exact legs for certain spectral laws (e.g. those tied to HEG or weakly correlated tiles), so that frontier spectral constraints and K0 data remain essentially fixed. Beyond this regime, non-central curvature grows as gaps close and correlation sets in, signalling departure from integrable directions. • Deformations preserving local d-shell physics. In transition-metal complexes, deformations of ligand geometry that preserve the local symmetry and splitting pattern of the d-shell can be viewed as integrable directions: K0 classes of d-shell projectors and certain spectral quantities remain invariant along such deformations. Universal leaves in this case correspond to families of complexes sharing the same d-shell universal object. In all these cases, the geometric language of exact legs and integrable directions provides a structural way to identify when and where SpectralXC constraints can be transported reliably without re-solving the full many-body problem, and when curvature signals the need for new tiles or corrective terms. IV. UP–SPECTRALXC: FORMAL DEFINITION A. Category of SpectralXC schemes and RG functors We now formalise the UP–SpectralXC framework by encoding its structural content in the language of category theory. The aim of this subsection is twofold: 1. to define a category R whose objects are SpectralXC schemes, i.e. concrete realisations of the data (39) at a given scale and descriptor; 33 2. to exhibit, for each scale µ , a category Cµ of SpectralXC schemes at fixed µ , and to define RG functors Fµ→µ0:Cµ−→ Cµ0, µ > µ0,(156) which implement spectral coarse-graining in a way compatible with the universal objects Uµ,θ introduced in (138). The categorical perspective provides a clean organisational principle for the multiscale, multi-descriptor, and multi-scheme nature of SpectralXC. Instead of tracking individual Hamiltonians and functionals, we work with objects and morphisms in a category, and RG becomes a family of functors between these categories [ 43 ]. This will be crucial when we articulate universality as an isomorphism property of universal objects under functorial transport. Objects: SpectralXC schemes as structured tuples We recall the notion of a UP–SpectralXC scheme at scale µ from (39) . For the purposes of this subsection, it will be convenient to work with a slightly streamlined version that omits redundant data (such as A=C∗(L), which is determined by L). We therefore define: Definition (SpectralXC scheme). ASpectralXC scheme is a tuple S:= µ, θ, L, F,Φ,φ,(Aµ, Bµ),(157) where: 1. µ > 0is a scale parameter (energy, momentum, resolution, or more abstract coarse-graining index); 2. θ∈Θis a point in the descriptor manifold (96); 3. L=L[n;θ]is a self-adjoint KS-like generator on a one-particle Hilbert space H1, L:H1→ H1, L∗=L, (158) with associated C∗-algebra Aθ=C∗(L)and spectral calculus (52); 4. F = {fk}m k=1 is a finite family of spectral probes as in (127) , each fk a bounded (often holomorphic) function on R, with associated operators fk(L)and spectral quantities Sfk[n;θ]defined by (133); 5. Φ : Aθ→Cis a normalised, positive, Ad-invariant state as in (131)–(132); 6. φ = ( φ1, . . . , φm )is a vector of target values (spectral laws) as in (136) , coming from tiles (HEG, SCE, TLL, Hubbard) or exact constraints; 7. ( Aµ, Bµ )is the restriction at scale µ of the higher connection on Θintroduced in (103) , evaluated at θ, which governs infinitesimal deformations of L[n;θ]in descriptor space. We will often write S = ( µ, θ, L )when the additional data ( F, Φ ,φ, ( Aµ, Bµ )) are understood; they are part of the structure but, for notational simplicity, we suppress them where no confusion can arise. Morphisms: basis changes, embeddings, and parameter flows We now define the category R whose objects are SpectralXC schemes. Intuitively, morphisms in R capture the allowed transformations between SpectralXC descriptions: changes of basis, embedding/- downfolding, resolution changes, and descriptor flows. Each of these has a clear physical meaning: • basis changes correspond to different representations of the same one-particle problem (e.g. plane waves vs local orbitals); • embedding/downfolding maps correspond to integrating out or projecting onto subsets of degrees of freedom, as in effective Hamiltonians, Feshbach projection, or Löwdin partitioning [45, 46]; 34 •resolution changes correspond to changes in µ, i.e. coarse-graining in energy or length scale; these will later be codified by RG functors Fµ→µ0; • parameter flows correspond to changes in θ along paths in Θ(geometry changes, field strength variations, interaction scaling). To capture all of these within a single notion of morphism, we define: Definition (Morphisms in R). A morphism α:S−→ S0,(159) between SpectralXC schemes S= (µ, θ, L, F,Φ,φ,(Aµ, Bµ)), S0= (µ0, θ0, L0,F0,Φ0,φ0,(Aµ0, Bµ0)),(160) consists of a tuple α:= αscale, αdesc, αHilb, αalg, αprobe, αstate, αconn,(161) satisfying the following conditions: 1. Scale map. αscale : {µ}→{µ0} is simply the assignment µ7→ µ0 , indicating whether the morphism changes scale (e.g. µ0 = µ for pure basis changes or descriptor flows, and µ0< µ for RG/coarsegraining moves). 2. Descriptor map. αdesc : Θ → Θis a smooth map (typically a diffeomorphism or an inclusion of a submanifold) such that αdesc(θ) = θ0. It encodes parameter flows in descriptor space. 3. Hilbert-space map. αHilb :H1→ H0 1is a bounded linear map, typically: •a unitary isomorphism for pure basis changes; • a partial isometry for embedding/downfolding (injecting a smaller Hilbert space into a larger one or projecting onto a subspace); • a composition of these with scale-dependent embeddings in multiscale schemes (e.g. tensornetwork coarse-graining [47]). 4. Algebra map. αalg : Aθ→A0 θ0 is a unital ∗ -homomorphism between the C∗ -algebras generated by L and L0, making the diagram L(H1)L(H0 1) AθA0 θ0 X7→αHilbXα∗ Hilb αalg (162) commute in the appropriate sense. For instance, for a pure basis change αHilb ( X ) = UXU−1 and αalg ( X ) = UXU−1 , while for Feshbach-type downfolding, αalg is given by a projection onto a subalgebra associated with an effective Hamiltonian [45, 46]. 5. Probe map. αprobe is a map between probe families, αprobe :F −→ F0, fk7−→ f0 k,(163) such that αalgfk(L)=f0 k(L0)in A0 θ0.(164) This may involve, for example, rescaling of energy variables to account for changes in µ. 6. State map. αstate is a natural relation between Φand Φ0such that, for all X∈Aθ, Φ0(αalg(X)) = Φ(X),(165) at least for X in a suitable subalgebra (e.g. the spectral subalgebra relevant at scale µ0 ). For basis changes, (165) is automatic with Φ 0 ( Y ) = Φ( U−1Y U ); for embedding and coarse-graining, Φ 0 is typically the push-forward of Φunder a conditional expectation or partial trace [48]. 35 7. Connection map. αconn specifies how the higher connection ( Aµ, Bµ )is transported: it is a rule taking ( Aµ, Bµ )at ( µ, θ )to ( Aµ0, Bµ0 )at ( µ0, θ0 ), consistent with the pull-back under αdesc and with the structure of RG functors (see below). In practice, αconn is given by parallel transport with respect to a chosen higher-gauge structure. Two remarks are in order: •The conditions above ensure that spectral quantities transform naturally under α: S0 fk[n0;θ0] := Φ0f0 k(L0)= Φ0αalg(fk(L))(165) = Φfk(L)=Sfk[n;θ],(166) at least in cases where α represents an equivalence (basis or scheme change). For true coarse-graining (RG), we will allow inequalities and relations rather than equalities, leading naturally to defect functionals. •Composition of morphisms is defined component-wise: if α:S→S0and β:S0→S00, then (β◦α)Hilb =βHilb ◦αHilb,etc. (167) Identity morphisms are given by trivial maps on each component. It is straightforward to check the category axioms (associativity of composition, identity laws) using the corresponding properties of bounded operators, ∗-homomorphisms, and smooth maps between manifolds [43]. We denote by R the category whose objects are SpectralXC schemes and whose morphisms are tuples of the form (161). Fixed-scale categories Cµ The category R is too large to directly encode RG flows, because it mixes objects at different scales. It is more convenient to work with fibres of Rat fixed µ. For each scale µ, we define a full subcategory Cµ⊂ R,(168) whose objects are SpectralXC schemes Swith fixed scale µ: Ob(Cµ) := S= (µ, θ, L, F,Φ,φ,(Aµ, Bµ)) ∈Ob(R),(169) and whose morphisms are those αin Rthat preserve scale, αscale(µ) = µ. (170) Physically, Cµ collects all SpectralXC descriptions at a given resolution µ , allowing basis changes, descriptor flows, and scheme changes, but not RG-induced changes in scale. In the language of Subsection II B, the family {Cµ}µ>0 is precisely the collection of categories referred to in (74) , now instantiated for SpectralXC. Objects in Cµ are, roughly speaking, all the different representations and parametrisations of KS-like generators and spectral probes at scale µ. For example, Cµmay contain: • SpectralXC schemes for a fixed molecule or material in different basis sets (plane waves, Gaussians, numerical orbitals); • Schemes for the same physical system but with different choices of probe families F (emphasising different spectral features); • Schemes differing by embedding choices (different partitionings of active and environment spaces). Morphisms in Cµ then express the relationships between these descriptions, and universal objects Uµ,θ defined in (138) become functorial constructions on Cµ. RG functors as spectral coarse-grainings We now define RG functors Fµ→µ0:Cµ−→ Cµ0, µ > µ0,(171) which implement spectral coarse-graining: integrating out or projecting away high-energy or short-distance degrees of freedom while retaining universal spectral structure [44, 48]. 36 Action on objects. Let S = ( µ, θ, L, F, Φ ,φ, ( Aµ, Bµ )) be an object of Cµ . We define Fµ→µ0 ( S )as follows. 1. Descriptor and scale. The descriptor is kept fixed: θ0:= θ, (172) while the scale changes from µto µ0. 2. Hilbert-space coarse-graining. Choose a decomposition of H1 adapted to scale µ and the spectral regions of interest, for instance H1=Hlow ⊕Hhigh,(173) where Hlow carries states with energies |λ|.µ0 (the low-energy subspace) and Hhigh carries the rest. Let P : H1→ Hlow and Q : H1→ Hhigh denote the corresponding orthogonal projections, with P + Q = 1 . We construct an effective low-energy generator Leff on Hlow by a suitable projection scheme, e.g. Feshbach or Löwdin partitioning [ 45 , 46 , 48 ]. For instance, the energy-dependent Feshbach effective Hamiltonian is Leff(z) := PLP +PLQ(z−QLQ)−1QLP, (174) and one can fix z in a relevant energy window near the Fermi level or approximate Leff ( z )by an energy-independent operator L0 that reproduces the low-energy spectrum. For our purposes, we assume we have constructed a self-adjoint L0 = L0 µ0 [ n ; θ ]on a coarse-grained Hilbert space H0 1 (identified with Hlow) which approximates the low-energy behaviour of Lto a desired accuracy. 3. Algebra and probes. Let A0 θ:= C∗(L0). We define a ∗-homomorphism αalg :Aθ→A0 θby αalg(X) := P XPHlow ,(175) which restricts operators to the low-energy subspace. We then define new probes F0 = {f0 k} on L0 by f0 k(λ) := fk(λ)for |λ|.µ0,(176) possibly with additional smoothing or rescaling to reflect the change in resolution. The requirement is that for zin a relevant low-energy domain, αalgfk(L)≈f0 k(L0),(177) in a sense that can be made precise (e.g. norm estimates in operator topology). 4. State and spectral constraints. We define Φ0 θ:A0 θ→Cas the push-forward of Φθalong αalg, Φ0 θ(Y) := Φθe Y,e Y∈Aθsuch that αalg(e Y) = Y. (178) In practice, for Y=f0 k(L0)we choose e Y=fk(L), so that S0 fk[n;θ] := Φ0 θf0 k(L0)= Φθfk(L)=Sfk[n;θ],(179) modulo errors controlled by the accuracy of the coarse-graining. The target values φ0 are inherited from φ ; at the level of defect functionals, this corresponds to requiring that the spectral laws hold as closely as possible at the coarse scale. 5. Connection. The higher connection ( Aµ0, Bµ0 )at scale µ0 is obtained by projecting the connection at µ onto low-energy degrees of freedom and reparametrising the descriptor dependence accordingly. Formally, this corresponds to a pull-back of ( Aµ, Bµ )by the inclusion of the low-energy subbundle, followed by a rescaling in the scale coordinate. The detailed construction will be given in later sections when we study curvature and defect monotones. We thus define Fµ→µ0(S) := µ0, θ, L0,F0,Φ0,φ0,(Aµ0, Bµ0)∈Ob(Cµ0).(180) 37 Action on morphisms. Given a morphism α : S→S0 in Cµ (so µ0 = µ ), we define Fµ→µ0 ( α )as the morphism Fµ→µ0(α) : Fµ→µ0(S)−→ Fµ→µ0(S0),(181) with components induced by the same type of constructions used for S and S0 , now restricted to the low-energy subspaces and the coarse-grained algebras. Concretely: •Fµ→µ0 ( α ) Hilb is the composition of αHilb with the projections P, P 0 onto the respective low-energy subspaces; •Fµ→µ0(α)alg is induced by restricting αalg to the low-energy spectral subalgebras; •Fµ→µ0 ( α ) probe maps the coarse-grained probes f0 k according to the original probe map and the identification of spectral windows at low energy; •Fµ→µ0(α)state ensures compatibility of Φ0and Φ00 in the sense of (165), now applied at scale µ0; •Fµ→µ0(α)conn is obtained by projecting αconn onto the low-energy connection data. It is straightforward, though notationally heavy, to check that this definition preserves composition and identities, and therefore that Fµ→µ0is a functor Cµ→ Cµ0in the sense of (171) [43]. Semigroup property and spectral universality. Finally, we impose the semigroup property Fµ1→µ0◦Fµ→µ1=Fµ→µ0, µ > µ1> µ0,(182) which corresponds to the compositionality of coarse-graining: integrating out modes between µ and µ1 , then between µ1 and µ0 , is equivalent to integrating out all modes between µ and µ0 in one step. At the level of effective generators, this translates into the requirement that the low-energy effective L0 obtained via two-step Feshbach/Löwdin projection agrees (up to controlled errors) with that obtained via a single-step projection. At the level of universal objects Uµ,θ , (182) ensures that the defect functionals introduced in (37)–(44) satisfy the monotonicity property (46) along RG flows. Physically, the functors Fµ→µ0 express in rigorous categorical terms the intuition underlying block-spin transformations, decimation, and tensor-network coarse-grainings in many-body physics [ 44 , 47 ]. What is new here is that the coarse-graining is performed at the level of spectral XC schemes, with explicit attention to probe families, states, and universal objects. RG is no longer just a flow of couplings: it is a functorial operation on SpectralXC schemes, constrained to preserve (up to controlled defects) the spectral universals that define XC universality. In subsequent subsections we will use this categorical and functorial framework to define and analyse spectral defect functionals, relate them to traditional c-theorems, and construct explicit UP–SpectralXC schemes for concrete physical systems. B. Spectral universal objects as UP–RG universals In Subsection III B we defined, for each scale µ and descriptor point θ∈ Θ, a spectral universal object Uµ,θ built from spectral quantities, K0 classes of projectors, and (coarse-grained) spectral measures of the KS-like generator L [ n ; θ ], see (138) . In Subsection IV A we then organised SpectralXC schemes into fixedscale categories Cµ and defined RG functors Fµ→µ0 : Cµ→ Cµ0 that implement spectral coarse-graining. We now complete the formal setup by: 1. defining a universal functor Uµ:Cµ−→ Uµ(183) which associates to each SpectralXC scheme its spectral universal object; 2. defining XC universality classes as equivalence classes of objects in Cµ whose universal objects are related by isomorphisms in Uµ; 3. exhibiting a universal property: for fixed probe constraints, certain spectral universals appear as terminal objects in appropriate diagrams, making universality literally a universal property in the categorical sense. This construction ties together the geometric, spectral, and categorical structures introduced so far, and gives a precise meaning to the slogan: “universality means sharing the same universal spectral object.” 38 The spectral universal functor Uµ We begin by refining the category of spectral universal objects introduced informally in Subsection III B. Fix a scale µ. Recall that for each object S= (µ, θ, L, F,Φ,φ,(Aµ, Bµ)) ∈Ob(Cµ), we defined the universal object Uµ,θ =S[n;θ],K[L[n;θ]], νn,θ,(184) where S [ n ; θ ]is the vector of spectral quantities (135) , K [ L [ n ; θ ]] is the set of K0 classes of relevant projectors (139), and νn,θ is the spectral measure (140). We now package these universal objects into a category Uµ: Definition (category of spectral universals). For fixed scale µ , the category U µ is defined as follows: •Objects are triples U= (S,K, ν),(185) where S∈Cm , K is a finite or countable subset of K0 ( Aθ )for some C∗ -algebra Aθ , and ν is a finite Borel measure on R(or a coarse-grained representative thereof). •Morphisms h:U→U0are triples h= (hS, hK, hν),(186) where: 1. hS : Cm→Cm0 is a linear map respecting the probe structure (typically a coordinate projection, injection, or unitary change of basis); 2. hK : K → K0 is a group homomorphism (respecting addition in K0 ) compatible with the embedding or projection of algebras induced by the underlying SpectralXC morphisms; 3. hνis a Markov kernel or push-forward map between measures, implementing coarse-graining or reparametrisation in spectral space. Composition of morphisms is defined component-wise. Intuitively, U µ collects all possible “shapes” of spectral data that can arise from SpectralXC schemes at scale µ , stripped of representation-dependent details. Objects encode what we consider physically universal for XC (spectral laws, robust projector classes, spectral weight distributions); morphisms encode how such data transform under changes of probes, algebras, and coarse-graining. We now define the universal functor Uµ: Definition (spectral universal functor). For each µ, the functor Uµ:Cµ−→ Uµ(187) is defined as follows: 1. On objects, for S= (µ, θ, L, F,Φ,φ,(Aµ, Bµ)), set Uµ(S) := Uµ,θ as in (184).(188) 2. On morphisms, for α:S→S0in Cµ, define h:= Uµ(α)as follows: •hSmaps S[n;θ]to S0[n0;θ0]by the rule Sfk[n;θ]7−→ S0 f0 k[n0;θ0],(189) where fk7→ f0 k and Φ 7→ Φ 0 are related by αprobe and αstate as in (161) . For equivalence morphisms (e.g. basis changes), hS is the identity; for coarse-graining-like morphisms, it may be a projection onto a subset of constraints. 39 •hKsends [PI(L)] ∈ K[L]to [PI(L0)] ∈ K[L0]via the induced map on projectors: PI(L)7−→ αalg(PI(L)) = PI(L0),(190) with the understanding that projectors may be merged or split under coarse-graining. •hνis the push-forward of spectral measures under the algebra map αalg: νn,θ(Λ) = Φθ(En,θ(Λ)) 7−→ ν0 n0,θ0(Λ) := Φ0 θ0E0 n0,θ0(Λ).(191) Lemma. Uµis a functor: it preserves identities and composition, Uµ(idS) = idUµ(S),Uµ(β◦α) = Uµ(β)◦Uµ(α),(192) for all composable α, β in Cµ. Proof. Identity preservation is immediate from the definitions: if α = idS , then all components of α are identity maps, hence all components of h = Uµ ( α )are identities on S,K, ν , so Uµ ( idS ) = idUµ,θ . For composition, let α : S→S0 and β : S0→S00 , with components αHilb, αalg, . . . and βHilb, βalg, . . . . Then the composite β◦α has components ( β◦α ) Hilb = βHilb ◦αHilb , and similarly for the algebra and state maps. Because the definitions of hS, hK, hν are functorial in these components (linear maps compose, group homomorphisms compose, and push-forward of measures is functorial), we have Uµ(β◦α) = Uµ(β)◦Uµ(α) component-wise.  Thus Uµ systematically extracts from any SpectralXC scheme at scale µ exactly the data we deem universal for XC. It is this functor, rather than the raw space of couplings or densities, that will carry the structure of UP–RG universals. XC universality classes as isomorphism classes of universals We can now formalise the notion of an XC universality class at a fixed scale µ as an equivalence class of SpectralXC schemes whose universal objects are isomorphic in Uµ. Definition (isomorphism of spectral universals). Two objects U = ( S,K, ν )and U0 = ( S0,K0, ν0 ) in U µ are said to be isomorphic if there exists a morphism h : U→U0 with an inverse h−1 : U0→U , i.e. if there exists a triple h= (hS, hK, hν) such that all three components are bijections (with appropriate inverses). We then write U≃U0. Definition (XC universality at fixed scale). Let S, S0∈Ob ( Cµ )be two SpectralXC schemes at scale µ . We say that S and S0 are XC-universal at scale µ , and write S∼µS0 , if their universal objects are isomorphic in Uµ, Uµ(S)≃Uµ(S0).(193) The equivalence class of Sunder this relation is denoted by [S]µ:= {S0∈Ob(Cµ)|S0∼µS},(194) and is called the XC universality class of Sat scale µ. Proposition. The relation ∼µ defined by (193) is an equivalence relation on Ob ( Cµ ), and the collection of equivalence classes {[S]µ}partitions Ob(Cµ). Proof. Reflexivity: For any S , the identity morphism idUµ,θ is an isomorphism in U µ , so Uµ ( S ) ≃Uµ ( S ) and S∼µS . Symmetry: If S∼µS0 , then there exists an isomorphism h : Uµ ( S ) →Uµ ( S0 ), which has an inverse h−1 : Uµ ( S0 ) →Uµ ( S ); hence S0∼µS . Transitivity: If S∼µS0 and S0∼µS00 , then there exist isomorphisms h : Uµ ( S ) →Uµ ( S0 )and h0 : Uµ ( S0 ) →Uµ ( S00 ), and their composition h0◦h is an isomorphism Uµ ( S ) →Uµ ( S00 ), so S∼µS00 . Therefore ∼µ is an equivalence relation, and its equivalence classes partition Ob(Cµ)as claimed.  From the physical point of view, this formalism states that: 40 • Two SpectralXC schemes S, S0 at the same scale µ are in the same XC universality class if they encode the same spectral laws S (up to invertible linear transformations of probes), the same robust projector structure (up to K0 isomorphism), and the same spectral weight distribution (up to invertible measure transformations). In other words, they agree on all aspects we have declared to be universal for XC. • Standard notions of universality (e.g. “same critical exponents”, “same gap structure”, “same topological class”) become derived properties of Uµ,θ , rather than primary definitions. XC universality is now a statement about isomorphism of universal objects, not about equality of particular scalar invariants. In this sense, the universal spectral object Uµ,θ is the carrier of XC universality at scale µ : all SpectralXC schemes sharing the same Uµ,θ form a universality class. Universal property: terminal objects of probe-constrained diagrams We now explain in what sense universality in UP–SpectralXC is literally a universal property in the categorical sense. The key idea is to view Uµ,θ (or, more precisely, a canonical representative of its isomorphism class) as a terminal object in a diagram of spectral data produced by compatible KS-like generators subject to fixed probe constraints. Probe-constrained diagrams. Fix a scale µ , a descriptor region W⊂ Θ, and a set of target spectral laws φ = ( φ1, . . . , φm ), defined on W (e.g. inherited from tiles and transported along exact legs as in Subsection III C). Consider the full subcategory Cφ µ(W)⊂ Cµ,(195) whose objects are those SpectralXC schemes S = ( µ, θ, L, . . . )with θ∈W that satisfy the spectral laws exactly: Sfk[n;θ] = φk(θ), k = 1, . . . , m, (196) and whose morphisms are those in Cµ preserving the constraints (i.e. mapping solutions of (196) to solutions). The restriction of the universal functor Uµto this subcategory is written Uφ(W) µ:= UµCφ µ(W). We can regard Uφ(W) µas a diagram in Uµindexed by Cφ µ(W): Uφ(W) µ:Cφ µ(W)−→ Uµ,(197) sending each constrained SpectralXC scheme to its universal object, and each morphism to the induced morphism of universals. Terminal objects and cones. In category theory, a terminal object T of a category D is an object such that for every object X there exists a unique morphism X→T . More generally, given a diagram D : I → D , a cone from D to an object T consists of morphisms ηi : D ( i ) →T for all i∈ I making all relevant triangles commute; a cone is terminal (or a limit) if it is universal among cones: any other cone factors uniquely through it [50, 69]. Definition (probe-constrained universal spectral object). Assume that the category U µ is complete (has all small limits). For fixed ( µ, W, φ ), we define the probe-constrained universal spectral object Uφ(W) µas a limit of the diagram (197), Uφ(W) µ:= lim ←− Uφ(W) µ,(198) together with a family of morphisms ηS:Uµ(S)−→ Uφ(W) µ, S ∈Ob(Cφ(W) µ),(199) forming a terminal cone. By the universal property of the limit [ 69 ], for any other object V in U µ and any cone {βS:Uµ(S)→V}S, there exists a unique morphism h:Uφ(W) µ→Vsuch that βS=h◦ηS,∀S∈Ob(Cφ(W) µ).(200) 41 Equation (198) makes precise the idea that Uφ(W) µ is the most universal spectral object realising the probe constraints φ on W . Every constrained SpectralXC scheme in Cφ(W) µ maps to Uφ(W) µ via a morphism that is compatible with all other such maps; and any attempt to map all these schemes into another spectral object Vnecessarily factors uniquely through Uφ(W) µ. In particular, if the diagram (197) is connected and non-empty, the existence of the limit implies that: • up to unique isomorphism, there is a canonical spectral object that gathers all spectral data consistent with the probe constraints in W; • any two constrained SpectralXC schemes in Cφ(W) µ become equivalent when viewed through the lens of Uφ(W) µ. Thus, in the UP–SpectralXC setting, universality is literally a universal property: the universal spectral object Uφ(W) µ is characterised by being a terminal object in the diagram of spectral data generated by KS-like schemes satisfying the same probe constraints. Physical meaning of the universal property Let us unpack the universal property (198)–(200) in physical terms. • The objects S∈ Cφ(W) µ represent all possible SpectralXC descriptions (choices of basis, embedding, gauge, internal parametrisation) of systems in the descriptor region W that satisfy the same spectral laws φat scale µ. • The images Uµ ( S )collect the spectral quantities, projector classes, and spectral measures that we consider universal invariants for XC. • The cone maps ηS : Uµ ( S ) →Uφ(W) µ express how each concrete spectral data set relates to the canonical universal object. For example, in a trivial case, ηS might be the identity; in more realistic situations, it might mod out representation-dependent redundancies or integrate out high-energy details. • The universality condition (200) says that any observable or diagnostic that is compatible with all constrained SpectralXC schemes factors through Uφ(W) µ . In particular, any attempt to define XC defect functionals, monotones, or scheme diagnostics that are truly universal must be functions of Uφ(W) µalone. In this light, the universal objects Uµ,θ introduced in (184) can be seen as local or pointwise representatives of these global limits: at each ( µ, θ ), a small enough neighbourhood W can be chosen so that the constrained diagram is simply connected and admits a limit, and Uµ,θ is isomorphic to the restriction of Uφ(W) µat that point. Moreover, as RG functors Fµ→µ0 act on Cµ and induce functors on U µ , the universal property is stable under scale transformations: if U µ and U µ0 are complete and the induced functors preserve limits (or, more weakly, send limits to objects with controlled defect), then b Fµ→µ0Uφ(W) µ≃Uφ(W) µ0,(201) up to controlled corrections, making universality a property extending across scales. In summary, UP–SpectralXC realises the RG intuition in a categorical, spectral way: instead of declaring universality by inspection of couplings or exponents, we construct universal spectral objects as limits of probe-constrained diagrams, and declare two descriptions universal if they share isomorphic universal objects. This perspective will be central when we construct spectral defect monotones and analyse their behaviour under UP–RG flows in subsequent sections. 48 Adjoint curvature and irreversibility An RG step from scale µto a coarser scale µ0< µ is represented by a functor Fµ→µ0:Cµ,σ −→ Cµ0,σ,(241) for some fixed scheme σ . In general, Fµ→µ0 is not invertible: high-energy or short-distance spectral information is integrated out and cannot be reconstructed uniquely from the coarse description. However, in many constructions of the exact RG [ 54 , 55 ], one can define approximate “reconstruction” maps that embed coarse descriptions back into a finer-scale space, at least up to controlled errors. Categorically, this suggests the existence of an approximate adjunction between Fµ→µ0and a functor Gµ0→µ:Cµ0,σ −→ Cµ,σ,(242) which we interpret as a lift or reconstruction functor. Adjunction data. Suppose Fµ→µ0 and Gµ0→µ form a (possibly approximate) adjoint pair. Then there exist natural transformations (unit and counit) η: IdCµ⇒Gµ0→µ◦Fµ→µ0, ε :Fµ→µ0◦Gµ0→µ⇒IdCµ0,(243) satisfying the usual triangle identities of an adjunction [ 56 ]. For each object S∈Ob ( Cµ )and each S0∈Ob(Cµ0), we have components ηS:S−→ Gµ0→µFµ→µ0(S), εS0:Fµ→µ0Gµ0→µ(S0)−→ S0.(244) In an ideal, fully reversible coarse-graining, ηS and εS0 would be isomorphisms, implementing equivalences Gµ0→µFµ→µ0≃IdCµ, Fµ→µ0Gµ0→µ≃IdCµ0.(245) For genuine RG flows, however, these maps typically fail to be invertible, and their deviation from isomorphisms encodes the irreversibility of RG. Adjoint curvature at the spectral level. We now push the adjunction data to the level of spectral universal objects via the functors Uµ and Uµ0 introduced in (183) . Given S∈ Cµ and S0∈ Cµ0 , we denote U:= Uµ(S), U0:= Uµ0(S0), U0:= Uµ(Gµ0→µFµ→µ0(S)), U0 0:= Uµ0(Fµ→µ0Gµ0→µ(S0)).(246) Let b Fµ→µ0 and b Gµ0→µ be the induced functors on universals. The unit and counit induce natural maps ˆηU:U−→ b Gµ0→µb Fµ→µ0(U),ˆεU0:b Fµ→µ0b Gµ0→µ(U0)−→ U0.(247) We can now quantify the failure of reversibility via the adjoint curvature functional RF(U) := Defectµ b Gµ0→µb Fµ→µ0(U)−Defectµ(U)+Defectµ0b Fµ→µ0b Gµ0→µ(U0)−Defectµ0(U0),(248) where U0 := b Fµ→µ0 ( U ). Since Defectµ ( U )is non-negative and vanishes on ideal tiles, (248) can be interpreted as the extra defect introduced by the round-trip RG maps U−→ b Gµ0→µb Fµ→µ0(U), U0−→ b Fµ→µ0b Gµ0→µ(U0).(249) Proposition (adjoint curvature and reversibility). If Fµ→µ0 and Gµ0→µ define an exact adjoint equivalence between suitable full subcategories of Cµ,Cµ0that contain Sand S0, then RF(U)=0. Conversely, if RF ( U ) = 0 for all U in a given universality class and Defectµ separates universal objects, then the induced maps ˆηU and ˆεU0 are isomorphisms in U µ, U µ0 , and the RG step is spectrally reversible on that universality class. Sketch of proof. If F and G define an equivalence, the unit and counit are isomorphisms, and the induced morphisms ˆηU,ˆεU0 are also isomorphisms. In that case U and b Gµ0→µb Fµ→µ0 ( U )lie in the same universality class and hence have the same defect; similarly for U0 and b Fµ→µ0b Gµ0→µ ( U0 ), so (248) vanishes. Conversely, if RF ( U ) = 0 for all U in a universality class and the defect functional is faithful 49 (i.e. Defectµ ( U ) = Defectµ ( V )and Defectµ0 ( b F ( U )) = Defectµ0 ( b F ( V )) imply U≃V ), then the equality of defects before and after round-trips, together with monotonicity, forces ˆηU and ˆεU0 to be isomorphisms. Thus F, G restrict to an equivalence on that universality class.  Physically, RF ( U )quantifies how much spectral universality is lost when we coarse-grain from µ to µ0 and then attempt to reconstruct the fine-scale description. A large adjoint curvature indicates strong irreversibility: high-energy spectral features that cannot be reconstructed without violating tile constraints or increasing the defect. A small adjoint curvature, on the other hand, indicates that the chosen RG step is nearly lossless on the spectral universals we care about, and that the effective description at scale µ0 retains essentially all relevant XC structure. Loop curvature and scheme dependence Adjoint curvature quantifies irreversibility along a given RG direction. To diagnose scheme dependence, we must examine curvature associated with loops in the scale–scheme manifold B . Consider a closed loop γ: [0,1] → B, γ(0) = γ(1) = b, (250) composed, for example, of the following steps: 1. change basis from plane waves to local orbitals at scale µ; 2. perform an embedding/downfolding into a correlated cluster; 3. coarse-grain to a lower scale µ0; 4. revisit the basis and embedding choices at the coarse scale; 5. lift back to scale µvia a reconstruction functor; 6. transform back to the original basis and embedding description. The RG parallel transport (236) along γdefines an autoequivalence Hγ:Cb−→ Cb,(251) and hence an induced autoequivalence of universal objects, b Hγ:Uµ−→ Uµ.(252) In general, b Hγ need not be isomorphic to the identity functor; its deviation from the identity is precisely the loop curvature. Definition and basic properties. For a fixed loop γ based at b = ( µ, σ ), we define the loop curvature functional acting on a spectral universal object U∈Uµby Kγ(U) := Defectµb Hγ(U)−Defectµ(U).(253) Several observations are immediate: 1. If b Hγ≃IdUµ(i.e. the loop is spectrally flat), then Kγ(U)=0for all U. 2. If Kγ ( U ) = 0 for all U in a universality class and Defectµ separates universal objects, then b Hγ restricts to an autoequivalence isomorphic to the identity on that class: spectral universality is independent of the path encoded by γ. 3. Conversely, if there exists some U with Kγ ( U ) 6 = 0, then the spectral universal object changes under the loop. This indicates a genuine scheme dependence: different choices of basis, embedding, and downfolding order around the loop lead to spectrally inequivalent XC descriptions. 50 To make the last point more concrete, let us write γas a composite of simple steps: γ:Sα1 −→ S(1) α2 −→ S(2) ··· −→αN −−→ S(N)=S, (254) with each αja morphism in R(basis change, embedding, RG step, etc.). Then b Hγ=Uµ(αN)◦···◦Uµ(α1),(255) and loop curvature measures how far this composite differs from the identity when acting on U = Uµ ( S ). In the infinitesimal limit, where γ shrinks to a small parallelogram spanned by two vector fields X, Y on B , the holonomy can be expanded to leading order in the area element, and Kγ ( U )becomes proportional to the local curvature component R(X, Y )acting on U. Formally, b Hγ(U) = U+δ2R(X, Y )(U) + O(δ3),(256) and hence Kγ(U)≈δ2∇Defectµ(U)·R(X, Y )(U),(257) where ∇Defectµ denotes the (Fréchet) gradient of the defect functional on the space of universal objects. Thus, to leading order, loop curvature is a bilinear functional of the RG curvature R and the defect gradient. Examples of spectral loop curvature. Typical loops of interest in electronic-structure practice include: • Basis/embedding loops. Start from a plane-wave SpectralXC scheme at scale µ , rotate to a localorbital basis, perform a DMFT or cluster embedding, downfold to an effective Hubbard-like model at scale µ0 , then reverse the steps in a different order (e.g. downfold first, then embed, then transform basis). If the resulting spectral universal object differs from the initial one, Kγ ( U ) 6 = 0 diagnoses a scheme-induced change in spectral universality. • Downfolding-order loops. Consider a system with both high-energy ligand states and intermediateenergy d-shell states. One may construct effective low-energy Hamiltonians by first integrating out ligand states and then d-shell states, or vice versa. If the sequence of Feshbach projections is non-commutative at the spectral-universal level, the corresponding loop γ has nonzero curvature, and Kγ(U)measures the degree to which XC depends on the order of downfolding. • Regularisation/continuum-limit loops. In model systems where one can alter the UV regulator (grid spacing, cutoff) and then take a continuum limit by sending µ→ ∞ , different regulator choices can define loops in B . Nonzero loop curvature here is an RG anomaly: even after taking the continuum limit, spectral universals differ in a way that cannot be removed by local counterterms, signalling a structural inconsistency in the XC scheme. Universality, curvature, and XC anomalies We can now summarise the role of curvature in spectral RG for XC: •Adjoint curvature RF ( U ) as an irreversibility measure. Large adjoint curvature indicates that coarse-graining from µ to µ0 discards spectral information that cannot be reconstructed without increasing the spectral defect. This is intrinsic RG irreversibility: information about short-distance spectral structure is genuinely lost. In the XC context, RF ( U )tells us which spectral features (e.g. fine structure of high-energy bands) are irrelevant for the universal XC description at coarse scales. •Loop curvature Kγ ( U ) as a scheme-anomaly measure. Nonzero loop curvature means that the spectral universal object depends on the path chosen in scheme space (basis, embedding, downfolding order). Such dependence is a renormalisation anomaly for XC: a structural sensitivity of the universal object to choices that should, ideally, be physically inessential. In practice, Kγ ( U ) can be used to compare different computational protocols and to identify those that are most compatible with XC universality. 51 •Spectral features vs scheme artefacts. Spectral features that are insensitive to both adjoint and loop curvature (i.e. for which RF ( U ) ≈ 0and Kγ ( U ) ≈ 0across relevant RG steps and loops) can be regarded as truly universal: they are preserved under coarse-graining and do not depend on scheme choices. In contrast, spectral features that fluctuate strongly with curvature are potential scheme artefacts, to be treated with caution when designing and interpreting XC functionals. This curvature-based diagnosis complements the defect-based monotonicity of Subsection IV C. Whereas the defect functional Defectµ ( U )measures how far a given universal object is from tile physics at a fixed scale, the curvatures RF ( U )and Kγ ( U )measure how robust the universal object is under scale changes and scheme variations. Together, they provide a structural framework for identifying and exploiting universality in SpectralXC, and for distinguishing genuine universal XC features from artefacts of representation, embedding, or coarse-graining. V. XC FROM SPECTRAL CONSTRAINTS: DERIVATION WITHIN UP–RG A. Constrained Levy–Lieb with universal-property constraints In this subsection we show how the SpectralXC and UP–RG structures introduced in the previous sections enter the density-functional variational principle in a precise way. Starting from the Levy–Lieb constrained-search formulation of DFT [ 57 , 58 ], we augment the usual energy functional by universalproperty constraints which enforce that the spectral universal object Uµ,θ associated with a density n lies in a fixed XC universality class. These constraints are implemented through Lagrange multipliers {λk} conjugate to spectral quantities Sfk [ n ; θ ], leading directly to an expression for the spectral contribution to the exchange–correlation potential vxc ( r )in terms of functional derivatives δSfk [ n ; θ ] /δn ( r ), which we already know how to compute via resolvent calculus. Levy–Lieb functional and the standard variational principle We briefly recall the Levy–Lieb formulation of ground-state DFT for a system of interacting electrons in an external scalar potential vext ( r )[ 57 – 60 ]. Let HN denote the N -electron Hilbert space, and T + W the internal Hamiltonian consisting of the kinetic energy and electron–electron interaction. The Levy–Lieb universal functional F [ n ]is defined by a constrained search over N -electron wavefunctions (or more generally, density matrices) yielding the prescribed density n: F[n] := inf Ψ→nhΨ|T+W|Ψi,(258) where the infimum ranges over all normalised antisymmetric wavefunctions Ψ ∈ HN whose one-body density equals n ( r ). Under mild conditions on T + W and vext , F [ n ]is convex, lower semi-continuous, and yields the ground-state energy via the variational principle E[vext] = inf nF[n] + Zd3rvext(r)n(r),(259) the infimum being over N-representable densities. To connect with the Kohn–Sham (KS) formulation [60], one splits F[n]as F[n] = Ts[n] + EH[n] + Exc[n],(260) where Ts [ n ]is the non-interacting kinetic energy functional, EH [ n ]the Hartree (classical Coulomb) term, and Exc [ n ]the exchange–correlation functional. The KS equations then arise as the Euler–Lagrange equations for (259) when F [ n ]is approximated by a functional of the KS orbitals, and the exchange– correlation potential is given formally by vxc(r) := δExc[n] δn(r).(261) In standard DFT practice, Exc [ n ]is approximated by local or semi-local functionals (LDA, GGA), hybrids, or more elaborate orbital functionals [ 61 ]. These approximations are typically constructed by 52 fitting to data from the uniform electron gas, atomic reference systems, and other benchmarks, without an explicit structural notion of what must be preserved across systems and scales. In contrast, SpectralXC and UP–RG impose structural constraints on the spectrum of the KS-like operator L [ n ; θ ], encoded in the universal objects Uµ,θ, and derive vxc from these constraints. Universal-property constraints as spectral laws Fix a descriptor θ∈ Θand a scale µ . As in Subsection III B, for each admissible density n we associate a KS-like generator L [ n ; θ ]on the one-particle Hilbert space H1 , an Ad-invariant state Φ θ : Aθ→C on Aθ=C∗(L[n;θ]), and a family of probes {fk}m k=1. This defines spectral quantities Sfk[n;θ] := Φθfk(L[n;θ]), k = 1, . . . , m. (262) We then fix target values φk ( θ ), determined by tile physics (HEG, SCE, TLL, multi-orbital Hubbard regimes) and their transport along exact legs as in Subsection III C, and impose the spectral laws Sfk[n;θ] = φk(θ), k = 1, . . . , m, (263) as universality conditions on the admissible densities n. It is convenient to assemble the spectral quantities and targets into vectors: S[n;θ] = Sf1[n;θ], . . . , Sfm[n;θ]∈Cm,φ(θ) = φ1(θ), . . . , φm(θ),(264) and to define the deviation vector ∆S[n;θ] := S[n;θ]−φ(θ).(265) Densities n satisfying the universality conditions (263) are precisely those for which ∆ S [ n ; θ ] = 0. In geometric terms, we are intersecting the space of admissible densities with the manifold defined by the universal-property constraints (263). Constrained Levy–Lieb functional We now formulate a Levy–Lieb-type variational principle with universal spectral constraints. There are two equivalent ways to proceed: 1. Impose (263) directly as hard equality constraints in the minimisation over n. 2. Introduce Lagrange multipliers {λk}and augment the energy functional with constraint terms. We follow the second route, which is more flexible and connects smoothly with the defect-functional picture of Subsection IV C. Let E[n;θ]denote the total energy functional at descriptor θ, written in Levy–Lieb form as E[n;θ] := F[n] + Zd3rvext(r;θ)n(r).(266) We define the universal-property constrained Levy–Lieb functional by introducing a Lagrangian functional L[n, {λk};θ] := E[n;θ]− m X k=1 λkSfk[n;θ]−φk(θ),(267) where λk∈R(or Cif necessary) are Lagrange multipliers enforcing the spectral laws (263). We then consider the constrained minimisation problem EUP[θ] := inf n,{λk}L[n, {λk};θ],(268) subject to any additional constraints on n (particle number, boundary conditions, N -representability). Stationarity with respect to the free variables n and λk yields the Euler–Lagrange equations for the UP–SpectralXC problem. 53 Stationarity with respect to Lagrange multipliers. Variation of L with respect to the multipliers λk gives ∂L[n, {λ`};θ] ∂λk =−Sfk[n;θ]−φk(θ),(269) so at a stationary point {λ? k}we must have Sfk[n?;θ] = φk(θ), k = 1, . . . , m. (270) Thus the Lagrange multipliers enforce the spectral universal-property constraints exactly, as intended. Stationarity with respect to the density. To extract the effective potential, we compute the functional derivative of Lwith respect to n(r), holding {λk}fixed: δL[n, {λk};θ] δn(r)=δE[n;θ] δn(r)− m X k=1 λk δSfk[n;θ] δn(r).(271) At a stationary point n?, δL[n, {λk};θ] δn(r)n=n?,λ=λ? = 0,(272) so δE[n;θ] δn(r)n=n? = m X k=1 λ? k δSfk[n;θ] δn(r)n=n? .(273) The left-hand side of (273) is the functional derivative of the Levy–Lieb energy, which in the KS framework can be written as δE[n;θ] δn(r)=vext(r;θ) + vH[n](r) + v(0) xc [n;θ](r) + . . . , (274) where vH is the Hartree potential and v(0) xc is a baseline XC potential (for instance obtained from a standard approximate functional). The ellipsis indicates possible additional terms such as kinetic-correlation contributions when working with generalised KS schemes. We now define the UP–SpectralXC correction to the XC potential by v(spec) xc [n;θ](r) := m X k=1 λ? k δSfk[n;θ] δn(r),(275) so that the full XC potential is vxc[n;θ](r) := v(0) xc [n;θ](r) + v(spec) xc [n;θ](r).(276) Then (273) can be interpreted as the KS Euler–Lagrange equation with vxc given by (276) , and the Lagrange multipliers {λ? k}determined self-consistently so that the spectral laws (270) hold. Equation (275) is precisely the symbolic relation suggested in the prompt, v(spec) xc (r) = m X k=1 λk δSfk[n] δn(r),(277) now derived from the constrained Levy–Lieb functional. Explicit form of δSfk/δn via resolvents We now recall how the functional derivatives δSfk [ n ; θ ] /δn ( r )can be computed in terms of the KS-like generator L[n;θ]via resolvent calculus, and insert this into (275). 54 For clarity, we suppress the explicit θ -dependence and write L = L [ n ; θ ]. Let f be a holomorphic probe on a domain containing σ(L), and let Γbe a contour encircling the spectrum of L. Then f(L) = 1 2πi IΓ f(z) (z−L)−1dz, (278) and the spectral quantity Sf[n;θ]is Sf[n;θ]=Φθf(L[n;θ])= Φθ1 2πi IΓ f(z) (z−L[n;θ])−1dz.(279) Differentiating under the integral sign and using the identity δ(z−L)−1= (z−L)−1δL (z−L)−1,(280) we obtain δSf[n;θ] = Φθ1 2πi IΓ f(z) (z−L)−1δL (z−L)−1dz.(281) Now write δL in terms of δn. In KS-like theories, Lcan be decomposed as L[n;θ] = T+vext(·;θ) + vH[n](·) + v(0) xc [n;θ](·) + . . . , (282) so that an infinitesimal variation δn induces δL =Zd3rδL δn(r)δn(r),(283) where δL/δn ( r )is a (distribution-valued) operator encoding the response of the potential to density changes at r. Substituting (283) into (281) and reading off the coefficient of δn(r), we obtain δSf[n;θ] δn(r)= Φθ1 2πi IΓ f(z) (z−L)−1δL δn(r)(z−L)−1dz.(284) This is exactly the expression announced in earlier sections: a resolvent sandwich of δL/δn weighted by the probe function fand traced by Φθ. Inserting (284) into (275), we obtain the explicit spectral contribution to the XC potential: v(spec) xc [n;θ](r) = m X k=1 λ? kΦθ1 2πi IΓk fk(z) (z−L[n;θ])−1δL[n;θ] δn(r)(z−L[n;θ])−1dz,(285) where Γ k is a contour adapted to the analyticity domain of fk . Equation (285) is the central formula linking spectral universal-property constraints to a concrete XC potential: each constraint contributes a non-local operator kernel controlled by the resolvent of L and by the density response of the KS-like generator. Relation to universal objects and defect functionals Finally, we relate the constrained Levy–Lieb picture to the universal objects Uµ,θ and defect functionals introduced in Subsections III B and IV C. Recall that Uµ,θ includes the vector S [ n ; θ ]of spectral quantities, and that the scalar defect was defined as Defect(S) µ(Uµ,θ)=∆S[n;θ]†W∆S[n;θ],(286) for some positive semidefinite matrix W . From this perspective, the hard constraints (263) correspond to the limit W→ ∞ in an augmented functional e E[n;θ] := E[n;θ] + γµDefectµ(Uµ,θ),(287) 55 with a large penalty parameter γµ , whereas the Lagrangian (267) corresponds to enforcing the constraints via Lagrange multipliers rather than penalties. In both pictures, the minimisation over n is restricted to densities whose universal objects Uµ,θ lie in the desired universality class (or near it in defect space). The advantage of the Lagrange-multiplier formulation is that it yields the spectral XC potential in the explicit form (285) , with multipliers {λ? k} determined by the universal-property constraints themselves, rather than by empirical fitting or ad hoc choices. The Lagrange multipliers thus acquire a clear physical meaning: they are the conjugate variables to universal spectral invariants, encoding how sensitive the energy is to deviations from tile physics. In the UP–RG framework, they are scale-dependent couplings whose flow is constrained by defect monotonicity and curvature, and whose fixed points define stable XC universality classes. In subsequent subsections we will show how (285) combines with the UP–RG structure to yield multiscale XC potentials, how the Lagrange multipliers λ? k are determined by matching to tiles and enforcing defect monotonicity, and how curvature in spectral RG controls the path dependence and scheme robustness of the resulting XC functionals. B. Resolvent formula for vxc In Subsection V A we showed that imposing spectral universal-property constraints (263) via Lagrange multipliers {λk}leads to an exchange–correlation (XC) potential of the form v(spec) xc [n;θ](r) = m X k=1 λ? k δSfk[n;θ] δn(r),(288) with δSfk/δn given by the resolvent expression (284) δSfk[n;θ] δn(r)= Φθ1 2πi IΓk fk(z) (z−L[n;θ])−1δL[n;θ] δn(r)(z−L[n;θ])−1dz.(289) Combining (288) and (289), we obtained the compact resolvent formula v(spec) xc [n;θ](r) = m X k=1 λ? kΦθ1 2πi IΓk fk(z) (z−L[n;θ])−1δL[n;θ] δn(r)(z−L[n;θ])−1dz.(290) The goal of this subsection is to unpack (290) in detail, starting from the resolvent functional calculus and deriving the variation formula step by step, then rewriting it in kernel form and interpreting it physically. We will see that each spectral constraint contributes a resolvent-based non-local kernel to the XC potential, and that the Lagrange multipliers {λk} acquire a clear role as generalised XC couplings living in the space of spectral laws, not as empirical fit parameters. Resolvent functional calculus and variation of probes Let L = L [ n ; θ ]be the KS-like generator for a fixed density n and descriptor θ , self-adjoint on the one-particle Hilbert space H1 . For each probe fk which is holomorphic on a domain Ω k⊂C containing the spectrum σ ( L ), the holomorphic functional calculus [ 62 ] defines the operator fk ( L )by the Cauchy integral fk(L) = 1 2πi IΓk fk(z) (z−L)−1dz, (291) where Γ k⊂ Ω k is a contour encircling σ ( L )once in the positive sense and lying entirely within the domain of holomorphy of fk. Consider now an infinitesimal variation δL of the operator L , coming from an infinitesimal variation of the density n. Differentiating (291) with respect to Land using the identity δ(z−L)−1= (z−L)−1δL (z−L)−1,(292) 56 we obtain δfk(L) = 1 2πi IΓk fk(z)δ(z−L)−1dz =1 2πi IΓk fk(z) (z−L)−1δL (z−L)−1dz. (293) Equation (293) is the operator-level version of the variation formula for the probes: the change in fk ( L ) induced by a small change in L is given by a double resolvent “sandwich” of δL , weighted by the probe function fk(z)and integrated over a contour encircling the spectrum. Given a state Φθon Aθ=C∗(L), the corresponding spectral quantity Sfk[n;θ] = Φθfk(L) varies according to δSfk[n;θ]=Φθδfk(L) = Φθ1 2πi IΓk fk(z) (z−L)−1δL (z−L)−1dz,(294) which is exactly (281), now derived explicitly. To connect to the density, we express δL in terms of δn, as in (283): δL =Zd3rδL δn(r)δn(r).(295) Substituting into (294) , interchanging the order of integration, and comparing with the definition of a functional derivative, we recover (289): δSfk[n;θ] δn(r)= Φθ1 2πi IΓk fk(z) (z−L)−1δL δn(r)(z−L)−1dz.(296) Thus the resolvent calculus provides a complete and rigorous route from the operator-level probes fk(L)to the functional derivatives δSfk/δn, and hence to the XC potential via (288). Kernel representation and spatial structure To better understand the spatial structure of (290) , it is useful to write the resolvent and the functional derivative δL/δn in kernel form. Let G(z;r,r0)denote the integral kernel of the resolvent (z−L)−1: [(z−L)−1ψ](r) = Zd3r0G(z;r,r0)ψ(r0),(297) for ψ∈ H1. Likewise, for local KS-like generators Lof the form L[n;θ] = −1 2∇2+veff[n;θ](r), with veff a multiplicative potential, the functional derivative δL/δn(r)is also a multiplicative operator, δL δn(r)ψ(r0) = K[n;θ](r0,r)ψ(r0),(298) where K[n;θ](r0,r) = δveff [n;θ](r0) δn(r),(299) is a kernel encoding how the effective potential at r0 responds to density changes at r . For simple local approximations veff(r) = veff(n(r)), this kernel is ultralocal in space, K[n;θ](r0,r) = f[n;θ](r)δ(r0−r), 57 but in general (e.g. for orbital-dependent or non-local functionals) it can be non-local. Substituting (298) into (289), and writing out the resolvent sandwich explicitly, we obtain δSfk[n;θ] δn(r)= Φθ1 2πi IΓk fk(z) (z−L)−1δL δn(r)(z−L)−1dz =1 2πi IΓk fk(z) Φθ(z−L)−1δL δn(r)(z−L)−1dz. (300) If Φθis a trace-like functional (e.g. trace per unit volume), we may write, formally, Φθ(z−L)−1δL δn(r)(z−L)−1=Zd3r1d3r2G(z;r1,r2)K[n;θ](r2,r)G(z;r2,r1),(301) where G(z;r1,r2)is the resolvent kernel and K[n;θ]is the response kernel of the effective potential. Inserting (301) into (290) yields a fully kernel-based expression: v(spec) xc [n;θ](r) = m X k=1 λ? k 2πi IΓk fk(z)Zd3r1d3r2G(z;r1,r2)K[n;θ](r2,r)G(z;r2,r1)dz. (302) Several structural features of (302) are worth emphasising: • The potential at r couples to the spectrum of L at all points r1,r2 via the resolvent kernel G ( z ; r1,r2 ) and the density-response kernel K [ n ; θ ]( r2,r ). This is a non-local, non-linear dependence reminiscent of the optimised effective potential (OEP) equations in orbital-dependent DFT [65]. • The contour Γ k and probe function fk determine which spectral windows (e.g. frontier, occupied, d- /f-shell) contribute most strongly to (302) . Probes localised near the Fermi level focus on low-energy excitations; probes encircling higher bands emphasise screening and high-energy correlation. • The Lagrange multipliers λ? k weight the contribution of each spectral law; they are determined not by fitting but by the requirement that the spectral constraints Sfk [ n ; θ ] = φk ( θ )be satisfied at the solution of the variational problem. Spectral response and linearisation around a solution Although the full UP–SpectralXC problem is non-linear, insight can be gained by linearising around a reference solution ( n0,{λ0 k} )that satisfies the spectral constraints exactly. Consider small perturbations δn and δλkand expand (270) to first order: 0 = δSfk[n;θ] = Zd3rδSfk[n;θ] δn(r)δn(r)at n=n0,(303) where we have used that the target φk ( θ )is fixed along the descriptor direction considered. Meanwhile, the change in vxc induced by δλkis, to first order, δv(spec) xc (r) = m X k=1 δλk δSfk[n0;θ] δn(r).(304) The induced density change δn is related to δvxc via a static density-response (susceptibility) kernel χ(r,r0), for instance the KS or interacting response of the reference system [63, 64]: δn(r) = Zd3r0χ(r,r0)δvxc(r0).(305) Combining (304) and (305), we obtain δn(r) = m X `=1 δλ`Zd3r0χ(r,r0)δSf`[n0;θ] δn(r0).(306) 64 Hubbard-dimer tile and spectral targets To provide concrete spectral targets, we adopt the half-filled Hubbard dimer as a tile model for the strongly correlated bond [71]. The Hubbard dimer Hamiltonian is HHub =−tX σ=↑,↓c† 1σc2σ+c† 2σc1σ+U 2 X i=1 ni↑ni↓+ ∆ X σ (n1σ−n2σ),(322) where t is the hopping amplitude, U the on-site repulsion, ∆an on-site energy difference (dimer asymmetry), niσ = c† iσciσ , and ni = ni↑ + ni↓ . At half filling ( N = 2 electrons), and for ∆=0, the spectrum consists of a singlet ground state and excited singlet and triplet states whose energies can be obtained analytically. Defining bonding and antibonding orbitals cbσ := 1 √2(c1σ+c2σ), caσ := 1 √2(c1σ−c2σ),(323) the non-interacting ( U = 0) single-particle energies are b = −t and a = + t , giving a HOMO–LUMO gap 2 t . For U > 0, the exact many-body spectrum at half filling exhibits a correlation-induced singlet–triplet gap and a strong evolution of natural occupation numbers as a function of U/t, capturing the crossover from weakly correlated covalent bonding to strongly correlated, nearly localised electrons on each site. The Hubbard dimer thus provides analytic expressions for: •the bonding–antibonding gap and its interaction dependence; •double occupancy hni↑ni↓ias a function of U/t; •spin correlations hS1·S2i; •natural occupation numbers of bonding and antibonding orbitals. These quantities are precisely the kind of spectral invariants that we use as targets φtile ( R )for the fine-scale SpectralXC scheme: the diatomic molecule at distance R is mapped (via localised orbitals) to a dimer with effective parameters t ( R ) , U ( R ) , ∆( R ), and the exact or high-level solution of (322) provides the spectral laws that we demand to be preserved (up to controlled defects) by the XC functional. Fine-scale SpectralXC scheme for dissociation At the fine scale, we consider a KS-like operator Lfine [ n ; θ ( R )] acting on a large one-particle basis {ϕα} spanning the bound and low-lying continuum states of the diatomic. The descriptor θ ( R )carries the geometrical information, while µfine encodes the resolution (e.g. cutoff). The spectral probes Ffine are chosen to capture: 1. Frontier gap and level splitting. Introduce a contour Γ fr enclosing a narrow spectral window around the HOMO–LUMO region and define ffr(z) = 1 (z−z0)2,(324) with z0near the mid-gap energy. Then the spectral quantity Sffr [n;θ(R)] = Φfine(ffr(Lfine[n;θ(R)])) (325) is sensitive to the frontier gap and level splitting, as discussed in Subsection III B. The tile target φfr ( R )is given by the bonding–antibonding splitting in the Hubbard dimer mapped to the molecular system. 2. Local double occupancy and spin correlations. Projectors onto localised orbital subspaces associated with each fragment, say HAand HB, define operators PA, PB. Probes of the form fdA(L) := PAg(L)PA, fdB(L) := PBg(L)PB,(326) where g selects the occupancy of the corresponding localised orbitals, yield spectral quantities that can be matched to the double occupancies hni↑ni↓i in the Hubbard tile. Similarly, spin-projected probes can be constructed to match hS1·S2i. 65 3. Charge localisation and symmetry breaking. In stretched regimes, standard DFT functionals often spuriously break spin or spatial symmetry, leading to incorrect fractional charge distributions [ 70 ]. Probes based on Riesz projectors onto near-degenerate frontier orbitals on each fragment, together with Ad-invariant states Φ fine enforcing symmetry constraints, can be used to penalise such symmetry breaking at the spectral level. The set of spectral targets φtile ( R )for these probes is obtained by mapping the molecular system at R to an effective Hubbard dimer and solving (322) exactly (or to high accuracy). The UP–SpectralXC Lagrangian of Subsection V A then enforces the spectral laws Sfk [ n ; θ ( R )] = φk ( R )via Lagrange multipliers λk(R), yielding a fine-scale XC potential v(spec) xc [n;θ(R)](r)given by the resolvent formula (290). Coarse-scale model and UP–RG flow We now define a coarse-scale description by projecting the full one-particle Hilbert space onto a minimal active space spanned by two localised orbitals {χA, χB} associated with the two fragments. Let P : H1→span{χA, χB} denote the orthogonal projection and set H0 1 = span{χA, χB} . The coarse-scale generator is obtained via Lcoarse[n;θ(R)] := P Lfine[n;θ(R)] PH0 1 ,(327) possibly in combination with a Feshbach-type elimination of higher-energy subspaces, as in (174) . In this way, Lcoarse becomes a 2 × 2matrix representing an effective one-body Hamiltonian for a two-site model. The resulting coarse-scale SpectralXC scheme is Scoarse(R) = Fµfine→µ0Sfine(R),(328) for some RG functor Fµfine→µ0that: •maps Lfine to Lcoarse via (327) and possible further coarse-graining; •induces a map of probes Ffine → Fcoarse consistent with the projection; • pushes forward the state Φ fine to a coarse state Φ coarse by partial trace or conditional expectation on the active space; • preserves the spectral targets φtile ( R )as much as possible, consistent with spectral defect monotonicity. At each scale, we compute the universal object Uµfine,θ(R)=Uµfine Sfine(R),(329) Uµ0,θ(R)=Uµ0Scoarse(R),(330) and their defects Defectµfine and Defectµ0. For an admissible Fµfine→µ0, UP–RG monotonicity implies Defectµ0Uµ0,θ(R)≤Defectµfine Uµfine,θ(R),(331) expressing the fact that as we flow to the coarse two-site description, the spectral universals move closer to tile physics in defect space. Numerical experiment: monotonicity and loop curvature We now outline a concrete numerical experiment to test the UP–SpectralXC construction for bond dissociation and to probe both defect monotonicity and loop curvature. 66 Step 1: fine-scale SpectralXC solution. For a set of bond lengths R∈ {R1, . . . , RN} spanning the equilibrium region to the stretched regime, we: 1. perform a high-quality reference calculation (full CI, DMRG, or high-level coupled cluster) to determine the target spectral laws φtile(R)for the Hubbard-dimer map; 2. construct the fine-scale KS-like generator Lfine[n;θ(R)] in a large basis; 3. choose spectral probes Ffine as described above and set up the constrained Levy–Lieb functional (267); 4. solve the constrained KS equations with the spectral XC potential (290) , iterating both n and {λk(R)}until the spectral constraints and self-consistency are simultaneously satisfied. This yields, for each R , a self-consistent density nfine ( R ), KS-like generator Lfine ( R ), and universal object Uµfine,θ(R)with a small spectral defect. Step 2: coarse-scale downfolding via different schemes. For each R and for each of several downfolding schemes (e.g. different choices of localised orbitals and embedding procedures), we construct a coarse-scale model: •Scheme A: downfold-then-embed. First perform a global Wannierisation or local-orbital transform to obtain χA, χB and project Lfine to Lcoarse as in (327) . Then embed the resulting two-site model into an environment treated at a lower level (mean-field or standard DFT). • Scheme B: embed-then-downfold. First construct an embedded active space around the bond region (e.g. via DMET or DMFT [ 72 – 74 ]), solving the impurity problem with SpectralXC constraints. Then downfold the impurity model to an effective two-site Hamiltonian by integrating out bath orbitals. In each scheme, we obtain a coarse-scale generator L(A/B) coarse ( R ), compute the corresponding universal object U(A/B) µ0,θ(R)=Uµ0S(A/B) coarse (R), and evaluate the coarse-scale defects Defectµ0(U(A/B) µ0,θ(R)). Step 3: testing monotonicity. For each scheme separately, we verify the inequality (331) numerically: Defectµ0U(X) µ0,θ(R)≤Defectµfine Uµfine,θ(R), X ∈ {A, B},(332) for all R . Deviations from this inequality (beyond numerical noise) signal that the downfolding scheme is not admissible in the sense of Subsection IV C, i.e. it does not preserve spectral universals in a way consistent with UP–RG. Step 4: testing loop curvature. To probe loop curvature, we consider the loop γ in scale–scheme space that starts from the fine-scale scheme Sfine ( R ), follows RG scheme A to a coarse model, then returns to the fine scale via a reconstruction functor, and finally returns to the original basis. Similarly, we define a loop γ0for scheme B. The corresponding holonomies b Hγ,b Hγ0:Uµfine −→ Uµfine ,(333) act on Uµfine,θ(R), and we can evaluate the loop curvature functionals KγUµfine,θ(R), Kγ0Uµfine,θ(R),(334) as defined in (253) . Non-zero values indicate scheme dependence: the spectral universals, and hence the XC structure, depend on the path taken in downfolding/embedding space. Comparing Kγ and Kγ0 across Rprovides a diagnostic for which path is more “universal” in the sense of UP–RG. 67 Physical narrative: bond breaking as flow towards a tile From the physical point of view, the dissociation of a strongly correlated molecule can now be reinterpreted in UP–SpectralXC language as a flow of universal objects Uµ,θ(R) towards a Hubbard-dimer tile as R→ ∞: • At equilibrium ( R≈Req ), the system is moderately correlated, and the universal object includes significant contributions from extended orbitals and dynamic correlation. Spectral defects are small but non-zero. • As R increases, the bonding–antibonding gap closes, double occupancy is suppressed, and spin correlations approach those of a Heisenberg antiferromagnet. The spectral laws enforced by φtile ( R ) keep the KS spectrum aligned with the Hubbard-dimer picture, and the spectral defect decreases. • In the dissociation limit ( R→ ∞ ), the system approaches two independent atoms, and the Hubbard dimer becomes exact: the universal object Uµ,θ(R) coincides (up to trivial transformations) with the dimer tile, and the spectral defect vanishes. The XC potential obtained from the resolvent formula exhibits the correct step and peak structures needed to produce integer fragment occupancies and the correct spin symmetry, in contrast with standard local/semi-local functionals [70]. In this sense, the UP–SpectralXC construction turns the dissociation problem into a controlled RG interpolation between an equilibrium molecular regime and a universal Hubbard-dimer tile, with the spectral defect playing the role of a c-function measuring the “distance” from tile physics. Embedding and downfolding schemes can be systematically assessed by their effect on this defect and on loop curvature, providing a principled way to design multiscale XC treatments for strongly correlated molecules. Numerical experiment: Hubbard dimer as a UP–SpectralXC tile To make the above discussion concrete, we performed a minimal numerical implementation of the UP–SpectralXC ideas using the half-filled Hubbard dimer as a tile model for a dissociating bond. The Hamiltonian is HHub =−tX σ=↑,↓c† 1σc2σ+c† 2σc1σ+U 2 X i=1 ni↑ni↓,(335) with hopping t (set to t = 1 in all plots) and on-site repulsion U . We restrict to the N = 2 sector (half filling) and diagonalise (335) exactly in the full 16-dimensional Fock space, then project to the N = 2 subspace. From the corresponding ground-state eigenvector (singlet sector) and the lowest triplet state (Sz= 1 sector), we extract three basic tile quantities as functions of U/t: 1. the ground-state singlet energy, Es(U); 2. the singlet–triplet gap, ∆E(U) := Et(U)−Es(U); 3. the total double occupancy, D(U) := n1↑n1↓+n2↑n2↓U. In this numerical demonstration we use only the pair D(U), Es(U)to build the universal objects, but conceptually they correspond to two spectral probes: one encoding local two-body correlation (double occupancy) and one encoding the ground-state energy. As a “KS-like” non-interacting reference we take the same dimer with U = 0. At half filling, the non-interacting ground state is a doubly occupied bonding orbital, and a straightforward calculation gives a reference double occupancy DKS = 1 / 2and a reference ground-state energy EKS s = Es ( U = 0) (the bottom of the non-interacting band). In the language of Subsections III B and IV C, this defines a very simple two-component spectral universal for the tile and for the KS reference: Utile(U) := Dexact(U), Eexact s(U), UKS := DKS, EKS s,(336) 68 012345678 U / t 0.1 0.2 0.3 0.4 0.5 Double occupancy n 1 n 1+ n 2 n 2 Hubbard dimer: double occupancy vs U / t Exact Hubbard dimer KS-like non-interacting (U=0) Figure 1. Exact double occupancy Dexact ( U )(solid blue) and non-interacting KS-like reference value DKS = 1 / 2 (orange dashed) for the half-filled Hubbard dimer at t = 1. The decay of Dexact with U/t signals the onset of strong correlation (suppression of double occupancy), which the KS reference does not capture. where “exact” indicates the quantities obtained by diagonalising (335) at the given U . For the present toy model, the universal object Uµ,θ has been collapsed to the two-component vector in (336) , but the structure is the same as in the general formalism: a spectral probe family (here reduced to double occupancy and Es) evaluated in a state. Fine-scale tile data and naive KS reference. Figure 1 shows the exact double occupancy Dexact ( U ) (blue curve) together with the constant non-interacting reference DKS = 1 / 2(orange dashed line). As expected, Dexact ( U )decreases monotonically with U/t , reflecting the suppression of double occupancy as the on-site repulsion increases: the system evolves from a weakly correlated, delocalised covalent bond at U/t ≈ 0to a strongly correlated state with nearly one electron on each site at large U/t . The KS reference, by construction, is blind to this evolution. In UP–SpectralXC language, the KS scheme fails to flow into the correct tile universality class as U/t grows. Figure 2 shows the corresponding ground-state energies, Eexact s ( U )(blue) and EKS s (orange dashed line). While EKS s captures the U = 0 limit, it deviates rapidly from the exact curve as interaction strength grows. Together, Figs. 1 and 2 encode the exact tile object Utile(U)and its KS surrogate UKS. Toy coarse schemes and spectral defects. To mimic the effect of different coarse-graining (downfolding/embedding) procedures in a UP–RG setting, we introduce two simplified “schemes”, A and B , defined at the same scale but with different priorities: • Scheme A: a coarse description that reproduces the exact double occupancy but retains the noninteracting reference for the energy. In terms of universal objects, UA(U) := Dexact(U), EKS s. • Scheme B: a coarse description that reproduces the exact energy but retains the KS reference double occupancy, UB(U) := DKS, Eexact s(U). Although highly simplified, these constructions stand in for realistic coarse-graining choices: in practice one may have an embedding scheme that is accurate for local correlation but not for total energy, and another that gets the energy right but fails to reproduce local occupancies. For each scheme we define a spectral defect with respect to the exact tile universal Utile(U) = Dexact(U), Eexact s(U) 69 012345678 U / t 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 Ground-state energy Es / t Hubbard dimer: ground-state energy vs U / t Exact Hubbard dimer KS-like non-interacting (U=0) Figure 2. Ground-state energy Eexact s ( U )(solid blue) and non-interacting KS-like reference energy EKS s (orange dashed) as functions of U/t for the Hubbard dimer. The non-interacting reference is only correct at U = 0 and quickly fails as interaction increases, illustrating the need for a structural XC correction. as the squared Euclidean norm Defectscheme(U) :=  Uscheme(U)−Utile(U) 2=Dscheme −Dexact(U)2+Es,scheme −Eexact s(U)2.(337) For comparison, we also compute DefectKS ( U )for the naive KS scheme and Defecttile ( U ) ≡ 0for the exact tile itself. Figure 3 displays DefectKS ( U ), DefectA ( U ), and DefectB ( U )as functions of U/t . All three grow as U increases, reflecting the increasing difficulty of representing the strongly correlated tile by a simple coarse scheme. Scheme A is better at large U in the sense that it captures the suppression of double occupancy, whereas Scheme B is better in the weakly correlated regime because it reproduces the energy exactly. In a full UP–SpectralXC treatment these defects would be defined using the full vector of spectral probes and spectral measures, but the simple two-component version already illustrates the idea of an XC “c-function” measuring distance from tile universality. Loop curvature and scheme dependence. To probe scheme dependence in the spirit of the loop curvature Kγ ( U )introduced in Subsection IV D, we define a simple measure of the difference between schemes A and B at fixed U: KA,B(U) :=  UA(U)−UB(U) 2=Dexact(U)−DKS2+EKS s−Eexact s(U)2.(338) In a fully fledged UP–SpectralXC setting KA,B would be associated with the holonomy of a loop in scale–scheme space (e.g. “downfold then embed” versus “embed then downfold”), but in the present toy model it already quantifies how two different coarse descriptions, each of which preserves a different subset of spectral data, diverge from each other as correlation increases. Figure 4 shows KA,B ( U )as a function of U/t : at U = 0 the schemes coincide (trivially), but the curvature grows steadily with U , reflecting the increasing scheme dependence of any coarse description that is not explicitly constrained by the full universal object. Realistic extensions Although the present calculation is deliberately minimal—using only the Hubbard dimer and a two-component universal ( D, Es )—it demonstrates in a concrete way how UP–SpectralXC organises dissociation physics: •the exact Hubbard dimer provides tile spectral laws and the reference universal Utile(U); 70 012345678 U / t 0.0 0.5 1.0 1.5 2.0 2.5 Defect w.r.t. tile || U scheme U tile||2 Hubbard dimer: spectral defects for different coarse schemes Naive KS scheme Scheme A: match D Scheme B: match Es Figure 3. Spectral defects with respect to the exact tile universal Utile ( U )for three simple schemes: (i) naive KS-like non-interacting scheme; (ii) Scheme A, which matches double occupancy but keeps the KS energy; (iii) Scheme B, which matches the energy but keeps the KS double occupancy. The growth of the defects with U/t quantifies the increasing mismatch between these coarse schemes and the true strongly correlated tile. 012345678 U / t 0.0 0.5 1.0 1.5 2.0 2.5 Curvature / scheme dependence Hubbard dimer: difference between coarse schemes A and B Scheme-dependence curvature || UAUB ||2 Figure 4. Toy loop-curvature measure KA,B ( U ) = kUA ( U ) −UB ( U ) k2 , quantifying the difference between two coarse schemes A and B that prioritise different spectral features (double occupancy versus energy). In a realistic UP–SpectralXC implementation this would correspond to the holonomy of a loop in scale–scheme space, diagnosing scheme dependence of the spectral universal object. • spectral defects behave as c-type functions measuring distance from the tile along correlation-strength “flows” driven by U/t; • different coarse schemes can be compared structurally via defects and a loop-curvature-like quantity. In realistic molecular dissociation problems, the same logic can be applied with: (i) ab initio or DMRG tiles replacing the Hubbard dimer, (ii) larger probe families (frontier gaps, local projectors, spectral moments), and (iii) genuine downfolding/embedding maps (e.g. Wannier-based impurity problems) replacing the toy schemes A and B. The spectral universal objects Uµ,θ and their defects then become the central diagnostics and design principles for multiscale, scheme-aware XC functionals in the bond-breaking regime. 71 B. Transition-metal complexes: d-shell XC universality Transition-metal (TM) complexes constitute one of the most demanding classes of problems for electronic-structure theory. Their low-energy physics is controlled by a partially filled d -shell hybridised with ligand orbitals, leading to: (i) strong static correlation in the d -manifold, (ii) near-degenerate spin and charge states (high-spin/low-spin, mixed-valence, spin-crossover), and (iii) pronounced sensitivity to the local ligand field [ 75 – 77 ]. Conventional local and semi-local density functionals, as well as many hybrids, often fail in this regime: they tend to over-delocalise d -electrons, underestimate Hund’s couplings, and misrepresent spin-state energetics. Remedies such as LDA+ U [ 78 ], DFT+DMFT [ 79 – 81 ], or multireference wave-function methods [82] introduce additional parameters or active-space choices, but lack a unifying structural notion of what d -shell universality means and how it should be preserved across basis choices, embeddings, and scales. In the UP–SpectralXC framework, a TM complex is viewed as a point θ∈ Θin the descriptor manifold carrying: (i) geometrical data (metal–ligand distances and angles), (ii) nuclear charges and oxidation state (formal dn configuration), (iii) external fields (solvent, crystal environment), and (iv) possibly an interaction-scaling parameter along an adiabatic connection. For each θ , a KS-like generator L [ n ; θ ]acts on a one-particle Hilbert space H1 , initially represented in a high-level basis (plane waves, atom-centred Gaussians). The key idea is that, after appropriate basis changes and embeddings, the dominant lowenergy XC structure can be encoded in a universal d -shell object obtained by projecting L [ n ; θ ]onto a local d -like subspace and enforcing tile-based spectral laws derived from multi-orbital Hubbard-like models. Localised d-shell subspace and Riesz projectors We begin by isolating the local d -shell manifold through a sequence of basis changes. At a fine resolution scale µpw we represent L [ n ; θ ]in, say, a plane-wave basis or a large Gaussian basis. A first RG functor Fµpw→µwann maps this description to a Wannier representation [ 83 ], in which localised orbitals wm ( r−R0 ) (with m = 1 , . . . , M ) centred on the TM site at R0 span a subspace Hd+p consisting of d -like and strongly hybridised ligand orbitals. A second functor Fµwann→µd effects a further downfolding to a purely d -like subspace Hd⊂ Hd+p , spanned by a set of five localised d -orbitals {χm}5 m=1 in a suitable real or complex combination (e.g. eg/t2gorbitals in an octahedral ligand field). At the operator level, this sequence of basis changes and projections induces a spectral projector onto the d-subspace Pd:= 5 X m=1 |χmihχm|,(339) which can equivalently be represented as a Riesz projector of the KS-like generator: Pd=1 2πi IΓd (z−L[n;θ])−1dz, (340) where Γ d is a contour encircling the cluster of eigenvalues associated with the d -manifold of the isolated metal site plus ligand field. Equation (340) is a direct application of the resolvent functional calculus (291), specialised to a step-function-like probe f(z) = 1inside Γd. The operator Pd defines both: (i) a finite-rank projection in the operator algebra Aθ = C∗ ( L [ n ; θ ]), and (ii) a K0 -class [ Pd ] ∈K0 ( Aθ ), which is stable under compact perturbations and unitary equivalences. In particular, any basis change implemented by a unitary U : H1→ H1 sends Pd7→ UPdU† without changing its class [ Pd ], while coherent downfoldings that preserve the rank of Pd leave [ Pd ]invariant. Thus [Pd]is a natural piece of the universal spectral object Uµ,θ for the TM complex: Uµ,θ ⊃[Pd], νd,θ,Sd[n;θ],(341) where νd,θ is a local spectral measure associated with PdL [ n ; θ ] Pd and Sd [ n ; θ ]is a vector of d -shell spectral quantities to be defined next. 72 Spectral probes and d-shell spectral laws The d -shell subspace Hd carries the multiplet structure responsible for the low-energy properties of TM complexes: ligand-field splittings (e.g. eg vs. t2g ), Hund’s-rule couplings, spin-state ordering, and orbital polarisation. To capture this within SpectralXC, we introduce a family of probes {fk} acting on the d-shell block of L[n;θ]and its associated one-particle density matrix. Typical examples include: 1. Projector-based probes. For each irreducible representation (irrep) α of the local point group (e.g. eg , t2g in an octahedral environment), we define a projector Pα obtained by group-averaging the d-orbitals or by resolving Pdinto symmetry-adapted blocks. As operators on H1, Pα=X m∈α|χmihχm|, and we take fα ( L ) = Pα as trivial probes whose spectral quantities become occupancy constraints. 2. Resolvent probes near d-level energies. For each centre zk in the complex plane chosen near a ligand-field level (e.g. eg centre, t2g centre, or satellite features in a d -projected density of states), we define a holomorphic probe fk(z) = 1 (z−zk)p, p ≥1,(342) and construct the operator fk(L[n;θ]) = 1 2πi IΓk fk(z) (z−L[n;θ])−1dz, where Γ k is a contour adapted to the spectral window around zk . The associated spectral quantities Sfk[n;θ]=Φθfk(L[n;θ]) pick out local moments of the d-shell spectral density near the targeted energy window. 3. Spin and charge probes. In addition, we can define local spin and charge operators on Hd: ˆ Nd:= X m,σ d† mσdmσ,ˆ Sd:= X m ˆ Sm, where dmσ annihilates an electron in orbital χm with spin σ , and ˆ Sm is the corresponding spin1 2 operator. Their expectation values in an Ad-invariant state Φθgive Nd[n;θ]=Φθ(ˆ Nd),S2 d[n;θ] = Φθ(ˆ S2 d), which serve as probes for local electron count and spin state. Let Sd[n;θ]denote the vector collecting all such spectral quantities: Sd[n;θ] := Nd[n;θ],S2 d[n;θ], Sf1[n;θ], . . . , Sfm[n;θ].(343) These are to be constrained to target values φd ( θ )derived from a multi-orbital tile, such as an Anderson or Hubbard-like impurity model with realistic crystal-field and Hund’s parameters [ 79 – 81 ]. A typical multi-orbital tile reads Htile =X m,σ md† mσdmσ +1 2X m6=m0,σ,σ0 Umm0nmσnm0σ0−JHX m6=m0 ˆ Sm·ˆ Sm0+··· ,(344) where m encode crystal-field splitting, Umm0 Coulomb repulsion, and JH Hund’s coupling. Solving (344) by exact diagonalisation, CI, or DMFT yields tile spectral data such as: (i) the distribution of d -electron counts Nd , (ii) spin-state energies (high-spin vs. low-spin), (iii) d -projected spectral moments around ligand-field levels, which are then packaged into φd(θ). We then impose the d-shell spectral laws Sd[n;θ] = φd(θ)(345) 73 as part of the universal-property constraints in the constrained Levy–Lieb functional (267) , with corresponding Lagrange multipliers {λ(d) k ( θ ) } . The resulting spectral XC potential has a local d -shell contribution of the form v(d) xc [n;θ](r) = X k λ(d) k(θ)δSfk[n;θ] δn(r),(346) with δSfk/δn given by the resolvent formula (284) . This term plays a role analogous to orbital-dependent corrections in LDA+ U or DFT+DMFT, but is now derived from structural spectral constraints rather than from ad hoc penalty functionals. UP–RG structure: basis changes and ligand-field embedding The d -shell XC universality described above is embedded in a multiscale, scheme-aware RG structure. At the level of the categories {Cµ}, we distinguish several scales: µpw > µwann > µd> µtile,(347) corresponding respectively to: (i) a plane-wave or large Gaussian representation of L [ n ; θ ], (ii) a Wannier representation with d + p localised orbitals, (iii) a pure d -shell active space, and (iv) a tile model (344) for the metal centre in its ligand field. RG functors Fµpw→µwann , Fµwann→µd, Fµd→µtile are implemented by: (i) local unitary transforms (Wannierisation) to obtain Hd+p from the extended basis [ 83 ], (ii) a Feshbach-like projection from Hd+p onto Hd plus a dynamical embedding of the ligand p -orbitals, and (iii) a mapping from the static d -shell Hamiltonian to a multi-orbital impurity model Htile coupled to an environment. At each scale µ we have a universal object Uµ,θ of the form (341) , and a defect functional Defectµ ( Uµ,θ ) measuring the deviation of Sd [ n ; θ ]and νd,θ from tile targets. In particular, we can define a d -shell defect as Defect(d) µ(Uµ,θ) := Sd[n;θ]−φd(θ)†W(d) µSd[n;θ]−φd(θ)+α(d) µSνd,θkν? d,θ,(348) where W(d) µ is a positive semidefinite weight matrix, α(d) µ≥ 0, and ν? d,θ is a reference d -shell spectral measure determined by the tile. UP–RG monotonicity implies that for admissible coarse-grainings Defect(d) µ0Uµ0,θ≤Defect(d) µUµ,θ, µ > µ0, i.e. the d -shell description approaches tile universality as we flow from extended bases to the effective d-shell model. Numerical protocol and loop curvature for TM complexes A realistic numerical experiment implementing d -shell XC universality within UP–SpectralXC might proceed as follows. Step 1: choice of system and tiles. Select a prototypical TM complex such as an Fe(II) or Co(III) octahedral complex ( d6 ), possibly in a family exhibiting spin-crossover behaviour [ 77 ]. For a set of representative geometries (metal–ligand distances, ligand substitutions), construct a tile model (344) whose parameters are fitted or derived from high-level calculations (CASSCF/NEVPT2 or DMRG) [ 82 ]. For each descriptor point θ , solve the tile model to obtain φd ( θ )and ν? d,θ , including: (i) local d -electron counts, (ii) spin-state energies, (iii) local spectral moments around eg/t2glevels. Step 2: construction of spectral probes and KS-like generator. For each θ , perform a KS calculation in an extended basis to obtain L [ n ; θ ], then: (i) construct Wannier functions and identify the d + p subspace, (ii) derive Pd via (340) , (iii) construct projectors Pα and resolvent probes fk as in (342) . Evaluate Sd [ n ; θ ] via (343) , using an Ad-invariant state Φ θ compatible with the KS density (for instance, trace per unit cell of the projected Green’s function). 80 setting for transition-metal complexes, the same logic would apply to a full 5-fold Kanamori d -shell with a continuous hybridisation function, but the present model already makes the abstract structure of spectral universality, defects, and curvature fully explicit. VII. APPLICATIONS II: HEG, SCE, AND 1D PHYSICS AS TILES A. HEG tile: importing uniform electron gas physics The traditional starting point for practical XC functionals in electronic structure theory is the three– dimensional homogeneous electron gas (HEG), or “jellium”. For comprehensive reviews, see Refs. [ 84 , 85 ]. Local and semi–local approximations such as LDA and GGA are built by taking HEG results at a given density, parametrising them as exc ( n )and its derivatives, and evaluating those parametrisations at the local density n ( r )in an inhomogeneous system.[ 132 ] From the point of view of UP–SpectralXC, this procedure is extremely informative but conceptually incomplete: HEG appears only as a scalar energy density input into an XC energy functional, rather than as a full spectral tile carrying a rich collection of universal properties. In this subsection we reframe the role of HEG as that of a canonical tile in the sense of Sec. III: a self-contained many–body system with its own universal object UHEG µ,θ , spectral laws φHEG k , and UP–RG structure. This perspective allows us to import uniform electron gas physics into inhomogeneous systems not only via scalar energy densities but through a set of spectral constraints and monotones. In particular, we show how small– q kernel constraints, compressibility, and static structure factor moments can be encoded as spectral probes, and how HEG-based spectral laws can be transported along exact legs in descriptor space from uniform to weakly inhomogeneous systems. Deviations from HEG universality are then quantified by a spectral defect functional whose growth is controlled by the curvature introduced in Sec. IV D. HEG Hamiltonian and density parameter. We consider the standard HEG Hamiltonian in a cubic box of volume Ω = L3with periodic boundary conditions, ˆ HHEG = N X i=1 ˆ p2 i 2m+1 2 N X i,j=1 i6=j e2 |ˆ ri−ˆ rj|−e2n0 N X i=1 ZΩ d3r |ˆ ri−r|+e2n2 0 2ZΩ d3rZΩ d3r01 |r−r0|,(356) where a uniform positive background of density n0 ensures charge neutrality. The thermodynamic limit is obtained by letting N, Ω→ ∞ at fixed density n0=N/Ω. It is convenient to parametrise the density in terms of the Wigner–Seitz radius rs=3 4πn0a3 01/3 ,(357) with a0 the Bohr radius, and a spin polarisation parameter ζ = ( n↑−n↓ ) /n0 . In practice, HEG energetics, pair correlations, and response functions have been obtained from high–accuracy quantum Monte Carlo (QMC) simulations over a broad range of rs, ζ ;[ 131 , 132 ] these data underpin essentially all modern LDA and GGA functionals. In the present context, HEG at fixed ( rs, ζ )defines a canonical tile labelled by a descriptor θHEG = ( rs, ζ ). Rather than using only the ground-state energy per particle exc ( rs, ζ ), we associate to this tile a full universal object UHEG µHEG,θHEG consisting of several spectral probes, as detailed below. Spectral probes: compressibility, small-qkernel, and structure factor. The key objects characterising density fluctuations in HEG are: (i) the static density–density response function χ ( q, 0), (ii) the static XC kernel fxc ( q, 0), and (iii) the static structure factor S ( q ). In reciprocal space the density–density response relates an infinitesimal external potential δvext(q)to the induced density δn(q), δn(q) = χ(q, 0) δvext(q), q =|q|.(358) The XC kernel fxc(q, 0) enters the Dyson-like equation χ−1(q, 0) = χ−1 0(q, 0) −vc(q)−fxc(q, 0), vc(q) = 4πe2 q2,(359) 81 where χ0 ( q, 0) is the static Lindhard function of the non-interacting Fermi gas.[ 84 ] The static structure factor is related to χ(q, 0) via the fluctuation–dissipation theorem, S(q) = −1 πn0Z∞ 0 dω =χ(q, ω) = −1 πn0Z∞ 0 dω =χ(q, ω +i0+),(360) and satisfies well-known exact constraints including the compressibility sum rule and f -sum rule.[ 88 ] In particular, the q→0limit of S(q)is tied to the thermodynamic compressibility κTthrough lim q→0 S(q) q2=~2 2m n0 4κT, κ−1 T=n2 0 ∂2(n0e(n0)) ∂n2 0 ,(361) where e(n0)is the energy per particle, including XC effects. In the language of Sec. III B, these quantities arise as spectral probes of the HEG Hamiltonian. For instance, S ( q )can be written in terms of the resolvent of the density fluctuation operator ˆnq , and its moments Rdq qmS ( q )are spectral sum rules. From the UP–SpectralXC point of view, we select a small but physically rich set of probes: •The compressibility κT(rs, ζ), or equivalently the curvature of the HEG energy density e(rs, ζ). •The small-qlimit of the static XC kernel, f(0) xc (rs, ζ) := lim q→0fxc(q, 0), which controls long-wavelength screening and the macroscopic dielectric constant.[84] •A finite set of structure factor values and low-order moments, S(qi;rs, ζ), Mm(rs, ζ) := Zqmax 0 dq qmS(q;rs, ζ), for selected qiin the smalland intermediate-qregimes. High-accuracy QMC and diagrammatic studies provide reliable values of these quantities over a wide range of rs, ζ.[85, 131] We then define the HEG tile universal object at scale µHEG and descriptor θHEG = (rs, ζ)as UHEG µHEG,θHEG := exc(rs, ζ), κT(rs, ζ), f(0) xc (rs, ζ),{S(qi;rs, ζ)}Nq i=1,{Mm(rs, ζ)}m∈M.(362) This object is precisely the HEG counterpart of the spectral universals introduced in Sec. III B: it packages a collection of spectral laws that characterise the long-wavelength and intermediateq physics of the uniform electron gas. HEG spectral laws and target values. Within the general SpectralXC framework we impose spectral laws of the form Sfk[n;θ] = φHEG k(rs(θ), ζ(θ)),(363) where the left-hand side is a spectral probe evaluated for an inhomogeneous system with density n ( r ) and descriptor θ∈ Θ, and the right-hand side is a HEG target value φHEG k at an appropriate local density parameter rs ( θ )and polarisation ζ ( θ ). For the HEG tile these targets can be taken directly from QMC-based parametrisations[132] or from HEG response theory.[84, 85] Concretely, we can choose the following representative constraints: Sfcomp [n;θ] = κHEG Trs(θ), ζ(θ),(364) Sfker [n;θ] = f(0),HEG xc rs(θ), ζ(θ),(365) Sfqi[n;θ] = SHEGqi;rs(θ), ζ(θ), i = 1, . . . , Nq.(366) The probes fcomp, fker, fqi are chosen so that their evaluation on a KS-like generator L [ n ; θ ]reproduces the corresponding response or structure factor features. In a practical SpectralXC implementation these would typically be realised using resolvent-based trace formulas for the non-interacting propagator plus an approximate vertex. The set of equalities (363) – (366) can be regarded as a finite-dimensional projection of the full HEG universal object (362) . When these spectral laws hold exactly along a path in descriptor space, that path is an exact leg in the sense of Sec. III C: the spectral data that define the HEG universality class are transported without defect from one point to another. 82 Path in descriptor space and UP–RG interpretation. We now embed HEG tiles into the broader descriptor manifold Θused in the SpectralXC construction. For HEG-based XC constraints a natural choice is Θ = n(rs, ζ, α, Q, . . .)o,(367) where rs, ζ parametrize a reference uniform density and spin polarisation, α controls the amplitude of an inhomogeneous modulation, and Q its wavevector. A simple example is an external potential of the form vext(r;α, Q) = αcos(Q·r),(368) whose amplitude α interpolates between the uniform HEG ( α = 0) and a weakly modulated electron gas ( |α|  1). For small |α| and small |Q| this system remains in the HEG universality class in the usual sense of RG: its long-wavelength behaviour is that of HEG, up to perturbative corrections. From the UP–RG viewpoint, we consider a family of categories {Cµ} indexed by a coarse-graining scale µ (e.g. a momentum cutoff or length scale) and a family of functors Fµ→µ0 : Cµ→Cµ0 implementing spectral coarse-graining. At each ( µ, θ ) ∈R+× Θwe attach a universal object Uµ,θ [ n ]. For θ in the HEG submanifold α = 0 and n ( r ) = n0 uniform, Uµ,θ reduces to the HEG universal UHEG µHEG,θHEG . Along a path γ: [0,1] →Θ, γ(t)=(rs, ζ, α(t),Q, . . .),(369) starting at α (0) = 0 and ending at some finite α (1), the universal object evolves according to the combined action of the external modulation and the RG flow. When the central 2-curvature introduced in Sec. IV D vanishes along γ , the HEG spectral laws are transported exactly: the path is an exact leg from HEG to the inhomogeneous state. When curvature appears, HEG universality is broken and the spectral laws must be relaxed. Spectral defect functional for HEG-based constraints. To quantify the degree to which an inhomogeneous system at (µ, θ)deviates from the HEG tile we introduce a HEG-specific spectral defect functional, DefectHEG µ(θ) = X k wkSfk[n;θ]−φHEG k(rs(θ), ζ(θ)) 2,(370) where the fk and φHEG k correspond to the compressibility, kernel, and structure factor constraints discussed above, and wk are positive weights chosen to normalise the contributions. In the language of Sec. IV C this is a specialisation of the generic quadratic defect functional to HEG tiles. The UP–RG monotonicity requirement asserts that for an admissible class of coarse-graining functors Fµ→µ0, DefectHEG µ0(θ0)≤DefectHEG µ(θ), θ0=Fµ→µ0(θ),(371) i.e. the HEG spectral defect cannot increase under RG flow. When DefectHEG µ(θ)≈0and its derivative along γ is small, the system is effectively HEG-like at large scales; when the defect grows or oscillates, HEG-based constraints (and hence LDA-like XC approximations) are being pushed beyond their regime of validity. Curvature and relaxation of HEG constraints. The loop and adjoint curvatures introduced in Sec. IV D provide a finer diagnostic of when HEG constraints must be relaxed. Consider, for example, two different paths γ1, γ2 in Θconnecting the same initial HEG tile θHEG = ( rs, ζ, α = 0 ,Q, . . . )to the same final inhomogeneous descriptor θinh . If the spectral universal objects obtained by transporting UHEG µHEG,θHEG along γ1 and γ2 differ by an amount measured by the loop curvature Kγ1,γ2 , then HEG universality is path-dependent; the HEG tile alone no longer suffices to characterise the IR behaviour, and additional tiles (e.g. SCE or 1D tiles) must be incorporated. Physically, this occurs when inhomogeneity drives the system into regimes dominated by, e.g., strong localisation or quasi-one-dimensional fluctuations. Mathematically, the central part of the curvature ( Fcent, Gcent )introduced in Eq. (232) measures the failure of the HEG spectral laws (363) – (366) to remain invariant around small loops in Θ. When these central curvatures vanish, HEG constraints are integrable and define exact legs; when they are nonzero, the HEG tile can at best provide a local approximation, and curvature corrections must be incorporated into the XC functional. In the language of practical DFT, this is the structural origin of gradient corrections and nonlocal kernels beyond LDA: they encode precisely the curvature-induced deviations from HEG universality. 83 Numerical experiment: HEG defect under weak inhomogeneity. To demonstrate these ideas in a concrete setting we propose a simple numerical experiment. Consider a spin-unpolarised ( ζ = 0) HEG at density corresponding to rs≈ 2in a cubic supercell with periodic boundary conditions. We introduce a weak sinusoidal modulation in one direction, vext(z) = αcos(Qz), Q =2π L,(372) and treat the resulting inhomogeneous system within a KS-like framework. For definiteness we discretise space on a uniform grid and represent the KS generator L [ n ; θ ]as a finite-difference approximation of the kinetic energy plus an effective single-particle potential, including Hartree and a simple XC term (e.g. LDA). The descriptor θ = ( rs, ζ, α, Q )is varied by tuning the modulation amplitude α from 0to a value where the density deviation δn(z)is no longer small. For each α we evaluate the local density n ( z ), the corresponding KS generator L [ n ; θ ], and the spectral probes Sfk [ n ; θ ]corresponding to compressibility, smallq kernel, and a few structure factor points S ( qi ) in the smallq regime. The HEG targets φHEG k ( rs, ζ = 0) for the underlying uniform density are taken from a QMC parametrisation of the HEG energy[ 132 ] and response functions.[ 85 ] We then compute the HEG spectral defect DefectHEG µ ( θ )of Eq. (370) for each α , and track its behaviour as a function of the modulation strength and coarse-graining scale µ (e.g. implemented via momentum cutoffs in the response). In the weakly inhomogeneous regime |α|  1the density modulation is small, the HEG-based constraints should be nearly satisfied, and DefectHEG µ ( θ )should remain small and monotone under coarse-graining. As |α| increases, the density develops sharper features; the local environment begins to resemble a quasi-Wigner crystal or quasi-one-dimensional wire in certain regions. In this regime we expect the HEG defect to grow, reflecting the breakdown of HEG universality, and the curvature to become non-negligible. The scale at which DefectHEG µ ( θ )exceeds a chosen threshold can then be used as a quantitative indicator of where HEG-based XC constraints must be relaxed by curvature corrections or supplemented by additional tiles (e.g. SCE or 1D tiles), in direct analogy with the discussion of Sec. VI B 1 for the finite 2 d+ 2 p cluster. From a practical DFT perspective, this numerical experiment translates into a systematic, UP–RGbased way of answering the question “how inhomogeneous is too inhomogeneous for HEG-based XC?”. Rather than relying on heuristic notions of slowly varying density, we measure the failure of HEG spectral laws directly through the defect (370) and its curvature. In subsequent subsections we will show how SCE and 1D tiles provide complementary universals for strongly inhomogeneous and quasi-one-dimensional regimes, and how UP–SpectralXC blends these tiles into a consistent multiscale XC description. B. SCE tile: strong-correlation asymptotics The homogeneous electron gas (HEG) tile discussed in Subsec. VII A captures the physics of a moderately correlated Fermi liquid at metallic densities. In the opposite limit of very low density (large rs ) or very strong electron–electron repulsion, the physics is qualitatively different: the Coulomb repulsion dominates the kinetic energy, electrons avoid each other in a highly correlated manner, and the many-body wavefunction develops strongly non-Gaussian structure. In this regime the relevant limiting object is not the HEG but the strictly correlated electron (SCE) system, i.e. the strong-interaction limit of the Hohenberg–Kohn functional.[ 89 , 90 ] In this subsection we treat the SCE regime as a second canonical tile in UP–SpectralXC: we build its universal object, outline its optimal–transport (OT) structure, identify spectral probes and laws, and discuss how SpectralXC + UP–RG interpolates between HEG and SCE tiles along density and geometry paths such as bond stretching or dilution of electron gases. SCE functional and optimal transport. For a given one-particle density n ( r )with Rn ( r ) d3r = N , the adiabatic connection formula expresses the universal Hohenberg–Kohn functional in terms of a coupling constant λ multiplying the electron–electron interaction.[ 89 ] In the strong-interaction limit λ→ ∞ (or, equivalently, rs→ ∞ at fixed shape of n ) one obtains the SCE functional VSCE ee [ n ], defined as the minimum possible Coulomb repulsion compatible with the density n: VSCE ee [n] := inf γ∈ΓnZγ(r1,...,rN)X 1≤i<j≤N 1 |ri−rj|d3r1···d3rN.(373) Here γ ( r1,...,rN )is an N -body probability density with one-body density n ( r ), and Γ n is the set of such γ . Equation (373) is a multi-marginal optimal transport problem with Coulomb cost.[ 91 , 92 ] Intuitively, 84 the SCE functional describes the minimal cost of distributing N classical charges with positions ri such that the resulting marginal density equals n(r). Under suitable regularity conditions and mild assumptions, the SCE problem admits a “Monge” solution in terms of co-motion functions.[90] One can write the SCE N-body density as γSCE(r1,...,rN) = 1 N!X πZd3rn(r) N N Y k=1 δrk−fπ(k)(r),(374) where the sum runs over all permutations π of { 1 , . . . , N} , and fi ( r ), i = 1 , . . . , N , are the SCE co-motion functions, with the convention f1 ( r ) = r . The fi map the position of a reference electron at r to the positions of the remaining electrons in an “optimal” way: given that one electron sits at r , the others must sit at fi ( r )to minimise the repulsion while preserving the density. They satisfy the density-preserving conditions n(r)d3r=n(fi(r)) d3fi(r), i = 1, . . . , N, (375) and nontrivial group-like relations encoding the indistinguishability of electrons.[90] In terms of the co-motion functions, the SCE functional is simply VSCE ee [n] = 1 2Zd3rn(r) NX 1≤i6=j≤N 1 |fi(r)−fj(r)|.(376) The SCE limit of the adiabatic connection provides the leading strong-coupling term W∞ [ n ]in the expansion of the coupling-constant integrand, and its first correction W0 ∞ [ n ]can be interpreted as the zero-point energy of small oscillations around the SCE configuration.[89, 93] SCE universal object and spectral probes. From the UP–SpectralXC point of view, the SCE tile is not described only by the scalar functional VSCE ee [ n ], but by a richer universal object that captures the OT geometry of the co-motion manifold and the spectrum of small fluctuations around it. For a given density n(r)we define the SCE tile universal at a strong-correlation scale µSCE as USCE µSCE,θ := VSCE ee [nθ], W0 ∞[nθ],{`α[nθ]}α∈A,{Λβ[nθ]}β∈B,(377) where nθdenotes the density for descriptor θ, and: •VSCE ee [nθ]is the leading SCE interaction energy (376). •W0 ∞ [ nθ ]is the subleading zero-point oscillation term obtained by expanding the adiabatic connection around the SCE manifold.[89, 93] •`α [ nθ ]are geometric OT probes, such as the distribution of pair distances |fi ( r ) −fj ( r ) | , or moments of the associated pair density. For example, `(p) ij [nθ] := Zd3rnθ(r) Nfi(r)−fj(r)p, p =−1,1,2,..., (378) encode the typical separation of electrons i and j in the SCE configuration. The p = − 1case reproduces the contribution of the (i, j)pair to VSCE ee . • Λ β [ nθ ]are spectral probes of small oscillations around the SCE manifold: they are the eigenvalues of the Hessian of the Coulomb cost constrained to the co-motion manifold, and correspond to “vibrational” modes of the strictly correlated configuration.[93] The collection of `α and Λ β captures how rigid or floppy the SCE arrangement is, i.e. how strongly the electrons are locked into a classical configuration versus how much quantum delocalisation remains in subleading orders. In the spectral language of Sec. III B, these quantities are obtained from the spectrum of an effective “internal” Hamiltonian describing motion relative to the SCE co-motion locus.[ 93 ] The moments `(p) ij [ n ]are spectral moments of the pair separation operator along this manifold, and the Λ β [ n ]are its normal-mode frequencies. Together they form the SCE spectral probes that enter our universal object. 85 SCE spectral laws and strong-coupling asymptotics. The SCE functional describes the asymptotic behaviour of the exact coupling-constant integrand as λ→ ∞: Wλ[n] = W∞[n] + W0 ∞[n] √λ+Oλ−3/2, W∞[n] = VSCE ee [n]−U[n],(379) where U [ n ]is the Hartree energy.[ 89 ] In the SpectralXC language we view W∞ [ n ], W0 ∞ [ n ]and selected `α[n],Λβ[n]as defining a set of SCE spectral laws, Sgk[n;θ] = φSCE k[nθ],(380) where gk are appropriate probe functionals (e.g. resolvent-based traces of an effective strong-coupling generator), and φSCE k [ nθ ]are target values obtained either analytically in model geometries or from dedicated SCE calculations for representative densities.[90, 93, 94] Heuristically, the SCE spectral laws enforce the following physics: • At very low densities or strong coupling, the electron–electron repulsion dominates and the electrons arrange in an almost classical configuration minimising the Coulomb energy subject to the density constraint. •Quantum fluctuations around this configuration are small and captured by a set of normal modes whose frequencies Λβ[n]scale as λ−1/2. • The internal distances and pair correlations between electrons, encoded in the `(p) ij [ n ], are fixed primarily by the geometry of the density nrather than by the kinetic energy. In UP–SpectralXC we impose SCE spectral laws as asymptotic constraints in the low-density regime, in direct analogy with HEG-based constraints in the high-density regime. Descriptor manifold and paths between HEG and SCE tiles. To interpolate between the HEG and SCE tiles we enlarge the descriptor manifold Θto include both density parameters and geometric deformations. For example, a simple choice is Θ = n(rs, ζ, λ, R, . . .)o,(381) where rs, ζ describe an underlying average density, λ is a coupling constant scaling the electron–electron interaction, and R is a bond length or interparticle separation (for molecular systems). In this space there are two natural families of paths: 1. Density paths: vary rs at fixed geometry (e.g. in a low-dimensional model of an electron gas), moving from the high-density HEG regime (rs1) to the low-density SCE regime (rs1). 2. Geometry paths: vary a bond length R at fixed particle number and total charge, as in a stretched diatomic or hydrogen chain, moving from a compact, moderately correlated geometry to a dissociated, strongly correlated geometry where SCE physics dominates.[93] In each case we consider a path γ: [0,1] →Θ, γ(0) = (rHEG s, ζ, λ = 1, R0, . . .), γ(1) = (rSCE s, ζ, λ 1, R1, . . .),(382) that connects a HEG-like point to an SCE-dominated point. Along this path the UP–SpectralXC machinery transports the universal objects, with HEG spectral laws approximately satisfied near t = 0 and SCE laws near t = 1. Exact legs in the sense of Sec. III C correspond to regimes where either HEG or SCE constraints remain valid over a range of t ; curvature signals crossovers between universality classes. Spectral defects and SCE–HEG interpolation. We introduce SCE-specific spectral defects analogous to the HEG defect (370) . Given a set of SCE probes gk and targets φSCE k [ nθ ]we define, at scale µSCE and descriptor θ, DefectSCE µSCE (θ) = X k ˜wkSgk[n;θ]−φSCE k[nθ] 2,(383) with weights ˜wk chosen to normalise the contributions. In strongly correlated regions (large rs , large R , large λ ) one expects DefectSCE µSCE to be small and monotone under coarse-graining, whereas the HEG 86 defect DefectHEG µ of Eq. (370) is large. In moderately correlated regions both defects are nonzero but of comparable magnitude; these are precisely the regimes where interpolating SpectralXC functionals benefit from imposing both HEG and SCE constraints simultaneously. In addition, one can define mixed defects that compare a given system not only to SCE or HEG tiles individually but to an interpolating combination of their universals, Umix µ,θ =α(θ)UHEG µ,θHEG(θ)+1−α(θ)USCE µ,θSCE(θ),(384) and minimise the distance between Uµ,θ and Umix µ,θ with respect to an interpolation parameter α ( θ ) ∈ [0 , 1]. In the simplest case α depends only on a local measure of density or inhomogeneity (e.g. a reduced gradient), providing a structural justification for interpolated functionals that mix HEG and strongcoupling limits.[94] Numerical experiment: bond stretching and low-density gases. To put these ideas to work we now sketch a numerical experiment that can be carried out within the SpectralXC framework. Consider a one-dimensional model of a diatomic molecule, such as a soft-Coulomb H2, with Hamiltonian ˆ H(R) = 2 X i=1 −1 2 d2 dx2 i− 2 X A=1 ZA p(xi−RA)2+a2!+1 p(x1−x2)2+a2+Z1Z2 p(R1−R2)2+a2,(385) where R1 = −R/ 2, R2 = + R/ 2and a is a softening parameter. For small R the system is moderately correlated; for large R it approaches a strongly correlated, almost dissociated limit where SCE physics is dominant.[93] Within SpectralXC, one constructs a KS-like generator L [ n ; θ ]for each bond length R and coupling constant λ , and selects spectral probes both of HEG-type (e.g. compressibility of the effective uniform reference gas at the average density) and SCE-type (e.g. moments of the co-motion distance between electrons and low-lying normal-mode frequencies of the internal Hamiltonian). Along the path γ ( t ) = ( rs ( R ( t )) , λ ( t ) , R ( t ) , . . . ), with rs obtained from the average density and λ ramped up, one evaluates DefectHEG µ and DefectSCE µSCE as functions of R and λ . In the compact geometry (small R ) one expects HEG-like constraints to be reasonably accurate (small HEG defect), with SCE defects relatively large; in the stretched regime (large R ) the roles should reverse, with SCE constraints nearly saturated and HEG defects large. The onset of static correlation and multi-reference behaviour is then identified not by an ad-hoc diagnostic but by the crossing of these structural defect measures. Similarly, in a low-density electron gas confined in a finite box or harmonic trap, one can vary the density parameter rs at fixed geometry, calculate SpectralXC universals along the path in Θ, and monitor how Uµ,θ moves from the HEG tile towards the SCE tile. In both cases, the curvature associated with transporting UHEG and USCE along different paths in Θprovides a quantitative measure of whether the system lies in a regime where HEG, SCE, or an intermediate tile gives the most robust description. From the perspective of practical DFT, the SCE tile thus plays the role of a strong-correlation anchor for SpectralXC-based XC functionals. It supplies exact asymptotic information on bond stretching, low-density electron gases, and other strongly correlated limits; UP–RG then tells us how to deform these asymptotics along paths in Θ, and SpectralXC uses spectral probes and defects to decide when and how to interpolate between HEG and SCE constraints. This elevates the role of SCE from a collection of limiting formulas to a fully fledged universal object in the category of SpectralXC tiles. C. 1D TLL tile: non-Fermi-liquid behaviour The HEG and SCE tiles introduced in Subsec. VII A and Subsec. VII B capture, respectively, the physics of a Fermi liquid at metallic densities and the strong-correlation limit in which electrons form a quasi-classical Coulomb arrangement. In one spatial dimension, however, even weakly interacting fermions generically fail to exhibit Fermi-liquid behaviour: instead, their low-energy physics is that of a Tomonaga– Luttinger liquid (TLL).[ 95 – 98 ] In a TLL, single-particle excitations are not well-defined quasiparticles, the momentum distribution lacks a Fermi step, and correlation functions decay as nontrivial power laws determined by interaction-dependent Luttinger parameters. Many quasi-one-dimensional (quasi-1D) materials and molecules—organic conductors, spin chains, nanowires, and conjugated polymers—are well described at low energies by TLL theory.[99] In the UP–SpectralXC language, a TLL provides a third canonical tile: a universal fixed point describing the low-energy sector of a broad class of 1D and quasi-1D systems. In this subsection we treat TLLs as tiles UTLL µTLL,θ , define appropriate spectral probes and laws (exponents Kρ , Kσ , and power-law tails of 87 the momentum distribution), and describe how these may be used as constraints for quasi-1D molecules or materials. We conclude with a numerical-experiment blueprint in which TLL-based SpectralXC predictions are compared to DMRG or exact diagonalisation (ED) data for 1D systems. Tomonaga–Luttinger Hamiltonian and bosonisation. For concreteness we begin with a spinful 1D fermion model, such as a Hubbard chain or a generic continuum model with short-range interactions.[ 98 , 100 ] At low energies and long wavelengths, the microscopic Hamiltonian ˆ H is mapped via bosonisation to an effective TLL Hamiltonian for charge and spin bosonic fields (φν, θν)with ν=ρ, σ: ˆ HTLL =X ν=ρ,σ vν 2πZdx Kν∂xθν(x)2+1 Kν∂xφν(x)2,(386) where vν are mode velocities and Kν are Luttinger parameters. The fields φν encode long-wavelength density fluctuations and θνare their conjugate phases.[98] For a generic 1D fermion system with SU(2)-symmetric spin interactions, the spin sector is gapless with Kσ = 1 and vσ determined by the underlying model; the charge sector has a Luttinger parameter Kρ and velocity vρ that depend on interactions and density. Repulsive interactions correspond to Kρ< 1; attractive interactions give Kρ>1.[99] The Hamiltonian (386) is quadratic in bosonic fields and thus exactly solvable. It is a stable RG fixed point: under coarse-graining, a wide variety of microscopic 1D models flow towards this form, with only the values of Kνand vνencoding model-specific details.[97] This is the sense in which TLL theory provides a “universal tile” for 1D physics. TLL correlation exponents and spectral properties. The key feature of a TLL is that all correlation functions exhibit power-law behaviour governed by Kν . For example, the equal-time single-particle Green’s function for right-moving fermions at large separation |x| → ∞ behaves as[98] GR(x) := hψ† R(x)ψR(0)i ∼ eikFx |x|η1, η1=η1(Kρ, Kσ),(387) with an exponent η1 determined by Kρ (and Kσ in the spinful case). The momentum distribution n ( k ), n(k) = hc† kcki=Zdx e−ikxhψ†(x)ψ(0)i,(388) has no discontinuity at kF ; instead, it exhibits a cusp with an interaction-dependent power-law singularity, n(k)∼n(kF) + Csgn(k−kF)|k−kF|α(Kρ), k →kF,(389) where α ( Kρ )is a known function of Kρ obtained from bosonisation.[ 98 , 99 ] For a non-interacting Fermi gas Kρ = 1, α (1) = 0, and a step discontinuity is recovered; for Kρ6 = 1, α ( Kρ ) > 0and the step is replaced by a cusp. Other TLL observables also show universal scaling: the density–density correlation function has powerlaw components at 2 kF and 4 kF with exponents that are simple functions of Kρ , and the local density of states exhibits a suppression (“zero-bias anomaly”) near the Fermi energy with exponent controlled by Kρ .[ 98 , 99 ] All these exponents are expressible in terms of Kρ and Kσ ; the velocities vν set the characteristic scales of dynamical response. TLL universal object and spectral probes. In UP–SpectralXC we regard the TLL as a tile at a lowenergy scale µTLL and define a TLL universal object for a homogeneous 1D system of density n and interaction parameters θTLL as UTLL µTLL,θTLL := Kρ, vρ, Kσ, vσ, αsp,{ηm}m∈M.(390) Here αsp is the exponent controlling the single-particle momentum-distribution tail (389) , and ηm denotes a finite set of correlation-function exponents for density, spin, and pairing correlations. In principle all ηm are functions of ( Kρ, Kσ ), but we treat them as separate components of the universal object for numerical purposes. The Luttinger parameters can be extracted from spectral probes that are directly accessible in microscopic models or DMRG:[98, 100] • The compressibility κ and Drude weight D determine Kρ and vρ . For spinless fermions, for example, one has K=π 2√κD, v =D πK ,(391) 88 where κ is the second derivative of the ground-state energy with respect to particle number, and D is the second derivative with respect to a twist in boundary conditions. Analogous relations hold for the charge sector of spinful systems.[98] • The exponent αsp can be obtained from the scaling of n ( k )near ±kF , or from the finite-size scaling of the single-particle Green’s function or local density of states. In practice, one defines a probe that samples the spectral function A(k, ω)near (kF,0) via a resolvent integral of the form Sfsp = Tr fsp(L) = 1 2πi IΓ dz fsp(z) Tr (z−L)−1,(392) where L is a KS-like generator and Γis a contour encircling the relevant spectral region. The scaling of Sfsp with system size and momentum window can be used to extract αsp. • Other exponents ηm are obtained from similar resolvent or correlation-function probes for density, spin, and pairing operators.[98] Thus, in the SpectralXC picture the TLL tile universal object consists of the set of Luttinger parameters and exponents that characterise the low-energy fixed point. TLL spectral laws and projections onto 1D-like regions. To use TLL tiles in an inhomogeneous setting we embed them into the descriptor manifold Θas in the HEG and SCE cases. For quasi-1D molecules or materials we may consider descriptors of the form Θ = n(nk, n⊥, gk, g⊥, λ, . . .)o,(393) where nk is the density along the chain direction, n⊥ encodes transverse confinement, gk and g⊥ measure longitudinal and transverse interactions, and λ is a projection parameter controlling how strongly the system is one-dimensional. In regions of space where n⊥ is small and confinement is strong, the local physics is effectively 1D and should lie near a TLL tile; in bulk-like regions it is better described by HEG-like tiles. As in the HEG and SCE cases, we impose a finite set of spectral laws Shk[n;θ] = φTLL k(θTLL),(394) where the right-hand side are the TLL target exponents ( Kρ, Kσ, vρ, vσ, αsp, ηm )for model parameters θTLL , obtained e.g. from analytic bosonisation or Bethe-Ansatz solutions of representative 1D models at the local density.[ 97 , 100 ] The left-hand side Shk [ n ; θ ]are resolvent-based probes of the type (392) evaluated on local projections of the KS-like generator L[n;θ]onto the 1D-like subspace. For example, in a quasi-1D nanowire embedded in three dimensions, one can define a projection onto transverse modes that are tightly confined, construct an effective 1D KS generator L1D for those modes, and apply the TLL probes hk to L1D . The resulting spectral quantities Shk [ n ; θ ]are then constrained to match the TLL target values φTLL kup to an allowed defect. TLL spectral defect and UP–RG interpretation. As in the HEG and SCE cases, we quantify the mismatch between a given system and the TLL tile via a spectral defect functional DefectTLL µTLL (θ) = X k ˆwkShk[n;θ]−φTLL k(θTLL(θ)) 2,(395) with weights ˆwk chosen to normalise each probe. In a quasi-1D system one expects DefectTLL µTLL ( θ )to be small in regions where confinement and interactions are such that the low-energy physics is TLL-like, and large where transverse modes are important or dimensional crossover occurs. From the UP–RG perspective, the TLL tile sits at a low-energy scale µTLL as a non-Fermi-liquid fixed point. Under coarse-graining, microscopic models with 1D character flow towards UTLL µTLL,θTLL ; the defect (395) is a c-type monotone that measures how far a given system sits from this fixed point. The loop curvature in TLL-based universals then diagnoses whether different embedding or downfolding schemes for quasi-1D systems yield consistent TLL physics: if the curvature in the space of schemes is small, the TLL description is robust; if it is large, the system may be near a dimensional crossover or in a regime where strictly 1D TLL theory is insufficient. 89 Application to quasi-1D molecules and materials. Quasi-1D molecules and materials, such as conjugated polymer chains, ladder compounds, or nanowires, provide natural arenas for TLL tiles. In such systems, the electronic structure at low energies is dominated by motion along one spatial direction, with transverse degrees of freedom gapped or weakly populated.[ 99 ] Standard 3D XC functionals often misrepresent these systems because they implicitly assume Fermi-liquid-like quasiparticles; they may underestimate correlation effects, fail to capture fractionalisation, or mispredict the scaling of spectral gaps with system size. Within UP–SpectralXC, one can proceed as follows: 1. Identify a quasi-1D subspace of the full descriptor manifold Θwhere the system behaves approximately as a 1D chain: for example, by projecting onto a set of Wannier orbitals localised along a particular direction. 2. For this subspace, build a KS-like generator L1D [ n ; θ ]and define TLL spectral probes hk (compressibility, Drude weight, single-particle exponents etc.) in terms of resolvents of L1D. 3. Use known TLL universals φTLL k for an appropriate reference model (Hubbard, extended Hubbard, etc.) at the same density to define spectral laws (394). 4. Impose these laws in a constrained Levy–Lieb framework (as in Sec. 4.1), resulting in Lagrange multipliers that generate a SpectralXC correction designed to enforce TLL exponents in the quasi-1D subspace. This procedure generalises the traditional strategy of fitting effective 1D models to ab initio data: instead of fitting coupling constants, we match spectral universals. Numerical experiment: comparison to DMRG and exact diagonalisation. As a concrete numerical test of a TLL tile in SpectralXC, consider the 1D Hubbard model, ˆ HHubbard =−tX i,σ c† iσci+1,σ + h.c.+UX i ˆni↑ˆni↓,(396) on a chain of length L with open or periodic boundary conditions. Away from half filling and for U > 0, this model is a paradigmatic realisation of a spinful TLL at low energies, with Kρ ( U, n )and vρ ( U, n ) known from Bethe–Ansatz and finite-size scaling.[98, 100] Step 1: Extract TLL universals from DMRG or ED. For a given U/t and filling n = N/L , one uses DMRG[101, 102] or high-precision ED to compute: • The ground-state energy as a function of particle number N and boundary twist Φto extract the compressibility κand Drude weight D. • The momentum distribution n ( k )near ±kF and correlation functions at large distances to obtain exponents αsp and ηmvia power-law fits. From κ and D one extracts Kρ and vρ using relations of the form (391) ; from n ( k )and correlators one obtains αsp and ηm . Together these form a numerical realisation of the TLL universal object UTLL µTLL,θTLL for the Hubbard model at the chosen (U, n). Step 2: Build a SpectralXC KS-like generator. In parallel, one constructs a KS-like generator L [ n ; θ ] for the same 1D system within a chosen DFT or SpectralXC approximation. This may be a lattice KS Hamiltonian or a continuum KS operator with an effective vxc ( x ). For each ( U, n )one solves the KS equation, obtains the density n ( x ), and computes the spectral probes Shk [ n ; θ ](compressibility, Drude weight, exponents) from L[n;θ]using resolvent formulas. Step 3: Spectral defects and comparison. One then evaluates the TLL spectral defect DefectTLL µTLL ( θ ) in the form (395) by comparing the KS-based probes to the DMRG extracted TLL targets. A naive functional (e.g. lattice LDA) is expected to produce significant defects (incorrect Kρ , αsp , etc.), whereas a SpectralXC functional constructed to match TLL exponents should yield a much smaller defect. Step 4: UP–RG and scheme dependence. Finally, one considers different coarse-graining or embedding schemes: for example, different choices of active region, different ways of coupling the 1D chain to higher-dimensional environments, or different lattice discretisations. For each scheme one extracts the TLL universals and computes loop curvatures as in Sec. 3.4, thereby quantifying the scheme dependence of the TLL tile in concrete simulations. Such numerical tests would show that TLL-based SpectralXC corrections can enforce non-Fermi-liquid exponents in 1D models and that the corresponding spectral defects behave as genuine monotones under 96 • Curvature-based c-functions Ccurv µ explicitly quantify scheme dependence: large loop curvatures reveal that different sequences of RG steps (e.g. downfolding then embedding vs. embedding then downfolding) lead to visibly different universals. In such cases, naive scalar c-functions can differ qualitatively between schemes, while the curvature provides an invariant indicator of non-integrability. Formally, suppose we have two RG schemes A and B inducing maps E(A) µ→µ0 and E(B) µ→µ0 on density operators. Even if the induced flows of some scalar observables o(A) µ , o(B) µ differ or oscillate, the relativeentropy c-functions satisfy Crel,(A) µ0(Uµ0,θ0) = SE(A) µ→µ0(ρµ,θ)kE(A) µ→µ0(ρ(T) µT,θT)≤Crel,(A) µ(Uµ,θ),(426) and similarly for scheme B . Thus, in each scheme separately, the structural monotone is non-increasing, even though the coordinate representation of the flow differs. The loop curvature compares these schemespecific c-functions and universals around closed paths and captures the intrinsic scheme dependence. Structural partial order and robustness of XC c-functions. The failures of naive scalar c -like quantities described above do not contradict the existence of structural monotones; rather, they highlight the need to formulate monotonicity at the level of the universal object and the underlying state, not at the level of a specific coordinate. In UP–SpectralXC we define a partial order on universals at fixed tile Tand scale µby Uµ,θ1Uµ,θ2⇐⇒ Cµ(Uµ,θ1)≥Cµ(Uµ,θ2),(427) where Cµ is any of the XC c-functions discussed in Subsec. VIII A. RG flows generated by admissible maps then always move “downward” in this partial order: Uµ,θ <Uµ0,θ0whenever (µ0, θ0) = Fµ→µ0(µ, θ).(428) Scalar projections cnaive µ = f ( Uµ,θ )may or may not be monotone with respect to this order; but Cµ itself is, by construction. From the standpoint of XC modelling, this means that even when standard heuristic indicators of correlation strength or “number of effective degrees of freedom” oscillate or are strongly scheme-dependent, we can still rely on structural defect functionals to tell us, in a coordinate-free way, whether we are moving closer to or further from a given tile (HEG, SCE, TLL, etc.). This robustness is a major advantage of UP–SpectralXC over ad-hoc scalar diagnostics: it turns the slogan “universality classes” into a genuine partial order on spectral universals, with well-defined monotones. In the subsequent subsections we will exploit this structural picture to construct explicit XC c-functions for realistic test cases (HEG to SCE interpolation, 1D TLL crossovers, and multiscale cluster embeddings), and to show how they can guide the design and assessment of SpectralXC functionals in situations where naive c-like scalars are misleading. C. Numerical illustration In Subsec. VIII A we introduced XC c -functions Cµ ( Uµ,θ )as defect functionals on the universal object Uµ,θ , and in Subsec. VIII B we explained why naive scalar “ c ”-like quantities often fail to be monotone in realistic non-relativistic and multiscale settings. We now outline concrete numerical protocols showing how these structural monotones behave in practice, using toy models that can be treated by DMRG or exact diagonalisation. The emphasis is not on reproducing a particular dataset, but on making explicit how one would demonstrate, in a controlled numerical experiment, that: 1. naive scalar diagnostics of “correlation strength” or “distance to universality” are generically non-monotone along RG flows or descriptor paths, while 2. defect-based XC c -functions defined on the full universal Uµ,θ remain contractive, and hence provide a robust monotone under coarse-graining. We present two classes of examples: (1) a one-dimensional Hubbard chain with a scale parameter that implements resolution coarse-graining, and (2) a simple molecular system (e.g. stretched H 2 or a few-site Anderson–Hubbard cluster) where embedding RG steps map an extended system to an effective impurity. In both cases, the numerical procedures are compatible with the UP–SpectralXC formalism introduced earlier, and they can be implemented using standard many-body tools such as DMRG and exact diagonalisation.[114–116] 97 Example A: 1D Hubbard chain and a TLL tile. Consider again the one-dimensional Hubbard model defined in Eq. (396) on a chain of length L with periodic boundary conditions, at a fixed filling n = N/L away from half filling and with repulsive interactions U > 0. At low energies, this model is a spinful Tomonaga–Luttinger liquid (TLL) characterised by charge and spin Luttinger parameters Kρ , Kσ and velocities vρ,vσ; the corresponding TLL universal object UTLL µTLL,θTLL was introduced in Eq. (390). We now define a family of effective descriptions indexed by a scale parameter k , which plays the role of an infrared cutoff in a functional RG (fRG) or smooth momentum-shell scheme.[ 114 , 115 ] At each scale k we have an effective action Γ k or, in the SpectralXC language, an effective KS-like generator Lk whose propagator Gk incorporates fluctuations down to momentum scale k . Schematically, k→ Λcorresponds to the bare model and k→0to the full interacting fixed point. Naive scalar: effective coupling or gap proxy. A common practice in fRG is to track a small set of effective couplings gk (e.g. forward-scattering, Umklapp amplitudes) or a scalar proxy for the gap or quasiparticle weight, such as cnaive k:= ∂2E0(k) ∂N2,(429) where E0 ( k )is the ground-state energy of the effective Hamiltonian at scale k and N is the particle number. One might hope that cnaive k is monotone as k→ 0; in practice, however, numerical fRG flows often show oscillatory or non-monotonic behaviour of such quantities, especially when couplings become large or the truncation of Γkis severe.[114] To mimic this situation in a fully controlled way, one can: 1. Construct a sequence of exact reduced density matrices ρk by partially tracing out high-energy modes from the full Hubbard chain state (obtained e.g. by DMRG). One partitions the Hilbert space into “low-energy” and “high-energy” sectors using a smooth momentum cutoff, and defines ρk:= Tr|q|>kρfull,(430) where ρfull is the exact ground-state or thermal state. 2. For each k , extract E0 ( k )from ρk by computing the expectation value of an effective Hamiltonian restricted to the low-energy sector, and then form cnaive kusing finite differences in N. There is no guarantee that the sequence cnaive k is monotone; in fact, it is easy to construct situations where it oscillates due to level crossings, finite-size effects, or truncation. Structural XC c-function: TLL spectral defect. In parallel, we define a TLL-based XC cfunction as in Eq. (395) using a set of TLL probes hk: CTLL k:= DefectTLL µTLL (θk) = X j ˆwjShj[nk;θk]−φTLL j(θTLL) 2,(431) where: •nkis the one-particle density extracted from ρk, •Shj [ nk ; θk ]are spectral probes evaluated on the KS-like generator Lk constructed from ρk (e.g. compressibility, Drude weight, exponents extracted from the momentum distribution), •φTLL j ( θTLL )are the TLL target values for the chosen ( U, n ), obtained from either Bethe–Ansatz or independent DMRG finite-size scaling, •ˆwjnormalise the contributions. The map ρfull 7→ ρk is a partial trace and hence a CPTP map; the map from ρk to the KS-like generator Lk (with its spectrum and probes) can be chosen to be contractive in suitable operator norms. Therefore, the data-processing inequality for quantum relative entropy and the contractivity arguments of Subsec. VIII A guarantee that CTLL kis non-increasing along the flow: CTLL k0≤CTLL k, k0< k. (432) Numerically, one would concretely: 98 1. Compute ρkfor a sequence of scales kj. 2. Construct Lkj, evaluate Shj[nkj;θkj], and evaluate CTLL kj. 3. Compare the behaviour of cnaive kjand CTLL kjas functions of log kj. In typical flows, one would observe that cnaive k shows small oscillations or even non-monotonic jumps (due to level crossings or changes in the effective parametrisation), whereas CTLL k decreases smoothly, saturating to a small value when the system has reached the TLL tile. This illustrates the central claim: scalar c-like quantities built from a single observable may fail, while structural defect c-functions on Uµ,θ remain monotone. Example B: Molecular dissociation and SCE/HEG tiles. As a complementary example, consider a simple molecular system such as H 2 or a few-site Anderson–Hubbard “molecule” embedded in a continuum environment. The descriptor now includes a bond length R and an embedding scale µ (e.g. the size of the active region in a DMFT or quantum embedding calculation).[ 116 ] For each ( µ, R ), an embedding calculation produces an effective impurity model with a self-consistent bath and an associated impurity state ρµ,R. Naive scalar: ad hoc “correlation indicator”. In many embedding studies, one defines a scalar “correlation indicator” such as: •the double occupancy on a given site, •the difference between restricted and unrestricted HF energies, • the magnitude of an effective on-site Ueff extracted by fitting a Hubbard model to the low-energy spectrum. Denote such a quantity by cnaive µ,R := fad hoc(ρµ,R).(433) As R increases and µ is varied (e.g. as the active region is enlarged), cnaive µ,R typically shows complicated, non-monotonic behaviour: it may increase with R up to a point and then decrease, or differ markedly between embedding schemes A and B even when the underlying physical dissociation path is the same. This reflects the strong scheme dependence documented, for instance, in comparative studies of DMFT, cluster DMFT, and other embedding techniques.[116] Structural XC c-function: mixed HEG/SCE defect. In UP–SpectralXC, the same dissociation process is described by a path γ ( t ) = ( rs ( R ( t )) , λ ( R ( t )) , R ( t ) , . . . )in the descriptor manifold, connecting a moderately correlated, HEG-like regime at small R to a strongly correlated, SCE-like regime at large R (see Subsec. VII B). At each ( µ, R )we associate a universal Uµ,R and define an XC c-function with respect to an interpolating HEG/SCE tile as in Eq. (384): Cmix µ,R :=  WµUµ,R −Umix µ,R  2 2,(434) where Umix µ,R contains a weighted combination of HEG and SCE spectral laws (compressibility and smallq kernel from HEG, co-motion and strong-coupling asymptotics from SCE), and Wµ normalises the components. Embedding RG steps that enlarge the active region or improve the bath description can be modelled as CPTP maps on ρµ,R (partial traces/inclusions combined with self-consistent updates). As in the TLL example, contractivity of these maps implies that Cmix µ,R decreases as µ is lowered (more complete embedding), for any fixed R , provided the tile interpolation Umix µ,R is constructed consistently across scales. Similarly, along the descriptor flow in R at fixed µ , Cmix µ,R behaves as a structural measure of how far the system is from the optimal HEG/SCE mixture. Numerically, one can: 1. Perform embedding calculations for several active-region sizes µ1> µ2>··· and bond lengths R. 2. For each ( µi, R ), construct Uµi,R (e.g. from the impurity Green’s function and self-energy) and Umix µi,R. 99 3. Evaluate both cnaive µi,R and Cmix µi,R. One will typically find that: •cnaive µi,R is highly scheme dependent, differing between embedding schemes and showing non-monotone behaviour as µiis changed; •Cmix µi,R decreases monotonically with decreasing µi for each fixed R , and provides a smooth measure of the approach to the mixed HEG/SCE tile as Ris varied. This is entirely analogous to the 2 d+ 2 p cluster behaviour seen in Sec. VI B 1, but now tied directly to HEG and SCE tiles. Interpretation and practical implications. The two numerical protocols sketched above illustrate the central virtue of XC c-functions in UP–SpectralXC: 1. They are defined on the universal object Uµ,θ , not on a particular choice of couplings or ad hoc scalar diagnostics. 2. Their monotonicity is tied to structural properties of the RG maps (contractivity, complete positivity), not to the details of a given truncation scheme. 3. They provide a partial order on universals that is robust under changes of scheme: different embeddings or coarse-grainings may produce different coordinate flows, but they cannot increase the defect relative to a given tile. In practice, this means that when designing and testing XC functionals within SpectralXC, one should: •track structural c-functions like CTLL µor Cmix µ,R alongside traditional scalar diagnostics, • diagnose failures of XC approximations by increases in these c-functions under operations that ought to be contractive (e.g. embedding, downfolding), • and use the behaviour of Cµ under descriptor flows to identify regimes where HEG, SCE, TLL, or other tiles provide the dominant universal anchor. While the examples presented here are toy models, the same logic extends to realistic ab initio systems: universals can be built from KS spectra, many-body Green’s functions, or DMFT self-energies, and structural XC c-functions can be evaluated using the same numerical infrastructure already employed in modern electronic-structure codes. This provides a concrete path for bringing RG-inspired monotones into the practical assessment and design of XC functionals. IX. DISCUSSION: FROM SLOGAN TO STRUCTURE A. Universality as a universal property The traditional RG picture of universality begins with a space of couplings (“ β -space”), a flow generated by β -functions, and fixed points that attract trajectories.[ 117 , 118 ] Two microscopic models are said to belong to the same “universality class” if their RG flows end at the same fixed point, and if they share the same set of critical exponents and scaling functions. This view has been extraordinarily successful in classical and quantum critical phenomena, but it leaves several structural questions open: • What is the object that is universal? Is it the full fixed-point theory, a limited set of exponents, or some minimal invariant data? • How should we treat universality in situations where there is no canonical coupling-space description (e.g. strongly inhomogeneous systems, non-Fermi liquids, non-invertible RG transformations, or multiscale embedding flows)? • How can we distinguish robust universality from “accidental” cancellations that occur in a particular scheme or parametrisation? 100 In UP–SpectralXC we answer these questions by importing the category-theoretic notion of a universal property.[ 119 – 121 ] Instead of defining a universality class as a set of microscopic Hamiltonians that happen to share exponents, we define it as an equivalence class of universal spectral objects Uµ,θ that enjoy a universal property with respect to a diagram of KS-like generators and RG functors. Universality becomes literally a universal property in the categorical sense: it is characterised by the existence and uniqueness of certain morphisms to or from a distinguished object, not by chance cancellations of nonuniversal details. In this subsection we formalise this idea and explain how it unifies the different tiles (HEG, SCE, TLL, local Anderson–Hund clusters) introduced in Secs. VII–VIII into a single structural notion of XC universality. Categories of SpectralXC schemes and diagrams. Recall from Sec. IV that at each scale µ we consider a category Cµwhose objects are SpectralXC schemes S= (µ, θ, L, {fk},Φ),(435) where L is a KS-like generator, {fk} a set of spectral probes, and Φan Ad-invariant state (trace-like functional) on the operator algebra A=C∗(L). Morphisms in Cµinclude: •basis changes (unitary conjugations on L), •embeddings and downfoldings (algebra homomorphisms and partial traces), •gauge transformations in probe space (linear relations among {fk}), •descriptor flows (changes in θinduced by parameter maps). Between different scales µ we have RG functors Fµ→µ0 : Cµ→Cµ0 , which implement spectral coarsegraining: integrating out high-energy modes, reducing resolution, or mapping extended systems to effective clusters. A typical physical situation gives rise to a diagram D in the 2-category of such scheme-categories: it is a collection of objects and morphisms representing different descriptions of the same underlying physics, linked by RG and embedding maps. For example: • A HEG tile, its downfolded low-energy model, and several approximations (RPA, local approximations) linked by projection functors. • A2 d+ 2 p Anderson–Hund cluster and its KS, cluster A/B, and impurity embeddings linked by downfolding and embedding functors (Sec. VI B 1). • A quasi-1D chain and its TLL description, DMFT embedding, and effective Hubbard model linked by RG and projection maps (Sec. VII C). The diagram Dorganises all these schemes into a single categorical object. Universal spectral objects and universal properties. From each scheme S∈Cµ we extract a universal spectral object Uµ,θ(S), as in Eq. (398), Uµ,θ(S) = {Sfk[n;θ]}k,{K0-classes}, νµ,θ, . . .,(436) where Sfk [ n ; θ ]are the spectral quantities induced by ( L, {fk}, Φ), the K 0 data encodes projectors onto relevant subspaces (e.g. d -shell projectors), and νµ,θ is a spectral measure of L (e.g. the density of states weighted by Φ). This extraction defines a functor Uµ:Cµ−→ Univµ,(437) from SpectralXC schemes at scale µ to a category Univµ of universal objects and their morphisms (e.g. natural transformations between probe collections, measure-preserving maps between spectral measures). Auniversal property arises when there exists an object U∗ µ∈Univµ such that the elements of D factor uniquely through U∗ µ in a suitable sense. Concretely, consider a diagram D of universal objects and maps ϕα : Uµ,θα→Uµ,θβ , obtained from a diagram of schemes by applying Uµ . A terminal universal for D is an object U∗ µtogether with a family of morphisms τα:Uµ,θα−→ U∗ µ,(438) such that: 101 1. For every morphism ϕαβ :Uµ,θα→Uµ,θβin D, τβ◦ϕαβ =τα.(439) 2. For any other object V and maps σα : Uµ,θα→V satisfying σβ◦ϕαβ = σα for all α, β , there exists aunique morphism u:U∗ µ→Vsuch that σα=u◦τα∀α. (440) In other words, U∗ µis a terminal object in the category of cones over D.[119, 120] In the UP–SpectralXC context we interpret U∗ µ as the universal spectral object associated with a tile: every scheme in the diagram D maps to U∗ µ in a way compatible with the RG and embedding maps, and any other attempt to summarise D must factor through U∗ µ . This is the precise sense in which HEG, SCE, TLL, or a local Anderson–Hund cluster is a “universal tile”: it is not merely a fixed point, but a universal object with a terminal property in the appropriate diagram of SpectralXC schemes. Universality classes as equivalence classes of universal objects. Given the category Univµ and the extraction functor Uµ , we can now define an XC universality class as an equivalence class of universal spectral objects under natural isomorphisms. Two universals Uµ,θ and U0 µ,θ0 are said to be universally equivalent at scale µif there exists an isomorphism in Univµ, η:Uµ,θ ∼ −→ U0 µ,θ0,(441) such that for each scheme S∈Cµin the relevant diagram η◦Uµ(S) = U0 µ(S),(442) where U0 µ is the extraction functor in an alternative but equivalent representation (e.g. different basis, different probe set related by a linear transformation). Equivalence classes under this relation form the universality classes at scale µ, denoted [U]µ:= U0 µ,θ0U0 µ,θ0≃Uµ,θ.(443) This definition generalises the usual notion of universality class based on fixed points and critical exponents: • In a relativistic critical theory, Uµ,θ can be taken to consist of the set of scaling dimensions and OPE coefficients of primary operators. The universality class is the equivalence class under basis changes in operator space. • In HEG-based XC physics, UHEG µHEG,θHEG (Eq. (362) ) contains the energy density, compressibility, smallq kernel, and structure factor moments. HEG universality is the class of systems whose universals are isomorphic to that of HEG under maps that preserve these probes. • In SCE-based XC physics, USCE µSCE,θ (Eq. (377) ) contains co-motion geometry and strong-coupling asymptotics; SCE universality groups systems whose universals share these features up to natural isomorphism. • In TLL-based XC physics, UTLL µTLL,θTLL (Eq. (390) ) contains Luttinger parameters and exponents; the TLL universality class collects all 1D systems whose long-wavelength spectral objects are isomorphic to this tile. Importantly, this definition does not depend on any particular choice of “couplings” or local functional form of β -functions; it depends only on the existence of isomorphisms in Univµ and on the universal property of U∗ µin the diagram of schemes. Universality preserved by functors, not by chance. A central advantage of the categorical viewpoint is that it shifts the burden of “universality” from accidental cancellations in specific parametrisations to structural properties of functors. In traditional RG applications one often argues that certain nonuniversal, microscopic details “drop out” as higher-order irrelevant couplings; but this is sometimes difficult to make precise, and the separation between universal and nonuniversal data can be scheme dependent.[ 117 , 118 ] In the UP–SpectralXC framework, universality is preserved whenever the relevant functors (resolution coarse-graining, embedding RG, descriptor flows) preserve the universal property of U∗ µ. Concretely: 102 • Resolution coarse-graining Fµ→µ0 induces a functor Uµ→µ0 : Univµ→Univµ0 that maps universal objects to universal objects and preserves terminal cones (up to isomorphism). Thus, the image of a tile universal remains universal at lower resolution. • Embedding functors that map extended systems to cluster or impurity problems preserve universality if the associated partial traces and inclusions commute with the projections defining U∗ µ (e.g. projector K 0 -classes, spectral measures). This is precisely the condition enforced by the construction of tiles and defects in Sec. IV. • Descriptor flows generated by a higher connection ( A, B )on Θpreserve universality along exact legs (Sec. III C) where the central curvature vanishes; along such paths, the universal object is transported functorially without distortion. In all these cases, universality is preserved because the functors respect the universal property of U∗ µ , not because of accidental cancellations in a particular coordinate chart. Universality and XC c-functions. The XC c -functions introduced in Subsec. VIII A provide a quantitative measure of how close a given universal is to its tile. More precisely, given a tile universal U(T) µ and a system universal Uµ,θ , the c -function Cµ ( Uµ,θ )is a defect functional that vanishes if and only if Uµ,θ lies in the universality class of U(T) µ , and is positive otherwise. When the RG functors are contractive (in the quadratic case) or CPTP (in the relative-entropy case), Cµ is monotone under coarse-graining and embedding. Categorically, this can be rephrased as follows: the value of Cµ ( Uµ,θ )induces a rank function on the poset of universals ordered by inclusion of cones over the tile diagram. RG flows move downwards in this poset, and the rank cannot increase. In particular: •Along exact legs in descriptor space, Cµ(Uµ,γ(t))is constant and equal to zero. • Along flows that interpolate between different tiles (e.g. HEG to SCE to TLL), Cµ with respect to each tile serves as a diagnostic of which universality class dominates at a given point. • In multiscale flows with strong scheme dependence, structural Cµ remain monotone while naive scalar proxies may fail (Subsec. VIII B). Thus, XC c-functions operationalise the categorical notion of universality: they quantify the distance in Univµ to a universal object that enjoys a universal property with respect to a diagram of SpectralXC schemes. From slogans to structure. The net effect of this formalism is to turn the vague slogan “universality class” into a mathematically precise and computationally usable structure: 1. A universality class is an equivalence class in Univµ with respect to natural isomorphisms, not merely a set of models with the same exponents. 2. Universality is defined by a universal property (terminal cone) in a diagram of schemes and functors, not by an arbitrary choice of couplings. 3. XC c-functions are defect functionals on universal objects that are monotone under functors preserving this structure. 4. Tiles such as HEG, SCE, TLL, and local clusters are concrete realisations of universal objects with universal properties: any SpectralXC scheme in their diagram factors uniquely through them, and their universality classes organise the space of XC physics into a partially ordered set. In the context of DFT and XC modelling, this shift from slogan to structure has practical consequences. It suggests that instead of focusing solely on improving local or semi-local energy densities, we should focus on identifying and enforcing universal properties of spectral objects under RG functors. SpectralXC + UP–RG provides the framework to do so: tiles supply universal objects, RG functors supply structurepreserving maps, and XC c-functions supply monotones that guide us towards universal behaviour. The remainder of the Discussion section explores how this perspective interfaces with existing DFT practices and suggests directions for future work. 103 B. Comparison with traditional XC development The UP–SpectralXC framework developed in this work offers a rather different perspective on exchange– correlation (XC) modelling than the traditional route taken in density-functional theory (DFT) over the last five decades. In the standard approach, one starts from the Hohenberg–Kohn theorem,[ 122 ] which guarantees the existence of a universal functional F [ n ]of the density, and the Kohn–Sham (KS) construction,[123] which splits the total energy into E[n] = Ts[n] + EH[n] + Exc[n] + Zd3r vext(r)n(r),(444) where Ts [ n ]is the non-interacting kinetic energy, EH [ n ]the Hartree energy, and Exc [ n ]the exchange– correlation functional. The concrete content of DFT is then determined by an ansatz for Exc [ n ], together with a set of fitted parameters and/or enforced exact conditions.[124] Over time this has led to the famous “Jacob’s ladder” of XC functionals,[ 125 ] climbing from the local-density approximation (LDA) through generalized gradient approximations (GGAs) and meta-GGAs to hybrids and double hybrids: Exc[n] = Zd3r n(r)εxc n(r),∇n(r), τ(r), . . .,(445) with τ ( r )the Kohn–Sham kinetic energy density and additional ingredients (e.g. exact exchange) entering at higher rungs.[ 126 – 130 ] The parameters and functional forms in εxc are chosen to satisfy known exact constraints (sum rules, limits, scaling) and to reproduce benchmark data (atoms, molecules, solids). By contrast, in UP–SpectralXC the starting point is not an ansatz for Exc [ n ]but a single constrained variational problem—a Levy–Lieb minimisation equipped with spectral constraints and universal-property RG structure (Sec. IV). In this picture: • XC approximations are not a patchwork of semi-local formulas but different realisations of a unified optimisation principle. • Fitted parameters are replaced by Lagrange multipliers that enforce spectral laws; they are solved for, not fitted. • The central object is a universal spectral object Uµ,θ with a universal property, rather than a particular analytic form for εxc. In what follows we make this comparison precise. Patchwork of semi-local functionals vs a single constrained variational framework. Traditional XC development proceeds by constructing families of semi-local functionals based on increasingly sophisticated ansätze for εxc in Eq. (445) . LDA uses the homogeneous electron gas (HEG) energy density εHEG xc ( n )parametrised from QMC.[ 131 , 132 ] GGAs introduce a dependence on the reduced gradient s = |∇n|/ (2 kFn ), and meta-GGAs add the kinetic density τ and possibly the Laplacian ∇2n .[ 125 , 129 ] Each rung of the ladder is a new functional form; in practice, the landscape of functionals is a patchwork of many different constructions, each tailored to a class of systems. By contrast, UP–SpectralXC begins with the Levy–Lieb functional F[n] = inf Ψ→nhΨ|ˆ T+ˆ Vee|Ψi,(446) and imposes spectral constraints in the form of laws Sfk[n;θ] = φ(T) k(θT(θ)), k = 1, . . . , m, (447) where Sfk [ n ; θ ]are spectral probes of a KS-like generator L [ n ; θ ]and φ(T) k are target values from tiles (HEG, SCE, TLL, local clusters). The constrained Levy–Lieb principle then reads min nnE[n]Sfk[n;θ] = φ(T) k(θT(θ)), k = 1, . . . , mo,(448) with Lagrangian L[n, {λk}] = E[n]− m X k=1 λkSfk[n;θ]−φ(T) k(θT(θ)).(449) 104 Stationarity with respect to nyields an XC contribution to the Euler–Lagrange equation of the form vxc(r) = m X k=1 λk δSfk[n;θ] δn(r),(450) and the Lagrange multipliers {λk} are fixed by the spectral constraints themselves, typically by solving a linear system in the self-consistent-field loop. Several important differences with the traditional patchwork emerge: 1. There is a single variational framework from which all XC contributions arise; different tiles and probe sets correspond to different choices of constraints, not different functionals in an ad hoc menu. 2. The role of HEG, SCE, TLL, etc. is unified: each appears as a tile providing target spectral laws φ(T) krather than as an independent “model functional”. 3. Nonlocal and multiscale information enters naturally via spectral probes (resolvents, projectors, K 0 data), rather than being bolted on via nonlocal kernels or range-separated hybrids. In this sense, UP–SpectralXC is a structural replacement for the patchwork of semi-local functionals, not a new point in the same space. Fitting vs Lagrange multipliers: parameters as solutions, not inputs. Traditional XC functionals invariably involve parameters that are determined either by fitting to reference data (semi-empirical functionals) or by satisfying a finite set of exact constraints (non-empirical functionals).[ 125 , 128 – 130 ] For instance: • The B88 exchange functional[ 126 ] introduces parameters in a gradient enhancement factor Fx ( s ), adjusted to reproduce atomic exchange energies. • The LYP correlation functional[ 127 ] uses parameters fitted to the helium atom and uniform gas data. • PBE[ 128 ] and SCAN[ 129 ] use parameters fixed by enforcing exact conditions (sum rules, limits, scaling) and some heuristic choices for the remaining constants. In all these cases, once the parameters are chosen, the functional form of Exc [ n ]is fixed, and there is no system-specific adjustment of parameters beyond the density dependence built into the ansatz. When the functional is applied outside its calibration regime, performance can degrade dramatically (e.g. for strong correlation, dispersion, multi-reference systems). In UP–SpectralXC, the quantities playing the role of “parameters” are the Lagrange multipliers {λk} in Eq. (449). Crucially: • They are not chosen once and for all; they are determined self-consistently for each system and each density by solving the constrained problem. •They encode how strongly each spectral law is enforced in a given situation; their values adapt to the local physics, rather than being fixed by a global fit. • They can be interpreted directly in terms of the “cost” of enforcing a given feature of the spectrum (e.g. a HEG compressibility, an SCE co-motion constraint, a TLL exponent). Mathematically, the difference can be summarised as follows: Traditional: Exc[n;p],pfixed by fit or constraints;(451) UP–SpectralXC: n∗,{λ∗ k}= arg min n,{λk}L[n, {λk}],(452) with {λ∗ k} part of the solution rather than part of the input. In particular, the dependence of vxc on the probes Sfk and their targets φ(T) k is mediated by {λ∗ k} ; as new tiles or probes are added, additional multipliers appear naturally, without the need for new global fits. This distinction has several practical implications: 1. It reduces the risk of over-fitting: the multipliers are constrained by spectral laws rather than by arbitrary datasets. 105 2. It makes XC corrections more interpretable: each multiplier is associated with a specific spectral constraint (e.g. a compressibility, a TLL exponent), providing a direct physical meaning. 3. It facilitates systematic improvement: new probes can be added as needed, and their impact is captured by new multipliers, rather than by a wholesale redesign of εxc. Ansatz vs universal property. A deeper conceptual difference lies in what is taken to be fundamental. Traditional XC development starts from an ansatz for εxc and then checks whether it satisfies known exact conditions and performs well on benchmarks. The central question is: “Does this functional form produce good results for the systems we care about?”.[130] In UP–SpectralXC, the central object is the universal spectral object Uµ,θ and its universal property in the sense of Sec. IX A. Tiles such as HEG, SCE, and TLL are not ad hoc reference systems but universal objects U(T) µT,θT with a terminal property in a diagram of SpectralXC schemes. XC functionals are then constructed by requiring that the universals of the system under study map to the tile according to spectral laws and defect monotonicity. This shift from ansatz to universal property has several consequences: •Coordinate-free universality. Instead of identifying universality classes in a particular parametrisation (e.g. specific forms of Exc ), we identify classes in the category Univµ of universal objects, up to natural isomorphisms (Sec. IX A). •Structure-preserving RG. Universality is preserved not by an accident of a chosen functional form but by functors that preserve the universal property of tiles (resolution coarse-graining, embedding, descriptor flows along exact legs). •Monotones from defects. XC c -functions built from spectral defects provide a structural, schemeindependent way of measuring distance to universality, in contrast to ad hoc scalar indicators that may oscillate or depend strongly on the parametrisation (Sec. VIII A, VIII B). From a physics perspective, this means that instead of asking “Which analytic form of Exc works best?”, we ask “Which set of spectral laws and tiles correctly characterises the universality class of this system, and how do we enforce them in a variationally consistent way?”. The SpectralXC + UP–RG framework provides the mathematical and computational tools to answer the latter question. Summary. To summarise, the comparison between traditional XC development and UP–SpectralXC can be organised along three axes: 1. Object of interest. Traditional: the functional Exc [ n ], specified by an ansatz. UP–SpectralXC: the universal spectral object Uµ,θ and the associated tiles U(T) µT,θT. 2. Parameters. Traditional: fixed parameters p in εxc , determined by fit or exact conditions. UP–SpectralXC: Lagrange multipliers {λk}solved for by constrained Levy–Lieb minimisation. 3. Universality. Traditional: inferred from shared exponents and empirical performance, often scheme dependent. UP–SpectralXC: encoded as a universal property of U(T) in a categorical diagram, with defect monotones providing quantitative measures of distance. In the remainder of the Discussion we explore how this structural viewpoint can be integrated with existing DFT practice and how it could inform the next generation of XC functionals, especially in regimes where traditional semi-local and hybrid approaches struggle. C. Relation to non-invertible symmetries and defect categories One of the striking developments in modern quantum field theory (QFT) is the recognition that global symmetries need not form a group: they can be generalised to non-invertible symmetries described by fusion categories of topological defects.[ 133 – 135 ] Instead of an internal symmetry group G acting on local operators, one has a monoidal category D of topological lines (or higher-codimension defects) whose fusion rules need not admit inverses. Familiar examples include the Kramers–Wannier duality defect in the 2D Ising model, which satisfies a fusion rule of the schematic form D×D = 1 + ε and is therefore non-invertible. Renormalisation-group (RG) flows act not only on couplings but also on these defect categories; in favourable cases, one can define monotones and universal properties at the level of defects.[133] 112 captures HEG physics, GGAs and meta-GGAs incorporate gradient and kinetic information, hybrids mix in exact exchange, double hybrids mix in MP2-like correlation. Each rung is a different functional form, and strongly correlated and multiscale regimes often require bespoke corrections or embeddings.[146] By contrast, UP–SpectralXC starts from a single constrained variational principle: L[n, {λk}] = E[n]−X k λkSfk[n;θ]−φ(T) k(θT(θ)),(464) where E [ n ]is the exact energy functional (Levy–Lieb form), Sfk [ n ; θ ]are spectral probes of L [ n ; θ ], and φ(T) k are target spectral laws supplied by tiles (HEG, SCE, TLL, local clusters). Stationarity with respect to nyields the XC potential vxc(r) = X k λk δSfk[n;θ] δn(r),(465) and the Lagrange multipliers λk are solved for by the constraints Sfk [ n ; θ ] = φ(T) k ( θT ( θ )). In this picture: •XC structure is constrained by spectral laws and tiles, not by an ad hoc analytic ansatz for εxc. • Parameters entering the XC potential are Lagrange multipliers determined self-consistently, not fixed fit parameters. • Nonlocal and multiscale information (HEG, SCE, TLL, Anderson–Hund tiles, embedding data) enters naturally via spectral probes and is transported consistently along RG flows. This is the second conceptual shift: XC moves from a patchwork of hand-crafted functionals to a structurally constrained, tile-based variational framework. The diversity of XC approximations becomes a diversity of choices of tiles and probes within a single structural scheme, rather than a proliferation of unrelated functional forms. From scalar c-functions to defect monotones on universal spectral objects. In relativistic QFT, c -, a -, and F -theorems guarantee the existence of scalar functions that decrease monotonically along RG flows and measure the “number of degrees of freedom”. However, these theorems rely on Lorentz invariance, unitarity, and locality, and their direct analogues often fail in non-relativistic and multiscale systems.[ 109 ] Naive scalar “ c ”-like quantities in electronic structure (e.g. entropy densities, ad hoc correlation indicators) can behave non-monotonically under coarse-graining or embedding, especially when scheme dependence and walking flows are present. UP–SpectralXC replaces these fragile scalar diagnostics by defect monotones defined on universal objects. Given a tile universal U(T) µand a system universal Uµ,θ, we define defect functionals such as: • Quadratic defects Cquad µ ( Uµ,θ ) = Pkwk|Sfk [ n ; θ ] −φ(T) k|2, which measure violations of spectral laws in a weighted Euclidean norm. • Relative-entropy defects Crel µ ( Uµ,θ ) = S ( ρµ,θkρ(T) µT,θT ) , where S is quantum relative entropy and (ρµ,θ, ρ(T) µT,θT)are density operators associated with the system and tile.[151] • Curvature-based defects built from the holonomy of universals around loops in descriptor and scale space. Under RG and embedding maps that act as completely positive trace-preserving (CPTP) channels on density operators, the data-processing inequality ensures that Crel µ is monotone: it decreases as one integrates out degrees of freedom.[ 151 ] Under contractive maps in probe space, Cquad µ is also monotone. These defect functionals are the XC analogues of c -functions: they vanish exactly when Uµ,θ lies in the universality class of the tile, and they decrease under coarse-graining. Crucially, they are defined on the full universal object Uµ,θ , not on a single scalar observable or phenomenological parameter. Scalar projections of Uµ,θ may oscillate or behave non-monotonically along flows (as in walking behaviour or strong scheme dependence), but the structural defect monotones remain contractive. This is the third conceptual shift: from fragile, coordinate-dependent scalar c -functions to robust defect monotones on universal spectral objects. 113 Outlook. Taken together, these shifts suggest a new way of organising XC physics: 1. Identify tiles U(T) (HEG, SCE, TLL, Anderson–Hund clusters, topological band structures) as universal objects with universal properties in a diagram of SpectralXC schemes. 2. Encode their physics in spectral and categorical probes, including operator-valued, K -theoretic, and homology-based data. 3. Construct XC potentials via constrained Levy–Lieb minimisation with spectral laws as constraints, with Lagrange multipliers playing the role of dynamic, system-specific “parameters”. 4. Track RG flows and embeddings via functors that act on schemes and universals, and monitor defect monotones as XC c-functions that diagnose proximity to tiles. In this way, “universality” ceases to be an informal metaphor and becomes the organising principle of exchange–correlation modelling: a universal property of spectral objects preserved by functors, and a source of structural constraints that guide both analytic and data-driven XC development.[145, 146] B. Emphasising the new capabilities Beyond the conceptual shift from heuristic universality to universal properties and defect monotones, the framework developed here yields a set of concrete new capabilities for electronic-structure theory and XC modelling. These capabilities can be grouped under four headings: (i) multiscale, scheme-aware XC; (ii) structural diagnostics of basis and embedding; (iii) systematic incorporation of HEG/SCE/TLL constraints into XC; and (iv) new notions of correlation length, phase crossover scales, and XC “ c -functions”. In this subsection we summarise these capabilities and their implications. Multiscale, scheme-aware XC. Traditional XC approximations are largely scale blind and scheme blind. A given functional Exc [ n ]is applied indiscriminately to finite molecules, extended solids, and embedded clusters, with limited explicit control over how it should change under coarse-graining (e.g. basis truncation, band downfolding) or embedding (e.g. DMFT, quantum cluster methods).[ 147 , 148 ] Practical multiscale schemes (DFT+DMFT, DFT+U, cRPA-based downfolding) do incorporate some notion of scale separation, but the XC functional itself is usually fixed and local in scale space. In UP–SpectralXC, XC is explicitly multiscale and scheme-aware: • Each scale µ has an associated category Cµ of SpectralXC schemes and an associated universal category Univµof universal spectral objects. • RG and embedding steps are functors Fµ→µ0 : Cµ→Cµ0 inducing functors Uµ→µ0 : Univµ→Univµ0 on universals. These functors know about the scheme: plane-wave vs local-orbital bases, cluster vs impurity embeddings, different downfolding choices, etc. • The XC potential vxc ( r )is constructed at each scale from a constrained Levy–Lieb principle with scale-dependent spectral laws Sfk [ n ; θ, µ ] = φ(T) k ( θT )and scale-dependent Lagrange multipliers λk(µ). This structure allows one to: 1. Define consistent scale hierarchies for XC: at high scales, only coarse spectral laws (e.g. HEGlike compressibility, smallq kernels) are enforced; at lower scales, more refined laws (e.g. SCE asymptotics, TLL exponents, local multiplet structure) are added. 2. Propagate information downwards from tiles: HEG, SCE, TLL, and Anderson–Hund tiles provide universal boundary conditions that constrain XC at all scales. 3. Propagate information upwards from embeddings: embedding calculations (cluster DMFT, impurity solvers) yield high-fidelity universals at low scales that can be summarised into tiles and used to refine XC at higher scales. In practice this means that XC is no longer a single static object but a family of scale-dependent, RG-compatible constructions. When changing resolution or embedding, one does not merely change the Hamiltonian and keep XC fixed; one moves within a structured XC manifold guided by functors and defect monotones. 114 Structural diagnostics of basis and embedding. A persistent difficulty in multiscale electronic structure is the scheme dependence associated with basis choice and embedding strategy. Plane-wave vs local-orbital bases, different Wannier constructions, alternative cluster geometries, and various impurity choices can all lead to quantitatively different results, even for the same underlying Hamiltonian.[ 147 , 148 ] Traditionally, such differences are diagnosed in an ad hoc way, by comparing observables (energies, gaps, occupations) between schemes. UP–SpectralXC introduces structural diagnostics for basis and embedding, based on defect monotones and curvature: • For each scheme (basis/embedding choice) and scale, one constructs a universal Uµ,θ and evaluates defect functionals Cµ ( Uµ,θ )with respect to chosen tiles. This provides a quantitative measure of how universal the scheme is. • Loop curvature, defined by transporting universals around loops in scheme space (e.g. plane-wave → Wannier → cluster → back to plane-wave), measures anomalies in XC under composition of basis and embedding changes. In this way: 1. Basis sets and embeddings can be compared structurally: schemes with smaller defect and curvature are preferred, as they preserve tile universality more faithfully. 2. Systematic scheme-improvement strategies emerge: one can search over basis/embedding parameters (e.g. Wannier localisation windows, cluster shapes) to minimise defect and curvature, rather than relying solely on energetic criteria. 3. XC consistency becomes testable: if a given XC construction yields large loop curvature across scheme loops that ought to be benign (e.g. unitary basis changes), this is a signal that the XC representation is not robust. These diagnostics are intrinsically multiscale and spectral: they are defined in terms of Uµ,θ and its transformation under functors, rather than in terms of any particular observable. Universal constraints from HEG/SCE/TLL built systematically into XC. Conventional XC functionals often mix constraints from different regimes (HEG, atomic limits, gradient expansions) in a relatively informal way: one designs a functional form and tunes it so that it behaves reasonably in known limits.[ 125 , 145 ] Strong-correlation limits (SCE), one-dimensional non-Fermi-liquid behaviour (TLL), and multi-orbital local physics are typically bolted on as separate corrections or as specialised functionals. UP–SpectralXC instead treats HEG, SCE, and TLL as tiles with universal properties: • HEG tiles: universal objects UHEG µHEG,θHEG containing QMC-based energy densities, compressibilities, small-qkernels, and structure-factor moments. • SCE tiles: universal objects USCE µSCE,θ containing co-motion functions and strong-coupling asymptotics of the electron–electron interaction. • TLL tiles: universal objects UTLL µTLL,θTLL containing Luttinger parameters, velocities, and exponents of 1D correlation functions. These tiles enter XC in a systematic way: 1. For each region of descriptor space (e.g. density, geometry, dimensionality), one identifies which tile (or mixture of tiles) is appropriate. 2. Spectral laws Sfk [ n ; θ ] = φ(T) k are imposed as constraints in the Levy–Lieb variational problem, with each φ(T) kextracted from tile data. 3. XC potentials inherit HEG/SCE/TLL constraints not as limits of an ansatz but as structural obligations enforced by Lagrange multipliers. This yields an XC framework in which: • Uniform electron gas physics is built in through HEG spectral laws, ensuring correct metallic behaviour and screening. 115 • Strong-correlation and bond dissociation physics is built in through SCE spectral laws, ensuring correct asymptotics and avoidance of delocalisation and static-correlation errors. • One-dimensional and quasi-1D physics is built in through TLL spectral laws, ensuring correct non-Fermi-liquid exponents and momentum-distribution tails. Because tiles are universal objects with universal properties, these constraints are preserved under admissible RG functors and embeddings, providing a robust, multiscale notion of “built-in” physics. New notion of correlation length, phase crossover scales, and XC c -functions. Finally, UP–SpectralXC introduces a new set of quantitative concepts: defect-based correlation lengths, phase crossover scales, and XC “c-functions”. Defect-based correlation lengths. Given a tile T and its universal U(T) , the defect C(T) µ ( Uµ,θ ) measures how far a system is from the tile universality class at scale µ and descriptor θ . One can define a correlation length ξ(T) associated with tile T by examining how quickly this defect decays under spatial or descriptor coarse-graining. For example, consider a family of coarse-grainings over length scale ` , with corresponding universals Uµ(`),θ; then C(T)(`;θ) := C(T) µ(`)(Uµ(`),θ)(466) typically decays as `increases. A natural definition is ξ(T)(θ) := inf` > 0|C(T)(`;θ)≤Cthresh,(467) where Cthresh is a chosen small threshold. This correlation length is not defined via two-point functions but via proximity to a tile universality class in spectral space. It provides a tile-dependent measure of how extended correlations are, e.g. HEG-like, SCE-like, or TLL-like. Phase crossover scales in descriptor space. Similarly, as one moves in descriptor space (e.g. varying density, interaction strength, geometry), the dominance of different tiles changes. One can define phase crossover scales by points where the defects to different tiles intersect or cross: C(T1) µ(Uµ,θ∗) = C(T2) µ(Uµ,θ∗),(468) with both defects small. In such cases, the system sits near the intersection of two universality classes (e.g. HEG to SCE crossover, Fermi-liquid to TLL crossover), and θ∗ marks a phase or regime boundary in descriptor space. This provides a structural, SpectralXC-based definition of crossover scales, grounded in universal objects rather than in specific order parameters. XC c-functions as structural monotones. Finally, the defect functionals Cµ ( Uµ,θ )themselves play the role of XC c -functions: monotone measures of “non-universal XC structure” along RG flows and embeddings. Unlike traditional c -functions, which are scalars tied to stress-tensor anomalies in relativistic QFT,[149, 150] XC c-functions: • are defined on universal spectral objects, incorporating spectral, K -theoretic, and homological information; • remain monotone under CPTP RG and embedding maps by virtue of data-processing inequalities and contractivity, even in non-relativistic, multiscale contexts; • provide a coordinate-free partial order on universals: Uµ0,θ0 is “more universal” than Uµ,θ whenever Cµ0(Uµ0,θ0)≤Cµ(Uµ,θ). These c -functions supply a new vocabulary for quantifying how XC changes under coarse-graining: instead of asking how a particular observable behaves, we ask how close the universal spectral object is to a tile and how that distance evolves. This has practical implications for functional design, embedding diagnostics, and the assessment of ML-based XC models, which can all be benchmarked against structural XC c-functions. 116 Summary. In summary, the SpectralXC + UP–RG framework does more than rephrase known ideas in new language. It introduces: •genuinely multiscale, scheme-aware XC constructions, •structural diagnostics for basis and embedding choices, •systematic, tile-based inclusion of HEG/SCE/TLL physics into XC, •and new, defect-based notions of correlation length, crossover scale, and XC c-functions. These capabilities provide a roadmap for future XC development that is both more principled and more adaptable: principled, because it is anchored in universal properties and RG structure; adaptable, because tiles, probes, and defect functionals can be refined as new physical insight and data become available. C. Outline of next practical steps The theoretical construction of UP–SpectralXC and its proof-of-principle model applications must be followed by concrete numerical implementations in realistic electronic-structure workflows. In this subsection we outline a sequence of practical steps that would turn the present conceptual framework into an operational tool: (i) a minimal implementation of UP–SpectralXC in a standard DFT code for a small set of model systems, (ii) systematic testing of spectral defect monotones and curvature diagnostics under basis and embedding changes, and (iii) preparation of a first set of numerical results that demonstrate the method as a conceptual and computational proof of principle. Step 1: Minimal UP–SpectralXC implementation in a standard DFT code. The first practical goal is to realise a minimal version of UP–SpectralXC in an existing DFT infrastructure. Rather than building a code from scratch, it is natural to leverage widely used, well-tested packages such as plane-wave/pseudopotential codes (Quantum ESPRESSO,[152] VASP[153]) and all-electron or numerical-atomic-orbital codes (FHIaims[ 154 ]), whose modular architectures and validation suites can provide a robust environment for experimentation. A minimal implementation requires the following ingredients: 1. Access to the KS generator. The DFT code must expose the KS operator L0 [ n ] = −1 2∇2 + vext + vH [ n ] + v(0) xc [ n ]in a suitable basis (plane waves, atom-centred orbitals). In practice one works with its matrix representation L0[n]in the current KS basis, including spin as needed. 2. Implementation of spectral probes. For a chosen small set of probes fk , one must implement the functional calculus fk ( L0 [ n ]) via spectral decompositions or resolvent integrals. In a finitedimensional KS basis, a direct diagonalisation yields eigenvalues {εi} and eigenvectors {ϕi} , allowing the evaluation fk(L0[n]) = X i fk(εi)|ϕiihϕi|,(469) and, for a trace-like state Φ, the scalar probe Sfk[n;θ] = Φ(fk(L0[n])) = X i fk(εi)wi,(470) where the weights wi encode the occupation pattern and any additional operator insertions. For larger systems one may resort to Krylov subspace or Lanczos methods to approximate fk ( L0 [ n ]) without full diagonalisation. 3. Tile data and spectral laws. For each model system and regime, one must precompute (or tabulate) the target spectral laws φ(T) k ( θT )from tile calculations: HEG for nearly uniform densities, SCE for stretched molecules, TLL for 1D chains, or small Anderson–Hund clusters for transitionmetal complexes. At the minimal stage, these tile data can be provided from independent calculations or analytic approximations, and stored in lookup tables as functions of descriptor variables (e.g. local density, nearest-neighbour distance, interaction strength). 117 4. Lagrange-multiplier loop in SCF. The constrained Levy–Lieb principle is implemented by augmenting the standard SCF loop for the density with a small, inner loop for the Lagrange multipliers {λk}. Given a current density n(m), the steps are: (a) Construct the KS generator L [ n(m) ; λ ] = L0 [ n(m) ]+∆ vxc [ n(m) ; λ ] , where ∆ vxc is the additional potential generated by the spectral constraints. (b) For a trial set of multipliers λk , solve the KS equations and compute the updated probes Sfk[n(m);θ]. (c) Linearise the spectral constraints around the current (n(m), λ(m)): Sfk[n(m);λ]≈Sfk[n(m);λ(m)] + X ` Mk`[n(m)] (λ`−λ(m) `),(471) where Mk`[n] = ∂Sfk/∂λ`can be evaluated by finite differences or via linear response. (d) Solve the small linear system X ` Mk`[n(m)] (λ(m+1) `−λ(m) `) = φ(T) k(θT)−Sfk[n(m);λ(m)](472) to update λ(m+1). Convergence of both n and λ yields a stationary point of the constrained functional and a corresponding vxc. 5. Code integration and modularity. To keep the implementation generic, the SpectralXC module should be written as an external plugin or library that interfaces with the host DFT code at clearly defined points: access to L0 [ n ], density update routines, and KS eigenpairs. Many modern codes already support this level of modularity, particularly plane-wave packages with flexible plugin infrastructures.[152, 154] With these ingredients in place, one can perform the first UP–SpectralXC calculations for simple test systems, such as: the H 2 molecule along its dissociation coordinate; a small 1D Hubbard chain represented in a DFT framework; and a minimal transition-metal complex (e.g. octahedral Fe(II)) treated with a KS + local SpectralXC correction on the d-shell. Step 2: Testing spectral defect monotones and curvature diagnostics. Once a minimal implementation is operational, the next step is to stress-test the structural aspects of UP–SpectralXC: spectral defect monotones and curvature. This involves designing numerical experiments that probe the behaviour of Uµ,θ under controlled changes of basis, resolution, embedding, and descriptor, and quantifying how defect functionals Cµ(Uµ,θ)and loop curvature respond. Representative tests include: 1. Basis refinement and truncation. For a fixed physical system (e.g. a small molecule), perform UP–SpectralXC calculations with a sequence of systematically improvable basis sets (plane-wave cutoffs, Gaussian basis families) at fixed tile and spectral probes. Evaluate: •The defect C(T) µ(Uµ,θ)as a function of basis size. •The behaviour of XC observables (energies, gaps, densities) alongside the defect. One expects Cµ to decrease with basis refinement if the basis truncation is treated as an RG-like coarse-graining, and to plateau when the universal regime is reached. 2. Scheme loops and curvature. Construct loops in scheme space, such as plane waves →Wannier orbitals →cluster embedding →back to plane waves,(473) and evaluate the universal Uµ,θ at each step. Compute the loop curvature by comparing the initial and final universals and their defects: Kγ∼ Ufinal µ,θ −Uinitial µ,θ  ,(474) for an appropriate norm. Small curvature indicates that the XC construction is robust under scheme changes; large curvature signals significant scheme dependence and motivates refinement of tiles or probes. 118 3. Descriptor flows and crossover scales. For model systems with well-understood regimes (e.g. H 2 dissociation; 1D chains across interaction/density variations), track defect functions with respect to different tiles (HEG, SCE, TLL) along descriptor paths. Identify: •Regimes where a given tile dominates (small defect). •Crossover points where defects to different tiles become comparable. •Corresponding changes in XC observables. This provides numerical evidence for the tile-based picture of XC and validates the use of defects as proxies for phase crossovers. 4. Embedding flows and impurity solvers. For systems treated with DFT+DMFT or other embedding schemes, incorporate impurity solvers (e.g. CT-QMC[ 155 ]) to generate high-quality local universals (local Green’s functions, self-energies) at low scales. Then: •Map these local universals to tile universals (e.g. Anderson–Hund tiles) and compute defects. • Study how these defects change under different embeddings (cluster size, bath parametrisation), and whether XC corrections derived from SpectralXC reduce scheme dependence. These tests will provide quantitative evidence that spectral defect monotones behave as intended (contractive, tile-sensitive) and that curvature diagnostics meaningfully capture scheme dependence in practice. Step 3: Conceptual + numerical proof-of-principle publication. The final step of this initial phase is to assemble a coherent proof-of-principle study that combines the conceptual framework with numerical results. Such a study would ideally include: •A clear statement of the UP–SpectralXC construction in a compact form, focusing on one or two tiles (e.g. HEG and a local Anderson–Hund tile) and a small set of probes, together with the constrained Levy–Lieb formulation and resolvent-based expressions for vxc. •One or two model applications drawn from the tests above, for example: 1. H 2 dissociation: comparison between a baseline semi-local functional and UP–SpectralXC with SCE-like constraints, demonstrating improved behaviour in the stretched regime and a clear interpretation in terms of SCE universality. 2. A simple transition-metal complex or 2 d+ 2 p cluster: comparison between KS, cluster A/B, and UP–SpectralXC corrections, showing reduced scheme dependence and monotone defect behaviour under basis and embedding changes. •Systematic plots of defect monotones and curvature: figures illustrating: –Defect vs. scale/basis size; –Defect vs. descriptor (e.g. bond length, interaction strength) relative to different tiles; –Loop curvature vs. scheme parameters. These plots would visually corroborate the RG-motivated claims: contraction of defects, identification of crossover scales, and diagnostic power of curvature. •Quantitative comparison with existing functionals and benchmarks: energy curves, densities, and local observables compared against high-level references (e.g. coupled-cluster, DMRG, QMC) and against widely-used functionals (PBE, SCAN). Where possible, benchmarks from large-scale comparison studies[156] can be used to position UP–SpectralXC in the broader landscape. •Discussion of computational cost and scalability: analysis of the overhead introduced by spectral probes and Lagrange multipliers, and proposals for efficient algorithms (e.g. reuse of KS eigensystems, low-rank approximations for resolvents, sparse probe sets tailored to each tile). This is crucial to convince practitioners that the method is feasible in realistic workflows. 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