Regularized Sector Functionals

• Regularized sector functionals are defined by combining sector decomposition with explicit regularization to control singular behavior across disciplines such as string theory, control theory, operator calculus, and more.
• They employ rigorous techniques like GSO projection, Riccati-type completions, and weighted area integrals to manage divergences, numerical instabilities, and cusp contributions.
• Applications range from the one-loop Type IIB torus vacuum and robust Lyapunov–Krasovskii functionals to regularized exchange-correlation functionals in electronic-structure theory.

Regularized sector functionals are functionals in which a sector decomposition, sector restriction, or sectorial calculus is combined with an explicit regularization prescription. In current arXiv usage, the exact expression occurs in the study of the one-loop Type IIB torus vacuum, where the four unprojected spin sectors are regularized individually before the final GSO projection (Wang, 16 Jun 2026). Closely related constructions appear in control theory as Lyapunov–Krasovskii functionals of robust type tailored to sector-based absolute stability (Scholl, 2023), in operator theory as regularized $H^\infty$ functional calculus and area integral norms for sectorial operators (1207.1174, Kunstmann et al., 2012), in probability as the relaxed sector condition for additive functionals of Markov processes (Horvath et al., 2012), and in electronic-structure theory as regularized exchange-correlation or adiabatic-connection functionals (Furness et al., 2021, Daas et al., 2023). This suggests a cross-disciplinary pattern rather than a single universal definition: sector data are kept explicit, and regularization is introduced to control cusp contributions, conservative robustness bounds, or numerically unstable derivatives.

1. Terminological scope

The literature does not use the term uniformly. In the superstring setting, “regularized sector functionals” denotes sector-resolved modular-integral functionals defined for the auxiliary torus blocks

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$

before the physical Type IIB combination

$\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$

is imposed (Wang, 16 Jun 2026). In control theory, the closest concept is the quadratic Lyapunov–Krasovskii functional of robust type, which is explicitly tailored to sector-based absolute stability and replaces the Lyapunov-equation template of complete-type functionals by an algebraic Riccati-equation template (Scholl, 2023).

Operator-theoretic papers use “sectorial” rather than “sector,” but they contribute a related notion: holomorphic functional calculus can be regularized by inserting a fixed function such as

$\psi(z)=\frac{z}{(1+z)^2},$

and area integral functions can be interpreted as a regularized family of sectorial functionals (1207.1174). In Banach-function-space theory, $\mathcal R_s$-sectoriality is the strengthening that makes $s$-power function norms stable under changes of auxiliary holomorphic kernels and supports a generalized Triebel–Lizorkin scale (Kunstmann et al., 2012).

A further, structurally adjacent, use of “sector” appears in probability. The relaxed sector condition introduces regularized operators

$$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$

for additive functionals of stationary and ergodic Markov processes, and uses their convergence to control the Kipnis–Varadhan martingale approximation and central limit theorem (Horvath et al., 2012). This is not a sector functional in the superstring or control-theoretic sense, but it belongs to the same family of techniques in which a sector-dependent object is softened by a parameter-dependent regularization.

2. Sector-resolved modular integrals in Type IIB theory

In the precise sense of (Wang, 16 Jun 2026), the starting point is the Type IIB torus vacuum

$Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$

with $\tau=x+\ii y$ and modular weight parameter $s=6$. Writing the integrand in $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$0-character form produces the four auxiliary sector blocks

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$1

These are not yet physical amplitudes; they are the sector-resolved inputs for the regularized modular integral.

The sector functionals are defined by expanding each block in Fourier modes,

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$2

and pairing each coefficient with a regularized mode block: $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$3 The key modular prescription is

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$4

so the compact-domain piece and the cusp tail are fixed simultaneously inside one regularized block (Wang, 16 Jun 2026).

The construction is based on splitting the fundamental domain as

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$5

with a compact keyhole region $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$6 and a cusp strip $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$7. In the Euclidean strip, horizontal orthogonality enforces diagonal projection,

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$8

so only $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$9 modes survive in the cusp contribution. The physical string prescription then replaces the long Euclidean tube by a Lorentzian continuation along the contour

$\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$0

and the vertical tail is evaluated through the identity

$\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$1

A central conceptual point is that the final cancellation is not built into the definition of the regularization. Using Jacobi’s abstruse identity,

$\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$2

the holomorphic blocks satisfy $\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$3, and hence

$\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$4

Therefore the regularized sector functionals are equal termwise,

$\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$5

and only afterward does the signed Type IIB combination vanish: $\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$6 This establishes that regularization and GSO projection commute in the vacuum problem (Wang, 16 Jun 2026).

The zero mode is evaluated exactly. Since $\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$7 and

$\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$8

one obtains the auxiliary-sector constant-mode amplitude

$\cZ_{\rm IIB}=\cZ_{VV}-\cZ_{VS}-\cZ_{SV}+\cZ_{SS}$9

This is not the full vacuum amplitude; it is the exact constant-mode normalization of one auxiliary sector.

3. Sector-based absolute stability and Riccati-type Lyapunov–Krasovskii functionals

In control theory, the closest analogue is the robust-type Lyapunov–Krasovskii functional for retarded functional differential equations with one discrete delay,

$\psi(z)=\frac{z}{(1+z)^2},$0

where the nominal system is

$\psi(z)=\frac{z}{(1+z)^2},$1

and the perturbation $\psi(z)=\frac{z}{(1+z)^2},$2 is locally Lipschitz and satisfies $\psi(z)=\frac{z}{(1+z)^2},$3 (Scholl, 2023). The goal is a robust absolute stability statement proved by a single quadratic LK functional that stabilizes the nominal delay system and a family of structured nonlinear perturbations.

The perturbation is written in feedback form

$\psi(z)=\frac{z}{(1+z)^2},$4

and the admissible perturbations are characterized by the quadratic sector form

$\psi(z)=\frac{z}{(1+z)^2},$5

The sector condition is

$\psi(z)=\frac{z}{(1+z)^2},$6

or more generally $\psi(z)=\frac{z}{(1+z)^2},$7. This framework includes the small-gain restriction

$\psi(z)=\frac{z}{(1+z)^2},$8

passivity-type sectors, and the circle-criterion form associated with a sector $\psi(z)=\frac{z}{(1+z)^2},$9.

The decisive modification is not the quadratic structure of the functional itself but the defining equation for its derivative along the nominal system. Instead of the Lyapunov-equation template used in complete-type functionals, the robust-type construction prescribes

$\mathcal R_s$0

with $\mathcal R_s$1 for an exact functional and $\mathcal R_s$2 for an inequality-based one. Here

$\mathcal R_s$3

is the finite-dimensional part of the derivative representation. The resulting monotonicity statement is

$\mathcal R_s$4

The paper then removes the mixed sector term $\mathcal R_s$5 by a Schur-complement/Aitken-type feedback transformation and splits the functional as

$\mathcal R_s$6

The remaining part $\mathcal R_s$7 satisfies an operator-valued algebraic Riccati equation on the infinite-dimensional state space

$\mathcal R_s$8

namely

$\mathcal R_s$9

This is the paper’s precise replacement of the Lyapunov-equation template by an ARE template.

Existence is characterized by an infinite-dimensional Kalman–Yakubovich–Popov lemma through the frequency-domain quantity

$s$0

where

$s$1

If $s$2 for all $s$3, then an LK functional of robust type exists; if $s$4 for some $s$5, then no such functional exists. For the small-gain sector this yields

$s$6

identified with the complex stability radius. For passivity-type sectors, the admissible bound becomes

$s$7

The paper is explicit that these functionals are designed to be less conservative than complete-type LK functionals (Scholl, 2023).

4. Sectorial operators, area integrals, and generalized Triebel–Lizorkin scales

In sectorial operator theory, regularization enters first through the holomorphic functional calculus. For a sectorial operator $s$8 and

$s$9

one may define $$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$0 for general $$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$1 by the regularized formula

$$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$2

with the natural domain

$$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$3

This is the paper’s explicit “regularized functional calculus viewpoint” (1207.1174).

On Hilbert spaces, area integral functions provide a second regularization. For $$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$4, a sectorial operator $$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$5 of type $$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$6, and $$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$7, the area integral norm is

$$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$8

The paper proves that these area integral functions are equivalent to the McIntosh square functions

$$B_\lambda=(\lambda I+S)^{-1/2}A(\lambda I+S)^{-1/2}$$9

and that bounded $Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$0 functional calculus is equivalent to the two-sided estimate

$Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$1

for some, hence every, nonzero $Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$2 (1207.1174). The paper explicitly interprets this as a kind of regularization of sector functionals: instead of evaluating $Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$3 only on $Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$4, one integrates the family $Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$5 over the whole sector $Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$6 with the weight $Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$7.

The Banach-function-space extension replaces Hilbertian square functions by $Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$8-power norms controlled through $Z^{\rm IIB}_{0,1} = \ii \cN \int_{\cF} \frac{\dd\tau\wedge \dd\bar\tau}{y^6}\,\cZ_{\rm IIB}(\tau,\bar\tau),$9-boundedness. A set $\tau=x+\ii y$0 is $\tau=x+\ii y$1-bounded if

$\tau=x+\ii y$2

and $\tau=x+\ii y$3 is $\tau=x+\ii y$4-sectorial if $\tau=x+\ii y$5 is $\tau=x+\ii y$6-bounded for every $\tau=x+\ii y$7 (Kunstmann et al., 2012). This vector-valued resolvent control is the mechanism that makes the generalized Triebel–Lizorkin norms

$\tau=x+\ii y$8

independent of the auxiliary holomorphic function $\tau=x+\ii y$9, and it yields bounded $s=6$0-calculus for the corresponding part operators on the associated scales.

A common misconception is to treat ordinary sectoriality as sufficient for these norm-equivalence results. The Banach-space paper is explicit that $s=6$1-sectoriality is a strictly stronger requirement than ordinary sectoriality because it asks for vector-valued uniform control of resolvents rather than operator-norm bounds alone (Kunstmann et al., 2012).

5. Relaxed sector conditions for additive functionals of Markov processes

For stationary and ergodic Markov processes $s=6$2 on $s=6$3, with Hilbert space $s=6$4, generator $s=6$5, symmetric part

$s=6$6

and antisymmetric part

$s=6$7

the object of interest is the additive functional

$s=6$8

for $s=6$9 with zero mean (Horvath et al., 2012). The Kipnis–Varadhan framework uses the resolvent

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$00

and the limits

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$01

If these hold, then the asymptotic variance

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$02

exists, there is a zero-mean $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$03-martingale $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$04 with

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$05

and the additive functional admits martingale approximation and, when $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$06, a central limit theorem.

The relaxed sector condition introduces the regularized skew-Hermitian operators

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$07

with formal limit

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$08

The main theorem assumes a dense core $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$09 on which $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$10 is essentially skew self-adjoint and

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$11

Under these conditions, the $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$12-condition

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$13

implies the Kipnis–Varadhan resolvent limits and hence the martingale approximation and central limit theorem (Horvath et al., 2012).

The significance of the RSC is twofold. First, it strictly generalizes the strong sector condition, which corresponds to boundedness of $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$14. Second, it recovers the graded sector condition when $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$15 acts diagonally in the grading. The paper emphasizes that the gain is a cleaner operator-convergence proof rather than merely a slightly weaker hypothesis. It also records a limitation: “So far we don't have convincing examples in this direction,” meaning that no concrete model is given in which RSC applies while earlier sector conditions fail (Horvath et al., 2012).

6. Regularization of functionals in electronic-structure theory

Electronic-structure theory uses “regularized functionals” in a different but technically related sense: a functional form is modified to remove divergences, restore selected exact constraints, or damp small-denominator pathologies. In the SCAN family, the exchange-correlation energy density is interpolated as

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$16

with iso-orbital indicator

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$17

Regularization begins in rSCAN, which introduces

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$18

with $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$19 and $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$20, and smooths the interpolation. This removes the asymptotic divergence in the XC potential and improves numerical stability, but breaks the uniform density limit, uniform coordinate scaling of exchange, and the slowly-varying density limit (Furness et al., 2021).

The subsequent sequence r++SCAN, r$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$21SCAN, and r$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$22SCAN restores selected exact conditions while maintaining a smooth interpolation. In r++SCAN the regularized indicator becomes

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$23

which restores the uniform density limit and uniform coordinate scaling. r$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$24SCAN then restores the second-order gradient expansions GE2X and GE2C while retaining the smooth interpolation and numerically benign derivative structure. r$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$25SCAN restores GE4X as well, but the additional $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$26-dependent corrections reintroduce oscillatory behavior in the derivatives of the enhancement factor and thus in the XC potential. The paper’s stated conclusion is that the greater smoothness of r$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$27SCAN seems to lead to better general accuracy than the additional exact constraint of SCAN or r$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$28SCAN does (Furness et al., 2021).

A parallel development appears in Møller–Plesset adiabatic-connection models. There the correlation energy is constructed by combining MP AC interpolation with regularized MP2 denominators and spin scaling, using

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$29

where the regularization parameters $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$30 and $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$31 damp small denominators $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$32. The low-cost model

$\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$33

takes $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$34, $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$35, and $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$36, retains formal $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$37 scaling, and is reported to be competitive with or better than dispersion-corrected double hybrids for non-covalent interactions (Daas et al., 2023). On the B30 anionic halogen-bond subset, the paper reports MAEs of $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$38 for MP2, $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$39 for B2PLYP-D3, $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$40 for B3LYP-D3, and $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$41 for $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$42-SPL2, and summarizes this as surpassing standard dispersion-corrected DFT by a factor of $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$43 to $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$44 for anionic halogen-bonded complexes (Daas et al., 2023).

These electronic-structure examples are not sector functionals in the modular or operator-theoretic sense. A plausible implication is that they nonetheless illuminate a broader modern use of regularization: exact constraints, asymptotic limits, or coupling-channel decompositions are retained as far as possible, but singular or excessively rough functional behavior is softened to improve robustness, smoothness, or transferability.

Regularized sector functionals therefore occupy a heterogeneous but coherent place across several fields. In superstring theory they are explicit linear functionals on unprojected spin-sector Fourier data; in control they are quadratic LK functionals whose nominal derivative is defined by a Riccati-type completion of squares; in operator theory they arise through regularized holomorphic calculus and sector-integrated norms; in probability they appear through regularized sector operators $\cZ_{VV},\qquad \cZ_{VS},\qquad \cZ_{SV},\qquad \cZ_{SS},$45; and in electronic structure they denote regularized functionals designed to preserve useful exact structure while suppressing numerical pathologies. The shared mathematical theme is not a single formula but the insistence that sector information remain explicit throughout the regularization.