Complexity Factor Formalism in GR

• Complexity factor formalism is a geometric framework that uses the trace-free scalar YTF to capture the balance between density inhomogeneity and pressure anisotropy in self-gravitating systems.
• Imposing YTF = 0 creates an exact relation between the density gradient and anisotropy, closing the stellar equilibrium problem without relying on arbitrary anisotropy assumptions.
• Extensions of the formalism address charged configurations, modified gravity, black holes, and wormholes, providing insights into stability, oscillations, and the tidal response of compact objects.

The complexity factor formalism is a geometric framework for characterizing self-gravitating matter configurations through scalars obtained from the orthogonal splitting of the Riemann tensor. In the static, spherically symmetric anisotropic case of general relativity, the formalism identifies the scalar $Y_{TF}$ as the complexity factor, with

$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$

so that complexity is tied to the combined effect of density inhomogeneity and local pressure anisotropy. The condition $Y_{TF}=0$ does not require homogeneity or isotropy separately; rather, it imposes an exact balance between them and thereby supplies an additional structural relation that can close the stellar equilibrium problem without an ad hoc anisotropy ansatz (Rincon et al., 2023).

1. Geometric origin of the formalism

The formalism begins with a static, spherically symmetric line element

$$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$

together with an anisotropic fluid characterized by energy density $\rho(r)$, radial pressure $P_r(r)$, tangential pressure $P_\perp(r)$, and anisotropy $\Pi(r)=P_\perp-P_r$. The orthogonal splitting of the Riemann tensor introduces the tensors

$$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$

and, in static spherical symmetry, the magnetic part $Z_{\alpha\beta}$ vanishes. Their scalar decomposition yields the structure scalars

$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$0

$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$1

In this representation, $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$2 and $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$3 encode trace information, whereas $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$4 and $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$5 encode trace-free structural content. The complexity factor is identified with $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$6, and its vanishing defines the class of vanishing-complexity configurations (Rincón et al., 2023).

The same decomposition may be written in terms of the electric Weyl tensor $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$7 and the anisotropic tensor $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$8: $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$9

$Y_{TF}=0$0

with $Y_{TF}=0$1. This makes explicit that the formalism is not an external diagnostic appended to the field equations, but a reorganization of curvature and matter information into invariant structural scalars (Rincón et al., 2024).

2. Vanishing complexity as a closure condition

Imposing

$Y_{TF}=0$2

produces the integro-differential relation

$Y_{TF}=0$3

Thus, zero complexity ties the anisotropy directly to the density gradient. In stellar applications this is the decisive step: once an equation of state is specified, the anisotropy is no longer a free function. The generalized Tolman–Oppenheimer–Volkoff system

$Y_{TF}=0$4

$Y_{TF}=0$5

becomes, after substituting the vanishing-complexity relation,

$Y_{TF}=0$6

An equivalent split is

$Y_{TF}=0$7

The boundary conditions commonly imposed are

$Y_{TF}=0$8

at the center, and

$Y_{TF}=0$9

at the surface (Panotopoulos et al., 2024).

The closure relation has been combined with distinct equations of state. For exotic or dark-energy stars, one adopts the Extended Chaplygin Gas equation of state

$$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$0

while for strange quark stars an interacting equation of state inspired by perturbative QCD up to $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$1 is used. In both cases the formalism replaces a prescribed anisotropy law by a density-gradient-driven relation, and numerical integration proceeds by choosing a central density and integrating outward until $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$2 defines the radius (Rincon et al., 2023).

3. Stellar structure, stability, and oscillations

In compact-star applications, the formalism has been used to compare isotropic models with anisotropic vanishing-complexity models for the same global parameters. A standard fourth-order Runge–Kutta “shooting” is used: choose central $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$3 to achieve the desired $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$4, integrate outward, and enforce $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$5 to fix $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$6. For dark-energy stars with $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$7 and $$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$8, the anisotropic case is defined entirely by

$$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2-r^2d\Omega^2,$$9

so no extra ansatz is needed (Panotopoulos et al., 2024).

The resulting interior profiles exhibit systematic differences. The metric potentials remain regular, with $\rho(r)$0 and $\rho(r)$1. The radial pressure decreases monotonically from the center to the surface, while vanishing-complexity stars show a modestly higher $\rho(r)$2 at interior points than isotropic ones. The sound speed $\rho(r)$3 remains in $\rho(r)$4, so causality holds, and the relativistic adiabatic index

$\rho(r)$5

is $\rho(r)$6 everywhere, ensuring stability against small radial perturbations; the anisotropic $\rho(r)$7 is slightly lower than in the isotropic case (Panotopoulos et al., 2024).

Radial pulsations are treated through the Chandrasekhar–Bardeen formalism with extra $\rho(r)$8-terms,

$\rho(r)$9

$P_r(r)$0

with $P_r(r)$1, $P_r(r)$2, finite $P_r(r)$3, and regularity at the surface. Numerically one shoots in $P_r(r)$4 so that $P_r(r)$5 remains finite. All $P_r(r)$6, and hence $P_r(r)$7, are slightly lower in vanishing-complexity anisotropic models than in isotropic ones, and the large frequency separation $P_r(r)$8 tends, for large $P_r(r)$9, to a constant whose anisotropic value is $P_\perp(r)$0 lower (Panotopoulos et al., 2024).

$P_\perp(r)$1	anisotropic $P_\perp(r)$2 (kHz)	isotropic $P_\perp(r)$3 (kHz)
0	6.19	6.73
1	14.50	15.23
10	83.17	86.66

The same framework has also been used to compute quadrupolar gravitoelectric tidal Love numbers. Under $P_\perp(r)$4 and the Extended Chaplygin gas equation of state, the gravitoelectric Love number $P_\perp(r)$5 is obtained by solving the Riccati equation for $P_\perp(r)$6 with $P_\perp(r)$7 and matching at the surface. Compared with conventional prescriptions such as $P_\perp(r)$8, the vanishing-complexity formalism yields more compact configurations and tidal Love numbers that are typically larger for a given compactness $P_\perp(r)$9 (Rincón et al., 2024).

Quark-star and exotic-matter studies use the same logic together with standard acceptability criteria: positivity, monotonicity, causality, energy conditions, and $\Pi(r)=P_\perp-P_r$0. These works report that imposing $\Pi(r)=P_\perp-P_r$1 tends to produce physically acceptable configurations and, compared with hand-chosen anisotropy laws, does so with fewer phenomenological parameters (Rincon et al., 2023).

4. Prescribed nonzero complexity and gravitational decoupling

The formalism is not restricted to the condition $\Pi(r)=P_\perp-P_r$2. In gravitational decoupling, the complexity factor can be prescribed as a supplementary condition that closes the system generated by the minimal geometric deformation

$\Pi(r)=P_\perp-P_r$3

where the total energy–momentum tensor is

$\Pi(r)=P_\perp-P_r$4

A generalization of the Tolman IV complexity is introduced through

$\Pi(r)=P_\perp-P_r$5

with $\Pi(r)=P_\perp-P_r$6 dimensionless and $\Pi(r)=P_\perp-P_r$7 of dimension $\Pi(r)=P_\perp-P_r$8. This leads to a first-order differential equation for the deformation $\Pi(r)=P_\perp-P_r$9,

$$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$0

after which one reconstructs the total density and pressures (Andrade et al., 2021).

This program has been implemented using Tolman IV, Wyman IIa, Durgapal IV, and Heintzmann IIa seed solutions. For these like-seed models, regularity, causality, dominant energy condition, surface redshift, and matching to Schwarzschild are imposed. Additional compactness restrictions appear model by model: $$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$1 for Model 1, $$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$2 and $$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$3 for Model 2, $$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$4 for Model 3, and $$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$5 for Model 4 (Andrade et al., 2021).

The same work compares the resulting density ratios with observational inputs for SMC X-1 and Cen X-3. Using the reported compactness parameters, the models yield density ratios $$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$6 that indicate SMC X-1 is best described by Models 3 and 4, whereas Cen X-3 is well fitted by Models 2, 3 and 4. In this formulation, the complexity factor is not merely a diagnostic of a finished solution; it is an explicit generating function for new anisotropic interiors (Andrade et al., 2021).

5. Extensions to charge, modified gravity, black holes, and wormholes

The formalism has been extended far beyond uncharged spherical fluids in general relativity. In charged spherical systems, the Einstein–Maxwell equations and the orthogonal splitting of the Riemann tensor yield

$$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$7

$$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$8

with

$$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta,\qquad X_{\alpha\beta}={}^{*}R^{*}_{\alpha\gamma\beta\delta}u^\gamma u^\delta,$$9

The charged vanishing-complexity condition becomes

$Z_{\alpha\beta}$0

and the presence of the electromagnetic field decreases the complexity of the system (Sharif et al., 2018).

In modified-gravity cylinders, the formalism changes in a theory-dependent way. In energy-momentum squared gravity, the complexity factor is $Z_{\alpha\beta}$1, whose explicit form contains density inhomogeneity, anisotropy, $Z_{\alpha\beta}$2-corrections, and nonlinear $Z_{\alpha\beta}$3-terms. For the model $Z_{\alpha\beta}$4, the vanishing-complexity condition reduces to

$Z_{\alpha\beta}$5

and the inclusion of additional terms of this modified theory leads to a more complicated system (Sharif et al., 2022).

In $Z_{\alpha\beta}$6 gravity with $Z_{\alpha\beta}$7, the cylindrical analysis identifies $Z_{\alpha\beta}$8, rather than $Z_{\alpha\beta}$9, as the complexity factor. Its expression contains the Weyl scalar, effective anisotropy, and terms involving $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$00, $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$01, and their derivatives. This shifts the criterion of “zero complexity” away from the general-relativistic spherical prescription and ties it to the nonminimal curvature–matter couplings of the theory (Yousaf et al., 2020). In Palatini $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$02 gravity, by contrast, the spherical static case retains $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$03, but curvature contributions enter both the generalized TOV equation and the integral relation defining the vanishing-complexity balance (Yousaf, 2020).

The formalism has also been recast for black holes. In a static, spherically symmetric AdS black hole with

$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$04

the Newman–Penrose treatment gives

$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$05

or, for a charged AdS black hole,

$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$06

At the event horizon $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$07, where $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$08,

$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$09

This quantity is interpreted thermodynamically as the extra “Van der Waals” support needed at the horizon (Bargueño et al., 2022).

For traversable wormholes, the relevant line element is the Morris–Thorne form,

$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$10

and the complexity factor becomes

$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$11

A simple analysis shows that any continuous auxiliary function $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$12 entering the zero-complexity solution must generate a singularity in $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$13. Hence no regular traversable wormhole can satisfy $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$14, and traversable wormholes are intrinsically non-zero complexity objects (Bhattacharya et al., 2023).

Setting	Complexity scalar	Distinctive result
Charged spherical GR	$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$15	electromagnetic field decreases the complexity
Cylindrical EMSG	$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$16	extra $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$17 and derivative terms increase complexity
Cylindrical $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$18	$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$19	nonminimal curvature–matter couplings enter the definition
Palatini $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$20 spheres	$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$21	curvature terms modify the zero-complexity balance
AdS black holes	$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$22	horizon value is $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$23
Traversable wormholes	$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$24	no regular zero-complexity solution exists

6. Conceptual interpretation and limitations

Several recurring points delimit the scope of the formalism. First, zero complexity is not equivalent to a homogeneous and isotropic fluid. The defining relation

$$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$25

shows that an inhomogeneous and anisotropic configuration may still satisfy $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$26 through exact cancellation (Rincón et al., 2023).

Second, the complexity factor is not universal across all geometries and theories. In spherical general relativity and many of its extensions it is $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$27, whereas in static cylindrical $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$28 gravity the identified scalar is $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$29. In energy-momentum squared gravity, even the conditions $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$30 and $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$31 do not guarantee $$Y_{TF}(r)=8\pi\,\Pi(r)-\frac{4\pi}{r^3}\int_0^r \tilde r^3\,\rho'(\tilde r)\,d\tilde r, \qquad \Pi(r)\equiv P_\perp(r)-P_r(r),$$32 unless further geometric fine-tuning is enforced (Yousaf et al., 2020). This suggests that the formalism is best viewed as a structural diagnostic tied to a given curvature decomposition and gravitational theory, not as a single theory-independent observable.

Third, vanishing complexity removes the arbitrariness of many anisotropy prescriptions, but it does not by itself solve the stellar model. An equation of state, central data, and matching conditions remain necessary. In the compact-star literature this is precisely the practical advantage of the method: once the equation of state is fixed, the anisotropy is fully determined by the density gradient, avoiding arbitrary ansätze (Panotopoulos et al., 2024).

Finally, the formalism does not assign the same physical meaning to all spacetimes. In compact stars it organizes hydrostatic equilibrium, stability, radial oscillations, and tidal response. In black holes it acquires a horizon-based Newman–Penrose and thermodynamic interpretation. In wormholes it becomes a classification tool whose zero-complexity branch is obstructed by traversability conditions (Bargueño et al., 2022). A plausible implication is that the complexity factor formalism is less a single model than a family of geometrically related constructions centered on structure scalars and on the balance between inhomogeneity, anisotropy, and, where present, extra curvature or electromagnetic sectors.