Vanishing Complexity Condition Overview

• Vanishing Complexity Condition is a context-dependent concept where a designated complexity measure is forced to zero, encapsulating diverse technical constraints.
• It appears in fields such as relativistic stellar structure, lattice decoding, query complexity, and Holant theory, highlighting its multidisciplinary significance.
• Applications include balancing anisotropy with inhomogeneity in stellar models and reducing algorithmic runtime in decoding and complexity analysis tasks.

Searching arXiv for papers using the phrase and closely related technical usages. Searching arXiv for "vanishing complexity" and related terms. “Vanishing Complexity Condition” is not a standardized cross-disciplinary term. In current arXiv usage it names, or closely tracks, several distinct vanishing constraints imposed on a quantity interpreted as a complexity measure. In relativistic stellar structure it usually means the vanishing of Herrera’s complexity factor $Y_{TF}$; in triangulated and singularity categories it refers to vanishing of the DHKK complexity $\delta_t(X,Y)$; in lattice decoding it denotes a vanishing complexity exponent together with a vanishing performance gap; in query complexity it refers to the regime of vanishing-error approximate degree; and in Holant theory it is expressed through vanishing signatures or quantum-vanishing rather than by the phrase itself (Arias et al., 2022, Araya et al., 2023, Singh et al., 2011, Sherstov et al., 2019, Cai et al., 13 Sep 2025). This suggests that the term is best understood as a context-dependent label for a zero condition on a complexity-measuring scalar, exponent, or functional, rather than as a single invariant.

1. Terminological scope and conceptual range

Several papers explicitly separate the phrase from the underlying technical notion. In random simplicial complexes, the relevant object is not called a vanishing complexity condition but a vanishing condition for cohomology, namely $j$-cohom-connectedness. In Holant theory, the corresponding notions are vanishing signatures, Bi-Holant-vanishing, and quantum-vanishing or quantum-nonvanishing. In equivariant coarse homology, the named condition is $E$-vanishing. In approximate degree, the relevant phrase is vanishing-error approximate degree rather than vanishing complexity (Cooley et al., 2018, Cai et al., 13 Sep 2025, Bunke et al., 2017, Sherstov et al., 2019).

Across these settings, the common structure is a constraint of the form “a complexity-bearing quantity is forced to be zero.” What changes is the object that carries complexity. Depending on the field, that object may be a structure scalar $Y_{TF}$, a cohomology group, a categorical mass functional $\delta_t$, a complexity exponent $c(r)$, an approximation parameter $\epsilon$, or the operational effect of a tensor inside all admissible contractions.

A recurring source of ambiguity is that “complexity” itself is not uniform. In relativistic fluid theory it is tied to internal structure, especially anisotropy and inhomogeneity. In coding and decoding it is asymptotic runtime growth. In Holant theory it is an obstruction to a converse theorem. In algebraic topology and homological algebra, vanishing conditions often concern cohomology, $\operatorname{Tor}$, or $\operatorname{Ext}$, and the word “complexity” enters through Betti-growth, decomposition complexity, or DHKK complexity rather than through a single universal definition.

2. Relativistic stellar structure and Herrera’s complexity factor

In static, spherically symmetric relativistic stellar models, the phrase usually denotes the condition $\delta_t(X,Y)$0, where $\delta_t(X,Y)$1 is the structure scalar identified with the complexity factor in Herrera’s formalism. For the metric

$\delta_t(X,Y)$2

with local anisotropy $\delta_t(X,Y)$3, one form used in the literature is

$\delta_t(X,Y)$4

while a metric form is

$\delta_t(X,Y)$5

The vanishing condition $\delta_t(X,Y)$6 therefore implies

$\delta_t(X,Y)$7

so it acts as a non-local equation of state relating pressure anisotropy to density inhomogeneity (Arias et al., 2022, Ratanpal et al., 2024, Contreras et al., 2022).

The crucial interpretive point is that $\delta_t(X,Y)$8 does not require isotropy or homogeneity separately. The cited papers repeatedly emphasize that the homogeneous isotropic fluid is the simplest case, but anisotropic and inhomogeneous configurations may also satisfy $\delta_t(X,Y)$9 if the relevant contributions cancel. In this sense, zero complexity is a balance law, not the absence of all structural features.

Charged models are treated in the same spirit. In like-Durgapal charged configurations, the same vanishing condition is imposed after passing to effective thermodynamic variables, and an additional ansatz ties anisotropy to charge,

$j$0

This leaves the role of $j$1 unchanged: it still governs the compensation between inhomogeneity and anisotropy, now with electromagnetic contributions absorbed into the effective matter sector (Contreras et al., 2022).

3. Constructive protocols, rigidity results, and dynamical extensions

A major use of the vanishing complexity condition in general relativity is constructive. In gravitational decoupling via extended minimal geometric deformation, the total metric is written as

$j$2

Once a supplementary condition determines the radial deformation $j$3, the vanishing complexity condition yields

$j$4

This turns $j$5 into an integrable relation for the temporal deformation. Tolman VII and Tolman IV seeds are used to illustrate the method (Contreras et al., 2022).

In like-Durgapal models, the procedure is even more explicit. Taking

$j$6

the vanishing complexity condition gives

$j$7

respectively. In the charged extensions, the same $j$8 condition is retained and the extra unknown $j$9 is fixed through the anisotropy-charge relation (Contreras et al., 2022).

The same condition can also be highly restrictive. For static anisotropic matter satisfying the Karmarkar condition and conformal flatness, the simultaneous requirements

$E$0

force

$E$1

which is exactly the Schwarzschild interior solution. In that setting, vanishing complexity is not generative but rigidity-producing (Ratanpal et al., 2024).

The condition has also been extended beyond static Einstein gravity. In evolving spherically symmetric fluids, the dynamical complexity factor remains $E$2, and the condition $E$3 is studied together with quasi-homologous evolution $E$4. In Rastall gravity, the analogous scalar is denoted $E$5, and the vanishing condition $E$6 is imposed in the presence of the effective density

$E$7

The paper formulates three stellar models under this condition and concludes that the results are consistent with general relativity, with model 2 giving more suitable results in Rastall theory (Herrera et al., 2020, Naseer, 10 Jul 2025).

4. Homological, categorical, and coarse-geometric formulations

In singularity categories, the relevant complexity is the Dimitrov–Haiden–Katzarkov–Kontsevich complexity $E$8, defined as the infimum of sums $E$9 over exact-triangle constructions of $Y_{TF}$0 from shifts of $Y_{TF}$1. The paper studies

$Y_{TF}$2

and proves boundedness of $Y_{TF}$3 in several classes of rings. The underlying mechanism is extension-theoretic generation: if $Y_{TF}$4 is a direct summand of an object in $Y_{TF}$5, then the nonvanishing range of $Y_{TF}$6 is bounded on one side. In complete intersections, the cited earlier theorem gives $Y_{TF}$7 for all nonzero real numbers $Y_{TF}$8 (Araya et al., 2023).

In module theory, “complexity” refers to Betti-number growth. The paper on the depth formula and vanishing of cohomology defines

$Y_{TF}$9

and uses reducible complexity to control vanishing of $\delta_t$0 and $\delta_t$1. If one module has reducible complexity and finite Gorenstein dimension, eventual vanishing of $\delta_t$2 yields the depth formula. If $\delta_t$3 has finite complete intersection dimension and complexity $\delta_t$4, then $\delta_t$5 vanishing values of $\delta_t$6 in an arithmetic progression with odd step force all higher $\delta_t$7 to vanish (Sadeghi, 2012).

In equivariant coarse homology, the named vanishing condition is $\delta_t$8-vanishing: $\delta_t$9 The paper proves that spaces with $c(r)$0-equivariant finite decomposition complexity and discontinuous $c(r)$1-action are $c(r)$2-vanishing for any weakly additive equivariant coarse homology theory with weak transfers. In a compactly generated target, this gives actual vanishing of the forget-control fibre (Bunke et al., 2017).

5. Topological, algebraic, and enumerative vanishing criteria

In random simplicial complexes, the closest analogue is a cohomological vanishing condition rather than a literal vanishing complexity condition. For the downward-closure $c(r)$3 of the binomial random $c(r)$4-uniform hypergraph, the paper studies $c(r)$5-cohom-connectedness,

$c(r)$6

It proves a sharp threshold at

$c(r)$7

a hitting-time theorem at the disappearance of the last minimal obstruction $c(r)$8, and a Poisson limit for $c(r)$9 inside the critical window (Cooley et al., 2018).

In positive-characteristic algebraic combinatorics, the relevant vanishing condition is membership in a finite-degree Zariski closure: $\epsilon$0 For symmetric sets in the Boolean cube, the paper characterizes when every degree-$\epsilon$1 polynomial vanishing on a set of Hamming layers must also vanish on another layer. For a single layer $\epsilon$2, with $\epsilon$3, the closure is

$\epsilon$4

In a large-$\epsilon$5 regime it also proves that ordinary finite-degree closure agrees with a closure defined using only symmetric polynomials, making the vanishing condition computationally tractable (Srinivasan et al., 2021).

A different algebraic usage appears in the vanishing ideal of a toric set. There the paper studies the degree-complexity of the vanishing ideal $\epsilon$6 with respect to reverse lexicographic order and proves that in the reduced Gröbner basis every variable exponent is at most $\epsilon$7, while every homogeneous binomial of degree at most $\epsilon$8 lying in $\epsilon$9 already lies in the toric ideal $\operatorname{Tor}$0. For uniform clutters it also classifies the complete intersection property of $\operatorname{Tor}$1 by linear algebra (Sarmiento et al., 2011).

6. Computational and counting-theoretic usages

In MIMO lattice decoding, the phrase refers to a vanishing complexity exponent together with a vanishing performance gap. The relevant quantities are

$\operatorname{Tor}$2

and the complexity exponent

$\operatorname{Tor}$3

For LR-aided MMSE-preprocessed lattice sphere decoding with timeout, the paper proves

$\operatorname{Tor}$4

so the decoder achieves a vanishing gap to exact regularized lattice decoding with subexponential complexity (Singh et al., 2011).

In query complexity, the corresponding notion is vanishing-error approximate degree. For a Boolean or partial Boolean function,

$\operatorname{Tor}$5

and the paper studies the regime $\operatorname{Tor}$6. It proves

$\operatorname{Tor}$7

$\operatorname{Tor}$8

$\operatorname{Tor}$9

and uses this vanishing-error regime to improve a QMA lower bound for permutation testing (Sherstov et al., 2019).

In Holant theory, vanishing means operational indistinguishability from zero. A finite set $\operatorname{Ext}$0 is Bi-Holant-vanishing iff it is Bi-Holant-indistinguishable from the all-zero signatures, equivalently

$\operatorname{Ext}$1

The newer invariant-theoretic paper shows that vanishing signatures are the only true obstacle to a converse of the Holant theorem, while the earlier dichotomy paper classifies symmetric Boolean vanishing signatures into the two classes $\operatorname{Ext}$2 and $\operatorname{Ext}$3, with

$\operatorname{Ext}$4

In both papers, vanishing is not algebraic zero but zero behavior in every admissible tensor-network evaluation (Cai et al., 13 Sep 2025, Cai et al., 2012).

A related decision-theoretic usage appears for Gromov–Witten invariants on partial flag varieties. There the vanishing problem is formalized as

$\operatorname{Ext}$5

and nonvanishing is reduced to solvability of an explicit polynomial system. Under GRH, the paper proves that nonvanishing lies in $\operatorname{Ext}$6, hence the vanishing problem lies low in the polynomial hierarchy (Pak et al., 21 Aug 2025).

The phrase therefore has a stable formal role only within particular subfields. Its most literal and developed use remains Herrera’s $\operatorname{Ext}$7 condition in relativistic stellar modeling, but the broader literature uses closely related vanishing constraints whenever a chosen complexity measure—cohomological, categorical, algorithmic, geometric, or operational—is required to collapse to zero.