Aperture-Mass Skewness in Weak Lensing

• Aperture-mass skewness is a third-order weak-lensing statistic defined by applying compensated filters to the shear or convergence field to isolate non-Gaussian features.
• It employs specific filtering techniques—such as compensated top-hat and Gaussian-like filters—to extract pure E-mode signals and suppress long-wavelength modes.
• Recent surveys like DES and HSC leverage aperture-mass skewness to compress three-point cosmic shear data, offering robust cosmological constraints and diagnostic tests.

Aperture-mass skewness is a third-order weak-lensing statistic built from the aperture mass $M_{\rm ap}$, a compensated filtering of the projected mass distribution or, equivalently, of the observed shear field. It is used to quantify non-Gaussian structure in the projected matter field and to compress three-point cosmic-shear information into an E-mode-clean scalar observable. In the literature, the term refers either to the reduced skewness,

$$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$

or, more loosely, to the third-order aperture moment itself; because compensated filters imply $\langle M_{\rm ap}\rangle=0$, the third moment and the connected third cumulant coincide in the equal-mean case (Barthelemy et al., 2020, Secco et al., 2022, Wielders et al., 24 Sep 2025).

1. Observable definition and normalization

The aperture mass is defined by applying a compensated circular filter $U$ to the convergence $\kappa$,

$$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$

with compensation condition $$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$. The same observable can be written directly in terms of the tangential shear $\gamma_t$,

$$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$

where

$$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$

This shear-space form is central observationally because $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$0 can be extracted directly from the observed shear field, up to reduced-shear corrections (Barthelemy et al., 2020, Secco et al., 2022).

The compensation condition has two immediate consequences. First, it removes sensitivity to constant or very slowly varying convergence modes, rendering $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$1 invariant under mass-sheet transforms and insensitive to long-wavelength modes. Second, it makes $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$2 a natural E/B-separating observable: with the standard construction, $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$3 isolates the E-mode while the corresponding $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$4 captures the B-mode component (Barthelemy et al., 2020, Secco et al., 2022).

The skewness is defined in more than one way across the literature. In the large-deviation treatment of tomographic aperture statistics, the reduced skewness is written as $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$5 with $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$6 and $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$7 (Barthelemy et al., 2020). In DES Year 3, the reduced skewness is defined as

$$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$8

and its redshift evolution is measured tomographically (Secco et al., 2022). By contrast, the HSC-Y3 joint cosmology analysis measures the third moment $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$9 itself and explicitly states that the normalized skewness $\langle M_{\rm ap}\rangle=0$0 is not used in that analysis (Sugiyama et al., 19 Aug 2025).

2. Filters, transforms, and estimator constructions

Several filter families are used in aperture-mass work. One theoretical treatment adopts a compensated top-hat difference,

$\langle M_{\rm ap}\rangle=0$1

with Fourier-space window

$\langle M_{\rm ap}\rangle=0$2

This choice is analytically convenient in the large-deviation formulation because it turns $\langle M_{\rm ap}\rangle=0$3 into a slope of the projected density between two concentric apertures (Barthelemy et al., 2020).

A widely used alternative is the compensated “exponential” or Gaussian-like filter of Crittenden et al. (2002),

$\langle M_{\rm ap}\rangle=0$4

with Fourier window

$\langle M_{\rm ap}\rangle=0$5

This filter appears in both survey analyses and covariance calculations and sharply peaks near $\langle M_{\rm ap}\rangle=0$6, which sets the effective multipole probed by a given aperture radius (Secco et al., 2022, Wielders et al., 24 Sep 2025).

A distinct line of work establishes that aperture mass is formally identical to a wavelet transform at a specific scale. In that formulation, convolving $\langle M_{\rm ap}\rangle=0$7 with a compensated analysing wavelet is formally identical to computing $\langle M_{\rm ap}\rangle=0$8 at the corresponding aperture scale. The isotropic undecimated starlet transform is emphasized because it is compensated by construction, isotropic, compact in real space, well localized in Fourier space, and implemented without truncation. On a $\langle M_{\rm ap}\rangle=0$9 image, the starlet computation is reported to be $U$0 times faster at $U$1 pixels and $U$2 times faster at $U$3 pixels than brute-force aperture-mass filtering, while also delivering all dyadic scales simultaneously (Leonard et al., 2012).

For third-order measurements, direct map filtering is not the only route. DES Year 3 measures the four three-point shear “natural components” $U$4 and transforms them to aperture statistics through real-space kernels $U$5 and $U$6, yielding the pure E-mode third moment through

$U$7

HSC-Y3 follows the same general strategy, evaluating the aperture-mass third moment as a Riemann sum over measured 3PCFs on a $U$8 grid using TreeCorr’s toSAS and calculateMap routines (Secco et al., 2022, Sugiyama et al., 19 Aug 2025).

3. Bispectrum and large-deviation descriptions

In harmonic space, the aperture-mass third moment is a filtered projection of the convergence bispectrum. For equal apertures, one standard form is

$U$9

while for general triplets $\kappa$0 the three filter transforms need not be equal. Equal radii primarily probe equilateral bispectrum configurations, whereas non-equal triplets capture non-equilateral shapes and therefore carry additional cosmological information (Secco et al., 2022, Wielders et al., 24 Sep 2025).

The large-deviation theory formulation replaces explicit bispectrum modeling by a description of the projected density field in independent line-of-sight slices. In the Limber/small-angle and Born approximations, concentric disks in each slice obey a large-deviation principle governed by a rate function $\kappa$1, with the scaled cumulant generating function obtained by Legendre-Fenchel transform. The nonlinear mapping from linear to late-time projected densities is modeled with 2D cylindrical collapse,

$\kappa$2

with $\kappa$3 chosen to match tree-level skewness in cylinders. The projected aperture cumulants are then additive along the line of sight,

$\kappa$4

and the reduced skewness follows from $\kappa$5 (Barthelemy et al., 2020).

Within this framework, the slice variance is

$\kappa$6

with $\kappa$7, and the projected variance and third cumulant are

$\kappa$8

$\kappa$9

Non-linear covariances in the slice integrands are taken from Halofit or the Euclid emulator, while higher-order cumulants are supplied by large-deviation theory at tree level (Barthelemy et al., 2020).

A related LDP treatment of the one-point aperture mass quantifies the difference between using the observable reduced shear $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$0 and the common approximation $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$1. For source redshift $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$2 in an Einstein–de Sitter background, the paper states $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$3 and finds that the maximal correction to $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$4 is $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$5; for the fiducial choice $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$6, the skewness shift is only a few percent (Reimberg et al., 2017).

4. Tomography, nulling, and mode control

Tomographic aperture-mass skewness extends the statistic from a single source population to multiple redshift planes or bins. A central development is the use of the Bernardeau–Nishimichi–Taruya nulling transform, or BNT transform, to construct reweighted kernels $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$7 that vanish below the nearest source plane. For three consecutive source planes, the BNT weights satisfy

$$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$8

with $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)\,\kappa(\boldsymbol{\theta}-\boldsymbol{\theta}'),$$9, and yield observables that depend only on a finite range of redshifts (Barthelemy et al., 2020).

This nulling has two distinct effects. The compensated aperture filter already removes long-wavelength modes. BNT additionally suppresses sensitivity to very small-scale lenses at lower redshift, where deeply non-linear evolution and baryonic effects are most severe. The paper reports that, for the nulled aperture-mass PDF, the exponential tails and bulk are modeled to better than percent-level in the bulk and within $$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$0 out to $$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$1, and that post-Born corrections to $$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$2 and $$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$3 become sub-percent with BNT for $$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$4 a few arcmin and $$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$5–$$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$6 (Barthelemy et al., 2020).

The same study gives practical guidance for tomographic use. At least three tomographic source planes are required to construct each nulled kernel, and the resulting kernel should have moderate width. It recommends choosing $$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$7 where perturbation theory is accurate, for example $$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$8 arcmin, and notes that a roughly scale-independent bias in nulled $$\int d^2\boldsymbol{\theta}'\,U_\theta(|\boldsymbol{\theta}'|)=0$$9 from discrete shell thickness largely cancels in reduced skewness (Barthelemy et al., 2020).

Tomographic survey analyses implement this compression differently. DES Year 3 uses two broad source bins, measures both $\gamma_t$0 and $\gamma_t$1, and reports clear redshift evolution of the reduced skewness, with lower-$\gamma_t$2 skewness larger than higher-$\gamma_t$3 at fixed angle (Secco et al., 2022). HSC-Y3 uses four tomographic bins, all 20 source-bin triplets, and an emulator for the redshift-dependent skewness $\gamma_t$4 that is projected tomographically through the lensing kernels (Sugiyama et al., 19 Aug 2025).

5. Survey measurements and cosmological information

DES Year 3 provides high signal-to-noise measurements of three-point shear correlations and the third moment of the mass aperture statistic from the first 3 years of data. For equal-aperture moments over $\gamma_t$5 arcmin, the non-tomographic $\gamma_t$6 is detected with $\gamma_t$7, while the combined tomographic data vector reaches a total $\gamma_t$8. The reduced skewness shows clear redshift evolution; at a representative scale $\gamma_t$9 arcmin, the ratio $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$0–$$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$1 (Secco et al., 2022).

The DES analysis also establishes several diagnostic properties of the statistic. Real parts of the four $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$2 are detected with $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$3–$$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$4 across $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$5 bins, the equilateral odd-parity signal $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$6 is consistent with zero, and parity-violating aperture combinations such as $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$7 and $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$8 are consistent with the null hypothesis. These results support the interpretation that the measured third-order signal is astrophysical and gravitational in origin (Secco et al., 2022).

HSC-Y3 carries the statistic into a joint cosmology analysis with second-order shear. The aperture-mass skewness is measured for $$M_{\rm ap}(\theta_0)=\int d^2\boldsymbol{\theta}'\,Q_\theta(|\boldsymbol{\theta}'|)\,\gamma_t(\boldsymbol{\theta}_0-\boldsymbol{\theta}'),$$9 arcmin in eight logarithmic points, with two largest-$$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$0 points per panel discarded, and the cumulative signal-to-noise ratio across retained radii and all tomographic triplets is $$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$1. In a flat $$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$2CDM joint analysis of 2PCFs and aperture-mass skewness, the reported constraints are

$$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$3

with a figure-of-merit improvement of $$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$4 relative to the 2PCF-only case, from $$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$5 to $$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$6 (Sugiyama et al., 19 Aug 2025).

These survey results confirm the role of aperture-mass skewness as a compressed representation of third-order shear information. DES emphasizes detection, E/B validation, and qualitative agreement with mock catalogs (Secco et al., 2022). HSC-Y3 shows that the compressed third-order statistic can be incorporated into a full likelihood with tomographic cross-covariances, MOPED compression, emulator-based theory evaluation, and joint nuisance marginalization, while finding no significant intrinsic alignment signal in that particular analysis (Sugiyama et al., 19 Aug 2025).

6. Covariance, contaminants, and current limitations

The limiting factors for aperture-mass skewness are not confined to raw signal-to-noise. Shape noise, post-Born effects, reduced shear, magnification bias, intrinsic alignments, and covariance modeling all enter at a level relevant for precision weak-lensing analyses. In the tomographic large-deviation study, Gaussian shape noise is modeled with per-pixel variance $$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$7 and map-level variance

$$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$8

with $$Q_\theta(\vartheta)=-U_\theta(\vartheta)+\frac{2}{\vartheta^2}\int_0^\vartheta d\vartheta'\,\vartheta' U_\theta(\vartheta').$$9 for regular $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$00 or $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$01 for nulled $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$02. For Euclid-like settings, the paper concludes that shape noise often dominates single nulled bins, so joint analysis of all nulled bins is needed to recover information (Barthelemy et al., 2020).

The same work quantifies post-Born and reduced-shear corrections. For the compensated top-hat difference with BNT nulling, lens-lens coupling and geodesic-deviation corrections to $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$03 and $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$04 are found to be sub-percent at the scales considered, and the leading reduced-shear correction is also sub-percent with BNT, though it is typically a few percent without nulling (Barthelemy et al., 2020). A separate LDP analysis of the one-point aperture mass reaches a compatible conclusion: reduced-shear corrections are usually only a few percent, but can approach $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$05 in the most optimistic EdS-like case (Reimberg et al., 2017).

Covariance modeling has become a subject in its own right. An analytical non-tomographic model for the cross-covariance between $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$06 and $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$07 decomposes into three contributions,

$$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$08

where the cross-covariance separates into terms governed by the power spectrum, bispectrum, and the five-point convergence polyspectrum, called the tetraspectrum in that work. The tetraspectrum-driven term $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$09 dominates at all tested scale combinations. Analytical contours are reported to be systematically tighter than those from numerical covariance, with a combined figure of merit $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$10 of the numerical case, rising to $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$11 when small-scale information is excluded (Wielders et al., 24 Sep 2025).

Intrinsic alignment contamination is an additional third-order-specific concern. A TATT-based treatment of the mass-aperture skewness computes the GGI, GII, and III bispectrum components and finds that changing from NLA to TATT introduces differences of around $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$12 between predictions for typical order-unity IA parameters. The same work reports that higher-order TATT contributions can damp the two-point function while boosting the mass-aperture skewness, and argues that joint $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$13PCF+$$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$14PCF analyses can help break degeneracies between cosmology and IA parameters (Gomes et al., 14 Jan 2026).

A final limitation is numerical rather than formal. The tomographic large-deviation study finds that, although cross-cumulants of top-hat-smoothed convergence agree with simulations at the sub-percent level, direct measurements of $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$15 can vary significantly with map resolution in the same realization and differ from theory by up to $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$16 in some cases. Because $$S_3=\frac{\langle M_{\rm ap}^3\rangle_c}{\langle M_{\rm ap}^2\rangle^2},$$17 involves subtracting nearly equal contributions, it is unusually sensitive to small numerical artifacts from discrete lens planes, map resolution, and line-of-sight summation. This suggests that precision use of aperture-mass skewness requires simulations specifically validated for higher-order weak-lensing observables rather than only for two-point statistics (Barthelemy et al., 2020).