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Weak-lensing Shear-Selected Galaxy Clusters from the Hyper Suprime-Cam Subaru Strategic Program: III. A precision cosmological sample enabled by optical confirmation

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Pith's one-line read This paper shows that a forced-mode red-sequence finder, fCAMIRA, measures photometric redshifts for 129 weak-lensing shear-selected galaxy clusters with bias ≈0.005 and scatter ≈0.008, and that these redshifts are precise enough for…

desk verdict Useful and honestly limited method paper, but the headline photo-z scatter is circularly validated and the cosmological forecast only tests the benign version of projection effects. read the letter →

arxiv 2608.02755 v1 pith:444OY52O submitted 2026-08-03 astro-ph.CO

classification astro-ph.CO
keywords weakgravitationallensinggalaxyclustersphotometricredshiftsredsequenceclustercosmologyshear-selectedprojectioneffectsHyperSuprime-Cam
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's goal is to give a clean redshift to each of the 129 weak-lensing shear-selected galaxy clusters found in the HSC-Y3 aperture-mass maps, so that cluster abundance can be modelled as a joint function of signal-to-noise and redshift instead of signal-to-noise alone. The tool it builds, fCAMIRA, forces the CAMIRA red-sequence finder to run at the fixed sky positions of the shear-selected peaks, calibrates a red-sequence galaxy model in a data-driven way (a metallicity–luminosity relation from X-ray-selected eFEDS clusters out to z≈1.3, then color-offset calibration with large spectroscopic samples), and ranks all optical counterpart candidates along the line of sight by their lensing scores. Against spectroscopic cluster redshifts, the resulting photometric redshifts show a mean bias of about 0.005 and scatter of about 0.008, with an outlier fraction of roughly 8% attributed to projection effects causing mis-identification of the optical counterpart. Mock abundance forecasts show that these systematics are negligible for HSC-Y3-like samples of about 220 clusters, which is why this matters: such a sample can now move from constraining only S8 to constraining Omega_m and sigma_8 simultaneously.

What carries the argument

fCAMIRA (forced-mode CAMIRA) is the algorithm that carries the argument: given a WL peak position, it builds two red-sequence richness maps in fine redshift bins—one, N_TI20, using the same truncated-isothermal filter that defines the aperture-mass maps, and the other, N_CAMIRA, using a spatial filter matched to a typical cluster size of R≈0.8 $h^{-1}$ Mpc—then finds redshift peaks at the WL center, locates optical cluster candidates within 7 arcmin, and assigns each candidate a lensing score S_opt = N_opt D_A(z_opt) D_A(z_opt, z_src)/D_A(z_src) with z_src≈1.3. Ranking candidates by S_opt assumes that richness scales linearly with halo mass (N_mem ∝ M), so the top-ranked counterpart is the halo that dominates the lensing signal; the ratio f_lens = S1/(S1+S2+S3) then quantifies how much of the WL signal comes from the best-matched halo, giving a per-cluster measure of projection contamination.

What would settle it

Take the 129 clusters, obtain complete spectroscopic redshifts for their candidate member galaxies (or for the roughly half with f_lens below 0.5), and compare the photo-z outlier fraction in the low-f_lens subset with the overall 8% rate; a significantly higher outlier fraction would indicate that the lensing-score ranking misidentifies projected halos as the true clusters.

Watch

Extended reading notes

Core claim

The central claim is that a forced-mode optical confirmation pipeline, built on a red-sequence model calibrated entirely from data, can assign photometric redshifts to shear-selected clusters with sub-percent accuracy and precision. Applied to the 129 clusters of the HSC-Y3 WL-selected sample, fCAMIRA achieves a mean redshift bias of ≈0.005 with scatter ≈0.008 relative to spectroscopic cluster redshifts, after correcting a mild bias; about 8% of the sample disagrees with positional cross-match redshifts by more than 0.15, which the authors attribute to projection effects along the line of sight rather than to failures of the red-sequence model. The authors further demonstrate, with mock cluster catalogs generated from the same WL selection, that a bias of 0.01, a scatter of 0.01, and an outlier fraction of 8% have negligible impact on cosmological constraints on Omega_m, sigma_8, and S8 for samples of about 220 clusters, while these systematics become comparable to statistical uncertainties for Stage-IV-like samples of about 2400 clusters.

Load-bearing premise

The method's ranking of which galaxy cluster corresponds to a detected lensing signal assumes that the number of red galaxies in a cluster grows in direct proportion to the cluster's mass; if that proportionality fails for a meaningful share of clusters, the assigned cluster distance and the claimed outlier rate could both be biased.

Editorial extensions

If this is right

  • The 129 HSC-Y3 shear-selected clusters now have uniformly measured photometric redshifts and richnesses from a single optical-confirmation procedure, providing the input needed to model cluster abundance as N(ν,z) rather than N(ν) alone.
  • The measured bias of about 0.005 and scatter of about 0.008 mean cluster-redshift systematics will not dominate parameter errors in HSC-Y3-like samples; the paper's forecasts show Ωm, σ8, and S8 each constrained to roughly 0.04 with about 220 clusters.
  • About 70 per cent of the sample has f_lens ≥ 0.5, and around 40 clusters have f_lens ≥ 0.9, indicating that the majority of WL detections are dominated by a single massive halo rather than by line-of-sight projections.
  • For Stage-IV-like samples of about 2400 clusters, a photo-z bias of 0.01, scatter of 0.01, and 8% outliers shift cosmological constraints by about 1 sigma, so next-generation surveys will need further improvement in cluster redshift measurements.
  • The lensing fraction f_lens is introduced as a practical tool to flag and potentially reduce line-of-sight contamination in shear-selected cluster samples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The f_lens ranking could be turned directly into a sample-selection cut: restricting to f_lens above a threshold such as 0.5 or 0.8 should reduce the projection-induced outlier fraction at the cost of sample size; the paper quantifies the f_lens distribution but does not re-run the cosmological forecasts under such a cut.
  • The same forced-mode confirmation logic should transfer to X-ray- or SZ-selected cluster samples, providing redshift and richness estimates on the same data-driven red-sequence model without entangling the selection functions of external catalogs.
  • The forecast that Stage-IV sample sizes will be sensitive to these redshift systematics suggests that future analyses will need to marginalize over an outlier plus redshift-scatter model or add spectroscopic calibration samples, rather than treating photo-z errors as negligible.
  • The agreement between fCAMIRA redshifts and spectroscopic BCG redshifts points to a low-cost validation path: a modest spectroscopic campaign targeting member galaxies of the low-f_lens clusters could directly test whether projection mis-identification is the dominant outlier channel.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper develops fCAMIRA, a forced-mode CAMIRA algorithm for optical confirmation of 129 weak-lensing shear-selected clusters from HSC-Y3. The red-sequence model is calibrated in two steps: a metallicity-luminosity relation from eFEDS X-ray clusters out to z≈1.3, and color offsets and intrinsic scatter from spectroscopic galaxies. fCAMIRA builds two richness maps, identifies redshift peaks at the WL centers, ranks optical counterparts by a lensing score (Eq. 11), and assigns the best-matched counterpart's refined redshift and richness. The paper reports a cluster photo-z bias of about 0.005 and scatter of about 0.008 relative to spectroscopic cluster redshifts, an outlier fraction of roughly 7–8% relative to external cross-matching, and uses mock forecasts to argue that these systematics have negligible impact on HSC-Y3-like N(ν,z) abundance cosmology.

Significance. If the claimed photo-z performance holds, this is a valuable ingredient for shear-selected cluster cosmology: it adds redshift information to the WL-selected sample, enabling N(ν,z) modelling that substantially improves constraints over N(ν) alone. The data-driven RS calibration is well conceived, the galaxy-level photo-z validation on a held-out spectroscopic sample is a real strength, and the planned public release of the confirmed cluster catalog is good practice. However, the cluster-level spectroscopic validation is not independent of the RS model, and the mock forecast does not reproduce the correlated nature of projection-induced misidentification. The central quantitative claims therefore need additional support before the paper can serve as the basis for precision cosmology.

major comments (3)
  1. [Section 5.1, Eq. (12)] The spectroscopic cluster redshift z_cl,spec used in Figures 8 and 9 is constructed from galaxies selected with w_mem > 0.1, where w_mem (Eq. 12) is built from the same calibrated RS model and evaluated at the fCAMIRA redshift z_cl that is being tested. If the RS colors have a systematic offset, or if the lensing score in Eq. (11) selects a foreground or background group, the selected "members" are preferentially the galaxies that agree with the trial redshift, so their spectroscopic redshifts will confirm z_cl by construction. The measured bias of about 0.005 and scatter of about 0.008 therefore largely quantify internal consistency rather than absolute accuracy. The only independent comparison in Figure 8, z_Chen25 versus z_cl,spec, shows roughly 3% scatter and outlier fractions of 7–15% depending on threshold, which is much larger than the claimed 0.008 scatter. Please re-derive z_cl,spec from members selected independently of the RS model, for example by taking all spectroscopic galaxies within a projected radius and a narrow velocity window, or by using an external cluster catalog, and re-report the bias, scatter, and outlier fraction for the same clusters.
  2. [Section 6, Eq. (14), Fig. 10] The forecast scenario "perturbed z_cl with outliers" adds Gaussian scatter and randomly shuffles cluster redshifts to produce an 8% outlier fraction. This is the benign, uncorrelated limiting case. In the real sample, projection-induced misidentification occurs because a rich, massive line-of-sight halo outranks the true counterpart in Eq. (11); the resulting redshift outliers are therefore correlated with richness, mass, and flens, and they affect the selection function in ways that random shuffling does not capture. The paper itself acknowledges at the end of Section 6 that projection effects in shear-selected clusters are more complex than in optically selected samples. As written, the mock does not validate the claim that the measured outlier population has negligible cosmological impact. Please inject outliers drawn from the actual flens-ranked distribution or from a mock containing projected halos, or explicitly restrict the conclusion to random contamination.
  3. [Section 4.2, Eq. (11)] The lensing score S_opt assumes N_mem ∝ M when ranking optical counterparts and choosing z_cl. The cited richness–mass relations are power laws with slope consistent with unity, but the paper does not test the sensitivity of the ranking to deviations from exact linearity. In particular, the roughly 16% of clusters with flens ≤ 0.4 are precisely the cases where the ranking must decide between candidates, and the self-consistent spectroscopic validation in Section 5.1 cannot detect a ranking failure because the members are selected using the same z_cl. A concrete test would be to repeat the counterpart selection with N_mem^α for α = 0.8 and 1.2 and report how many best-matched counterparts change and how the outlier fraction changes.
minor comments (5)
  1. [Section 7, Conclusions] "fCMAIRA" appears as a typo for fCAMIRA; the same section also contains "light-of-sight" where "line of sight" is intended.
  2. [Section 3.3] "potometric redshift" should be "photometric redshift".
  3. [Figure 2] The axis label "mRS(z, )" is incomplete; it should identify the plotted quantity, e.g., δm_RS as a function of rest-frame wavelength for the labeled redshifts.
  4. [Section 6] "the dimensionless aperture aperture mass peak" contains a duplicated word ("aperture aperture").
  5. [Abstract and Section 5] The abstract states "approximately 8% of the total sample exhibits redshift discrepancies greater than 0.15," while Section 5 reports 6.7% at the 0.15 threshold and 13–15% at thresholds of 0.10 and 0.08; the abstract and Section 7 should use the same definition and value as Section 5.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline cluster photo-z precision (bias ≈0.005, scatter ≈0.008) is validated against 'spectroscopic' cluster redshifts built from galaxies selected by the same RS model and the same z_cl being tested; the external cross-match comparison shows much larger scatter and outliers.

  1. self definitional [Section 5.1, Equation (12) and the paragraph defining z_cl,spec]
    "First, we obtain the member galaxy candidates of each individual shear-selected cluster by requiring the membership probability of w_mem > 0.1. ... With z_spec, we estimate the spectroscopic redshift z_cl,spec of the clusters using the bi-weighted location estimator ... w_mem = n(χ2,θ|z_cl) w_m(Δm) F(|θ−θ_opt|). ... We calculate the scatter of z_cl,phot with respect to the spectroscopic redshifts z_cl,spec, which is found to be at a level of ≲0.008."

    The benchmark z_cl,spec is not an external cluster redshift: it is the biweighted mean of spec-zs of galaxies admitted by w_mem>0.1, and Equation (12) evaluates w_mem at the very fCAMIRA photo-z z_cl being tested, using the same RS model. Galaxies whose colors disagree with z_cl are down-weighted or dropped before their spec-z can challenge the model. The reported scatter ≲0.008 therefore measures internal consistency of the RS-selected members with the model redshift, not absolute photo-z accuracy. The independent z_Chen25 comparison in the same figure shows roughly 3% scatter and 6-15% outliers, indicating that the self-selected validation misses the dominant error terms, especially projection-induced misidentification.

  2. other [Section 5.1, BCG agreement paragraph]
    "The excellent agreement is expected, because the cluster member identification relies on the RS model that is calibrated using the spectroscopic samples, which include bright galaxies as the BCGs."

    The paper itself states that the z_cl,spec versus z_bcg,spec agreement is expected from the shared RS calibration. This confirms that the Section 5.1 validation chain is an internal-consistency check rather than an independent benchmark: both redshift estimates are linked through membership weights built from the same calibrated model, so the agreement does not independently support the 0.008 scatter claim.

full rationale

Most of the pipeline is not circular. The RS model's metallicity-luminosity relation is calibrated on X-ray-selected eFEDS clusters, the color offsets on external spectroscopic samples, and the model is tested on a 10% holdout of spec galaxies (Figure 3: bias ~0.2%, scatter ~2.5%). The lensing-score ranking in Equation (11) rests on an explicit N_mem ∝ M assumption supported by external richness-mass calibrations; this is a modeling assumption with empirical support, not a constructed prediction. The circularity is concentrated in the cluster-level validation: z_cl,spec in Section 5.1 is the biweighted spec-z of galaxies selected by w_mem>0.1, with w_mem computed from the same RS model at the same z_cl being tested. This self-selection suppresses the measured scatter and prevents the test from detecting model-wide color offsets or misidentified counterparts; the paper even labels the BCG agreement 'expected' for this reason. The only partly external comparison (z_Chen25) shows much larger scatter and 6-15% outliers, and the abstract's claim of comparing fCAMIRA photo-z with direct positional cross-matching is not exactly what the body does, which compares z_cl,spec with z_Chen25. The cosmological-impact mock (Section 6) injects Gaussian scatter and random shuffles, the benign version of misidentification; it does not validate the true correlated projection error. Since the RS model has independent holdout support but the headline cluster precision claim is not independently benchmarked, partial circularity (6/10) is appropriate.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central redshift measurement depends on a richly calibrated red-sequence model with many fitted color offsets and intrinsic scatters, plus hand-chosen detection thresholds in fCAMIRA. The ranking principle assumes a linear richness-mass relation. The cosmological forecasts additionally assume perfect knowledge of the selection function, which the authors acknowledge is not yet available. No new physical entities are introduced.

free parameters (6)
  • A_metal, B_metal, sigma_color = -0.34 +/- 0.02, -0.17 +/- 0.01, 0.048 +/- 0.005
    These set the metallicity-luminosity relation and intrinsic red-sequence color scatter, fitted to stacked eFEDS cluster color-magnitude diagrams in Section 3.1.
  • Color offset polynomial coefficients and per-band intrinsic scatter per redshift bin = 5 coefficients plus 5 scatter values per bin, 100 bins
    Empirical calibration of the RS model to spectroscopic samples in Section 3.2; 10 parameters per redshift bin.
  • Cluster photo-z bias correction = 0.0047
    Measured relative to spectroscopic cluster redshifts in Section 5 and applied to all final photo-z values; this is a data-driven correction.
  • fCAMIRA peak-finding thresholds and z_src = N_mem > 5, S/N > 2, search radius 7 arcmin, z < 1.3, z_src = 1.3
    Chosen by hand in Section 4.2; the lensing score assumes sources at z_src = 1.3.
  • Mock WL mass-bias parameters = (A_WL, B_WL, gamma_WL, sigma_WL) = (1, 0.05, -0.1, 0.25)
    Assumed for the mock cluster catalogs in Section 6, based on prior X-ray cluster calibration, and not fitted in this paper.
  • Redshift outlier fraction in perturbed forecast = 8%
    Taken from the measurement in Section 5 and injected into the mock catalogs in Section 6 to assess cosmological impact.
assumptions (7)
  • domain assumption Richness scales linearly with halo mass, N_mem proportional to M
    Used in Section 4.2 to rank optical counterparts by lensing score; supported by power-law richness-mass relations but not exact per cluster.
  • domain assumption Baseline RS templates with z_f = 3, tau = 0.4 Gyr, Chabrier IMF, BC03 SEDs
    Section 3 adopts these to generate model SEDs before the empirical calibration; if wrong, the color-offset calibration must absorb the error.
  • domain assumption Absolute M* = -21.26 at z = 0.03 from Lan et al. 2016
    Used to anchor the characteristic magnitude of the RS model; external calibration from local cluster luminosity functions.
  • domain assumption eFEDS X-ray clusters are free from galaxy-property selection bias
    Section 3.1 relies on X-ray selection to calibrate the metallicity-luminosity relation without color-dependent selection.
  • domain assumption WL source selection and TI20 filter isolate clean shear-selected peaks
    Section 2.2 assumes the HSC-Y3 WL maps with conservative source cuts and core-excised TI20 filter produce a roughly 98% pure cluster sample.
  • domain assumption Mock forecasts assume perfect knowledge of the selection function C in equation 14
    Section 6 states the mock validation is optimistic; in practice the selection function must be modeled, which is deferred to future work.
  • domain assumption Halo mass function (Bocquet et al. 2016) and concentration-mass relation (Diemer and Kravtsov 2015)
    Standard astrophysical inputs used to build mock clusters and compute aperture masses in Section 6.

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Cite this review

Pith. "Pith review of Weak-lensing Shear-Selected Galaxy Clusters from the Hyper Suprime-Cam Subaru Strategic Program: III. A precision cosmological sample enabled by optical confirmation." pith.science (2026). https://pith.science/paper/444OY52O

@misc{pith2026260802755,
  author       = {Pith},
  title        = {Pith review of: Weak-lensing Shear-Selected Galaxy Clusters from the Hyper Suprime-Cam Subaru Strategic Program: III. A precision cosmological sample enabled by optical confirmation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/444OY52O}},
  note         = {Machine review of arXiv:2608.02755}
}
abstract

We develop fCAMIRA (forced-mode CAMIRA), a tool for optical cluster confirmation, and apply it to a sample of 129 weak-lensing (WL) shear-selected galaxy clusters identified in aperture-mass maps obtained from the Hyper Suprime-Cam Subaru Strategic Program Three-Year (HSC-SSP Y3) weak-lensing data. fCAMIRA is built upon the CAMIRA cluster-finding algorithm and relies on a red-sequence (RS) galaxy model that is calibrated in a data-driven way. The RS model adopts the metallicity-luminosity relation measured in this work using X-ray-selected clusters up to redshift $z\approx1.3$, followed by the calibration of color offsets using large spectroscopic samples. With the RS model, we build two types of galaxy richness maps, one obtained with a spatial filter matched to a typical cluster size of $R=0.8\,h^{-1}\,\mathrm{Mpc}$ and the other obtained with a fixed angular-size filter identical to that used in constructing the WL aperture-mass maps. The fCAMIRA algorithm utilizes these two richness maps, identifies all optical counterpart candidates along the line of sight of each shear-selected cluster, and measures the cluster photometric redshift from the highest-ranked counterpart. The ranking is determined by the fractional lensing contribution $f_{\mathrm{lens}}$ of each candidate. Using available spectroscopic cluster redshifts, we quantify the mean bias and scatter in the cluster photometric redshifts at levels of approximately 0.005 and 0.008, respectively, demonstrating excellent photometric-redshift performance. We compare the fCAMIRA photometric redshifts with estimates from direct positional cross-matching and find that approximately 8% of the total sample exhibits redshift discrepancies greater than 0.15. This outlier fraction is primarily attributed to projection effects, leading to the misidentification of the optical counterparts. (abridged)

Figures

Figures reproduced from arXiv: 2608.02755 by the authors.

Figure 1
Figure 1. — The calibration of the metallicity-luminosity relation using the eFEDS cluster sample. Left panel: The background-subtracted color-magnitude diagrams of cluster galaxies for the subsample at 0.26 ≲ 𝑧 ≲ 0.30. The red points and error bars show the mean colors and scatters obtained from Gaussian fits to the color distributions at fixed magnitudes, which are represented by the intensity of the underlying grey color. … view at source ↗
Figure 2
Figure 2. — The calibration of the color offsets of the RS model. The colored curves show the 1𝜎 confidence levels of the magnitude biases 𝛿𝑚resid as a function of rest-frame wavelength 𝜆rf at several redshifts, whose colors are indicated in the upper-right corner. and hence exhibit biases in colors with respect to those of observed galaxies. To mitigate these biases in this subsec￾tion, we further perform an empirical calibr… view at source ↗
Figure 3
Figure 3. — The validation of the photo-𝑧 derived using the calibrated RS model. Top panel: The comparison between the RS photometric redshifts 𝑧phot and spectroscopic redshifts 𝑧spec. The red dots are outliers defined by [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: — The flowchart of the fCAMIRA algorithm (see details in Section 4.2). The number parameter 𝑛 (𝑧) as a function of redshift 𝑧 is inferred from the chi-square estimate 𝜒 2 (𝑧) using the RS fitting for each galaxy in the catalog, following the procedure in Section 4.1. T…
Figure 5
Figure 5. Figure 5: — An unambiguous optical confirmation of a shear-selected cluster at 𝑧cl = 0.182. Left panel: The RGB image centered on the weak-lensing peak position 𝜽WL (blue square), overlaid with contours of the WL signal-to-noise ratio map 𝜈 (𝜽) in blue and the richness map 𝑁TI20…
Figure 6
Figure 6. Figure 6: — An optical confirmation of a shear-selected cluster at 𝑧cl ≈ 0.35 with projection effects quantified as 𝑓lens ≈ 0.75. The sample plotting style is used as in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: — The distribution of the photometric redshifts 𝑧cl and optical richnesses 𝑁mem of the 129 shear-selected clusters. The cluster redshifts 𝑧cl include the correction for the bias (see Section 5). The color scale indicates the lensing fraction 𝑓lens defined in equation (…
Figure 9
Figure 9. Figure 9: — The comparison between the photometric redshifts 𝑧cl,phot obtained with the fCAMIRA algorithm and the spectroscopic cluster redshifts 𝑧cl,spec. The cluster photometric redshifts 𝑧cl,phot include the correction for the bias (see Section 5). The data points are colored…
Figure 10
Figure 10. Figure 10: — The forecast cosmological constraints from the abundance modelling of mock shear-selected clusters. Left panel: The comparisons of the parameter constraints obtained with the HSC-Y3 like sample (𝑁cl ≈ 220) between the modelling 𝑁 (𝜈) without the cluster redshift in …
Figure 11
Figure 11. Figure 11: — The RS distribution of the passively evolving galaxies inside the eFEDS clusters in the redshift bins. Each subplot represents a stacked result in a redshift bin and is produced in the identical way as in the left panel of [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 12
Figure 12. Figure 12: — The stacked color distribution of the member galaxies in the eFEDS clusters. The open circles represent the distribution of the color offset ΔColor with respect to the predicted RS color normalized by the RS color scatter 𝜎RS. We include a systematic uncertainty (at…
Figure 13
Figure 13. Figure 13: — The cutout RGB images of the individual WL shear-selected clusters. Each subplot is made following the same manner as in [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: — Continued from [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: — Continued from [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: — Continued from [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.