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The core observation model and the definitions/units of the key Wiener-filter objects are not specified rigorously enough to verify correctness and enable reproduction (Sec. 2.2.1; also impacts Sec. 3.1–3.3). In particular, Eq. (2) is presented as $\mathbf{W}_\ell=(\mathbf{S}_\ell+\mathbf{N}_\ell)^{-1}\,\mathbf{s}_{\ell,{\rm tSZ}}$, with $\mathbf{s}_{\ell,{\rm tSZ}}$ described as a cross-spectrum vector, but the paper does not write an explicit multi-frequency mixing model (spectral responses, beam transfer functions, map units) that maps the simulated components into the channel maps. As written, it is unclear whether (i) $\mathbf{S}_\ell+\mathbf{N}_\ell$ equals the total data covariance $\mathbf{C}_{dd}(\ell)$ required by standard LMMSE, (ii) what exactly is included in “signal” vs “noise” bookkeeping (CMB/kSZ/tSZ/CIB/noise), (iii) whether spectra are computed on beam-convolved maps or after beam equalization/deconvolution, and (iv) whether the standardization of the y-map (Sec. 2.1) changes the meaning/units of $\mathbf{s}_{\ell,{\rm tSZ}}$ and the reconstructed $\hat y$.
Recommendation: In Sec. 2.2.1 (or an appendix), add an explicit harmonic-space data model, e.g. $d_{i,\ell m}=B_i(\ell)\,[g_i\,y_{\ell m}+a_i^{\rm CMB}c_{\ell m}+a_i^{\rm kSZ}k_{\ell m}+a_i^{\rm CIB}f_{\ell m}]+n_{i,\ell m}$, defining all coefficients (tSZ spectral factor $g_i$, assumed CMB/kSZ scaling, CIB convention), map units ($K_{\rm CMB}$ vs intensity), and how beams/bandpasses enter. Then define $\mathbf{C}_{dd}(\ell)$ and $\mathbf{C}_{dy}(\ell)$ and explicitly map them onto your $\mathbf{S}_\ell,\mathbf{N}_\ell,\mathbf{s}_{\ell,{\rm tSZ}}$ notation. Clarify whether spectra are computed with beams included, or after smoothing to a common beam. Finally, state explicitly how y-standardization is applied in the pipeline: whether $\mathbf{s}_{\ell,{\rm tSZ}}$ is built from standardized vs physical y, and how $\hat y$ is de-standardized before integrated-Y photometry (Sec. 2.3.2, 3.3.3).
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Treatment of CIB as “noise” vs its physical correlation with tSZ/halos is not disentangled, which matters for bias interpretation and for the claim that the observed $\sim$20–30% negative photometric bias is “Wiener attenuation” (Sec. 1, 3.3.3, 4). Because the CIB is correlated with large-scale structure and cluster environments, a correlated foreground can introduce bias terms beyond a simple (single-channel) $S/(S+N)$ intuition; additionally, if y–CIB correlations are present in the simulations, they may leak into $\mathbf{s}_{\ell,{\rm tSZ}}\equiv\mathbf{C}_{dy}(\ell)$ depending on how it is estimated. Without clarifying whether the simulation includes explicit tSZ–CIB cross-correlation and how it propagates into $\mathbf{C}_{dy}$ and $\mathbf{C}_{dd}$, it is hard to interpret whether the method is suppressing purely contaminating CIB, suppressing cluster-associated correlated emission in a mass/redshift/environment-dependent way, or partially fitting correlated CIB as “signal.”
Recommendation: Clarify in Sec. 2.1 and Sec. 2.2.1 whether the simulated skies include non-zero y–CIB cross-spectra (one-halo/two-halo terms), and explicitly report/plot $C_\ell^{y,{\rm CIB}_\nu}$ for at least a few $\nu$. Then, in Sec. 3.3.3, decompose the recovered-Y bias into (i) multiplicative attenuation from the filter response/transfer function and (ii) bias from correlated foreground structure (if present). If feasible, add a controlled test in Sec. 3: re-run with y–CIB correlation turned off (or scaled) to isolate how much of the gain (purity/scatter) and how much of the bias arises from covariance suppression vs true correlation with the target. Also revise the text in Sec. 3.3.3 to avoid the scalar $S/(S+N)$ statement for the multi-frequency case unless you explicitly state assumptions under which it reduces to that form.
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Covariance estimation and use ($\mathbf{S}_\ell$, $\mathbf{N}_\ell$, and/or total $\mathbf{C}_{dd}$) is under-specified and may be “simulation-trained” in a way that overstates robustness (Sec. 2.2.1, 3.1). The manuscript states auto/cross spectra are estimated from 500 simulated patches, but does not provide patch geometry/area, masking/apodization, multipole binning, pseudo-$C_\ell$ vs direct Fourier estimation, whether the covariance-estimation simulations are disjoint from those used for evaluation (risk of circularity), nor how spectra are smoothed/regularized. Given the method’s main selling point is exploiting detailed cross-frequency CIB covariance, sensitivity to finite-sample noise and to modest covariance mismatch is a central concern for real-data applicability.
Recommendation: Expand Sec. 2.2.1 and Sec. 3.1 with a reproducible covariance pipeline description: patch size/shape, number of patches, sky fraction, mask/apodization, $\ell$-range and binning, estimator used for spectra (and any debiasing), and any smoothing/fit applied to enforce positive-definiteness. State explicitly whether filter design uses simulations/patches disjoint from evaluation maps. Add at least one robustness study in Sec. 3.1–3.3: construct the MWF using one set of realizations/regions and apply it to another; and/or perturb the assumed covariances (e.g., scale CIB power by $\pm$10–20%, alter cross-band correlation coefficients, misestimate noise) and quantify impact on transfer functions (Sec. 3.2) and catalog purity/completeness and Y-bias/scatter (Sec. 3.3).
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The baseline ILC is not defined precisely enough to judge fairness/representativeness, and some reported ILC failure modes could reflect implementation choices rather than intrinsic limitations of modern ILC variants (Sec. 2.2.2, 3.2–3.3, 4). It is unclear whether the ILC is global vs $\ell$-dependent, whether it is localized (patch/needlet), how $\mathbf{C}_\ell$ is estimated, whether beams are equalized, whether CMB is nulled (often done in y-map construction), what regularization is used, and whether weights are derived from the same covariance-estimation setup as the MWF. Without these details, the comparison risks being interpreted as “MWF vs ILC in general,” which would be too broad.
Recommendation: In Sec. 2.2.2, give the explicit ILC weight formula and implementation domain (per-$\ell$, binned-$\ell$, patch-wise, needlet/NILC-like). Specify how $\mathbf{C}_\ell$ is estimated (same patches/masks/bins as MWF or not), how beams/noise are handled (common-resolution smoothing vs including beam factors in $\mathbf{C}_\ell$), and whether additional constraints are applied (e.g., CMB null, dust/CIB constraints). In Sec. 4, narrow the claim scope: state clearly that conclusions are relative to this specific baseline. If feasible, add at least one more competitive baseline (e.g., constrained ILC with CMB nulling and/or a scale-localized/needlet ILC) to better contextualize the gains.
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The simulated-observation specification is incomplete, making it hard to map results onto actual SO/Planck performance and to reproduce the study (Sec. 2.1). The manuscript does not provide a per-channel table of central frequencies, beam FWHM, pixelization/resolution, noise levels (e.g., $\mu$K-arcmin or noise $N_\ell$), sky coverage/masking, or whether noise is white/anisotropic/1/f. It is also unclear which astrophysical components are included/omitted (radio point sources, Galactic dust, beam/calibration uncertainties, bandpass mismatch), and this matters because covariance-based methods can be sensitive to missing/non-stationary components (Sec. 4).
Recommendation: Add a concise table (main text or appendix) in Sec. 2.1 listing for each channel: $\nu$, beam FWHM, map unit, pixel size (HEALPix $N_{\rm side}$ or flat-sky pixel scale), and noise model (amplitude and $\ell$-dependence). State sky geometry (full-sky vs tiles), masking/apodization, and how SO+Planck are combined spatially. Enumerate included components (CMB, tSZ, kSZ, CIB, instrumental noise) and explicitly list omitted ones (radio sources, Galactic dust, etc.). In Sec. 4, add a limitations paragraph assessing how these omissions could affect covariance estimation and cluster purity/bias in real data, and how the framework could incorporate additional components.
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Cluster detection/photometry methodology and selection-function characterization are not yet detailed/quantified enough for cosmology-facing interpretation (Sec. 2.3.2, 3.3.1–3.3.3). Key missing items include the explicit matched-filter definition (and which noise power spectrum enters), template/profile construction and size grid, SNR normalization, peak-finding/deblending, halo matching criteria, halo mass definition (e.g., $M_{500c}$), and mass/redshift ranges used. The fixed “3-pixel aperture” integrated-Y needs an angular-scale mapping and justification across varying cluster sizes; additionally, the paper reports a sizeable negative Y bias but does not present a practical calibration workflow (e.g., transfer-function correction or simulation-derived mapping) or quantify residual scatter after calibration.
Recommendation: In Sec. 2.3.2 and Sec. 3.3.1, write the matched-filter equations explicitly (Fourier/harmonic form), define the noise $C_\ell^{\rm noise}$ used in the denominator, and state whether the filter is recomputed separately for MWF vs ILC maps. Describe the cluster template (beam-convolved profile, parameter grid) and SNR normalization. Specify peak-finding/deblending and matching-to-halo rules (matching radius, handling of multiple matches). In Sec. 3.3.2–3.3.3, report completeness and purity as functions of true mass (and, if feasible, redshift) at the chosen SNR thresholds, not only vs recovered Y, and provide numerical scatter metrics (e.g., $\sigma_{\ln Y}$) with uncertainties. State the pixel size so the 3-pixel aperture corresponds to a well-defined angular scale; ideally test a size-adaptive aperture or profile-based flux estimator and verify conclusions are unchanged. Finally, outline and demonstrate a calibration scheme for the MWF attenuation/bias (e.g., fit $Y_{\rm true}$ vs $Y_{\rm rec}$ in simulations, or correct using the measured transfer function), and report the post-calibration scatter relevant for cosmological analyses.