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Ambiguity about what spectra enter $\rho_\ell$ (Eq. (2)) and how beam/transfer functions are handled (Sec. III.B–III.C; Eqs. (2)–(3); Figs. 1–4). It is not explicit whether $C_\ell^{aa}$, $C_\ell^{bb}$, $C_\ell^{ab}$ used in $\rho_\ell$ are (i) raw pseudo-spectra, (ii) mode-coupling-corrected but beam-convolved spectra, or (iii) beam-deconvolved spectra $\tilde C_\ell$. This choice directly affects the interpretation of high-$\ell$ behavior, the role of beam uncertainties, and the logic connecting $\ell_{\max}$ to the beam envelope.
Recommendation: In Sec. III.B/III.C, state unambiguously the full data model and processing sequence for the spectra used in Eq. (2): whether mode coupling is corrected (and how), whether pixel/window/filter transfer functions are applied, and whether nominal beams are deconvolved before forming $\rho_\ell$. Use a compact notation (e.g., $\hat C_\ell$ pseudo, $\bar C_\ell$ mode-decoupled, $\tilde C_\ell$ beam-deconvolved) and reference which one is used in each figure. If $\rho_\ell$ uses deconvolved spectra, add a short note on numerical stability when $B_\ell$ becomes small and any regularization/binning choices used to control noise amplification.
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Uncertainty/error-bar construction for $C_\ell$ and especially $\rho_\ell$ is under-specified, limiting interpretability and reproducibility (Sec. III.1–III.2; Figs. 1–3). The paper refers to “empirical split-cross scatter”, but does not define precisely how scatter is computed (over which cross combinations; with what weighting; per-bin or after smoothing), nor does it address that $\rho_\ell$ is a ratio of correlated estimators (numerator/denominator share sky, mask, and often noise/foreground fluctuations).
Recommendation: Expand Sec. III.1–III.2 to give a precise, reproducible recipe for uncertainties: (i) define the multipole binning (edges/widths and any within-bin weighting); (ii) specify how the $N$ split-cross spectra are combined into a final binned $C_b$ for each pair; (iii) define how the covariance of $C_b$ is estimated (scatter across cross combinations, jackknife/bootstrap over splits, or analytic Gaussian approximation), and then how this is propagated to $\mathrm{Var}(\rho_b)$ (either via error propagation with correlations stated, or via resampling that recomputes $\rho_b$ per realization). Clarify whether the plotted error bars include only split-to-split noise variability or also masked-sky sample variance, and briefly justify why sample variance largely cancels (or does not) in $\rho$.
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The algorithm/criteria used to define the stability windows $[\ell_{\min},\ell_{\max}]$ are currently heuristic and not quantitatively reproducible (Sec. III.4; Sec. IV.2–IV.7; Table I). Since Table I is a key deliverable, readers need to know what objective thresholds (S/N, beam-systematic tolerance, coherence threshold, slope change, etc.) were applied and how sensitive the windows are to reasonable variations of those thresholds.
Recommendation: In Sec. III.4, formalize the stability-window selection with explicit quantitative criteria. For example, define $\ell_{\min}$ via a minimum S/N on the relevant spectra (or on $\rho$), and define $\ell_{\max}$ using one or more thresholds such as: (a) $\rho$ dropping below a stated value (e.g., 0.99 for same-band, 0.98 for cross-band) or deviating from unity by $> X\sigma$; (b) beam-envelope-propagated fractional uncertainty exceeding $Y\%$; and/or (c) a foreground-mixture indicator exceeding a tolerance. Then, in Sec. IV (or an appendix), include a brief sensitivity test showing how $\ell_{\max}$ shifts when thresholds change (even $\pm100$–$200$ in $\ell$ is informative). Annotate Figs. 1–4 with vertical lines and numeric labels for $\ell_{\min}$, $\ell_{\max}$ per pair to make Table I directly traceable.
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Beam-shape envelope construction and its statistical meaning are not sufficiently specified, despite being central to the paper’s recommended $\ell$-cuts and systematics guidance (Sec. II.3; Sec. V.A; Fig. 6; Table II). The envelope is described as a maximum variation across beam splits, but the exact inputs (which splits), normalization, smoothing/regularization, and whether it is intended as an upper bound vs a $1\sigma$-like uncertainty are unclear.
Recommendation: In Sec. II.3 and Sec. V.A, explicitly document: (i) which beam-split products are used (elevation/PWV/time/detector subsets; ideally with DR6 product names/paths); (ii) how each split beam is normalized before comparison (e.g., $B_{\ell=0}=1$, fixed solid angle, or other); (iii) whether the per-$\ell$ maxima are smoothed or percentile-based to avoid noise spikes at high $\ell$; and (iv) whether the envelope includes only shape differences or also amplitude/normalization (calibration-like) differences. In Sec. VI.2 add a short “how to use” paragraph: what downstream users should assume about correlation of beam-shape errors across arrays/bands, and whether the envelope is best treated as a conservative bound or an effective uncertainty.
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Beam-to-coherence propagation (Eq. (6) and surrounding discussion) is currently not well justified and may be misleading without an explicit beam-error model (Sec. VI.2; Eq. (6)). For purely multiplicative, map-level beam transfer-function errors applied consistently to auto- and cross-spectra, $\rho_\ell$ cancels beam factors to first order; a residual impact generally requires split-dependent effective beams, non-factorizable beam systematics, or mismatched deconvolution/transfer functions across maps.
Recommendation: State explicitly the beam systematics model under which Eq. (6) is intended (e.g., different effective beams per split/map; beam errors that do not factorize as a single $B_\ell$; or anisotropic/elliptical effects that couple to masking/filtering). Provide a short derivation or a conservative bound with clearly stated assumptions. If Eq. (6) is meant as a heuristic upper limit rather than a first-principles result, label it as such and explain when beam effects cancel in $\rho_\ell$ to leading order.
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Pseudo-$C_\ell$ pipeline details and validation of the flat-sky approximation on a large sky fraction are insufficiently documented/quantified (Sec. III.1; Sec. VI.C). With $f_{\rm sky} \approx 0.46$ and $\ell_{\min}\sim 400$–$500$, curvature/mode-coupling/pixelization choices can matter for low-$\ell$ behavior and thus for window boundaries and coherence trends.
Recommendation: In Sec. III.1, specify the pseudo-$C_\ell$ implementation details needed to reproduce results (binning, apodization beyond “10′ cosine taper”, treatment of $M_{\ell\ell'}$ for cross-spectra, pixel window, filtering/transfer functions, and map projection/patching if any). In Sec. VI.C, add at least one quantitative validation: e.g., compare $\rho_\ell$ for a representative pair (such as pa4_f150$\times$pa5_f150) between the current flat-sky pipeline and a curved-sky pseudo-$C_\ell$ estimator over $400\leq\ell\leq1500$, or use simulations/literature bounds to show the induced bias is negligible relative to quoted deviations.
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Foreground-vs-instrument attribution of decoherence (especially for $90\times150$ high-$\ell$ and all 220-related pairs) remains largely qualitative; calibration/bandpass mismatch is also not explicitly bounded (Sec. IV.2–IV.6; Sec. VI.3). The paper’s “bigger picture” conclusions (why $\ell_{\max}$ differs by pair, and why 220 should be used mainly as a foreground diagnostic) would be more convincing with minimal quantitative cross-checks.
Recommendation: Add lightweight quantitative diagnostics to separate contributions to $|1-\rho_\ell|$: (i) a simple two-/three-component toy model (CMB + dust/CIB + tSZ/PS) showing expected $\rho_\ell(\ell)$ trends for $90\times150$ and $150\times220$ given plausible component fractions (anchored to Planck/SPT literature already cited); (ii) a mask-dependence test (e.g., slightly more/less aggressive Galactic/dust cut) demonstrating that 220-pair decoherence responds as expected if dust-driven; (iii) a null test using split-difference maps to bound noise/analysis-induced decoherence; and (iv) a short statement (with numbers, if available from DR6) of relative calibration uncertainty and how a small calibration/bandpass mismatch would manifest in $\rho_\ell$ and in the deconvolved spectral ratios. These do not need a full parametric fit, but should quantitatively support the dominant-effect claims used to motivate Table I.