This manuscript studies generative modelling of *correlated* extragalactic foreground maps from FLAMINGO, focusing on the joint tSZ+CIB field (and extensions to tSZ+CIB+kSZ and multi-frequency CIB). The core technical contribution is a generator-agnostic calibration/post-processing operator that alternates (i) an $N \times N$ per-$\ell$-bin Fourier-space whitening/re-colouring (implemented via matrix square-roots/Cholesky; Eq. (12)) to match the full auto- and cross-power spectra, with (ii) pixel histogram (rank/CDF) matching (paired and ensemble variants) to enforce 1-point marginals. The paper benchmarks this operator when applied to three upstream generators—Scattering Transform / ScatCov microcanonical synthesis, a jointly trained DDPM, and a Gaussian random-field baseline—and finds that after calibration they become effectively indistinguishable on a wide battery of diagnostics (Secs. 4–5, 7.3). A second, conceptually important contribution is a “non-by-construction” ladder (Sec. 6) that quantifies what multi-channel ScatCov can recover *without* truth-based projections: e.g. the tSZ$\times$CIB pixel cross-correlation saturates around $\sim$60% (with soft $C_\ell$ rescaling improving some 2-pt agreement but leaving tail/peak limitations). The manuscript is technically rich and empirically broad; the main opportunities for improvement are (i) sharper framing of what is *guaranteed* vs *learned* (and what that implies about discriminative power of diagnostics), (ii) clearer treatment of paired histogram conditioning and patch-to-patch variability, and (iii) more explicit numerical/statistical details around Eq. (12), uncertainty quantification, and recommended “which recipe for which science goal” guidance.

• ✖ The FLAMINGO simulation suite used as ground truth in this work is a $\sim 10^8$ CPU-hour hydrodynamical computation, yet even at that cost it provides only $\mathcal{O}(10^3)$ patches at $5^\circ$ scale, far below the sample size required for percent-level covariance estimation on tSZ$\times$CIB cross-spectra or for simulation-based inference training on full-sky patches (Schaye et al. 2023).
• _Reference(s):_ Schaye et al. (2023)
• _Justification:_ Schaye et al. (2023) analyzes idealized cloud–wind interactions across hydrodynamics solvers and does not discuss the FLAMINGO simulation suite, its CPU-hour cost, the number of $5^\circ$ patches, or requirements for tSZ$\times$CIB covariance estimation or simulation-based inference. Thus the statement is not supported by the attached paper.

• ⚠ The scattering transform provides a translation-invariant, phase-sensitive statistical description of non-Gaussian random fields through successive wavelet modulus averages, as introduced by Mallat (2012) and Bruna & Mallat (2013), and has been used to construct generative models that preserve higher-order moments beyond power spectra.
• _Reference(s):_ Mallat, 2012, Bruna and Mallat, 2013
• _Justification:_ Mallat (2012) defines the scattering transform as successive wavelet modulus operations followed by averaging, yielding translation-invariant representations and proving Lipschitz stability to deformations. It also shows that expected scattering coefficients of stationary (non‑Gaussian) processes depend on higher-order moments and can discriminate processes with identical power spectra (Mallat, 2012, Sec. 4; abstract). Bruna and Mallat (2013) reiterate these properties, show texture discrimination using scattering, and employ a PCA-based generative classifier, but do not construct generative models that preserve higher-order moments; moreover, the transform explicitly removes phase via the modulus, not phase-sensitive (Mallat, 2012, Sec. 2.2; Bruna and Mallat, 2013, Sec. 2–3). Thus the claim is only partially supported.

• ✖ Allys et al. (2020) pioneered the use of scattering-transform-based microcanonical synthesis for astronomical foregrounds, showing that LBFGS-optimised synthesis of dust maps matching first- and second-order scattering coefficients ($S_1$ and $S_2$) produces statistically indistinguishable synthetic fields, and emphasising that such unsupervised synthesis avoids inheriting simulation bias because it conditions only on the observed realisation.
• _Reference(s):_ Allys et al. (2020)
• _Justification:_ Allys et al. (2020) develop Wavelet Phase Harmonics (WPH) statistics and use a microcanonical maximum-entropy synthesis optimized with L-BFGS-B for projected large-scale structure density fields, not astronomical dust foregrounds or the scattering transform $S_1$/$S_2$ (see Sec. II and IV). While they mention related scattering-transform work (e.g., Cheng et al. (2020) and other refs.), this paper does not pioneer scattering-based synthesis, does not match first- and second-order scattering coefficients, and does not claim avoidance of simulation bias by conditioning only on an observed realization—in practice they target WPH statistics estimated from a set of 30 simulation maps (Sec. IV.B). Thus the statement is not supported by Allys et al. (2020).

• ✖ Prabhu et al. (2025) extended scattering-transform work to a denoising diffusion probabilistic model (DDPM) baseline for correlated tSZ$\times$CIB foregrounds, demonstrating that a score-based diffusion model trained on Planck data recovers auto-spectra but degrades cross-component correlations, with their key metric $r_{\mathrm{tSZ}\times\mathrm{CIB}}$ reaching 0.91 for ST synthesis versus 0.71 for DDPM (their Table 3).
• _Reference(s):_ Prabhu et al. (2025)
• _Justification:_ Prabhu et al. (2025) train a DDPM on Agora simulations of correlated CIB and tSZ at 150 GHz, not on Planck data, and they do not discuss or extend scattering-transform (ST) methods. They report that the DDPM reproduces both auto- and cross-power spectra within sample variance (see §4.3, Fig. 4), rather than degrading cross-component correlations. There is no Table 3 or metric $r_{\mathrm{tSZ} \times \mathrm{CIB}}$ with values 0.91 (ST) vs 0.71 (DDPM). Hence the statement is not supported by Prabhu et al. (2025).

• ✖ The ScatCov operator used here follows Prabhu et al. (2025, Section 2.3) by computing cross-coefficient statistics between second-order scattering moments, enabling phase-sensitive correlation tracking across components beyond what single-channel scattering coefficients provide.
• _Reference(s):_ Prabhu et al. (2025, Section 2.3)
• _Justification:_ Prabhu et al. (2025, Section 2.3) does not describe a ScatCov operator or scattering transforms. The paper focuses on DDPMs and evaluates power spectra, pixel histograms, Minkowski functionals, and bispectrum/trispectrum (see Sections 3.1–3.2). There is no Section 2.3 detailing cross-coefficient statistics between second-order scattering moments or phase-sensitive correlation tracking. Therefore, the statement is not supported by the attached paper.

• ✖ Following Allys et al. (2020, Eq. 3), the ScatCov synthesis loss weights each coefficient by the inverse of its reference amplitude, $w_q = 1/|\mathrm{ScatCov}_q(x_{\mathrm{ref}})|$, to equalise the contributions of S1–S4 terms during LBFGS optimisation.
• _Reference(s):_ Allys et al. (2020, Eq. 3)
• _Justification:_ Allys et al. (2020, Eq. 3) defines the covariance of wavelet transforms $C_{\xi_1,\xi_2}(\tau)$ and does not discuss any synthesis loss weighting scheme. In the synthesis section, the loss $L = d(\Phi(\rho), \Phi(\tilde{\rho}))$ is referenced to external works for its exact form, with no mention of weights $w_q = 1/|\mathrm{ScatCov}_q(x_{\mathrm{ref}})|$ or equalising S1–S4 terms during L-BFGS optimisation. Therefore, the stated weighting is not supported by Allys et al. (2020, Eq. 3).

• ✖ Spectral-matched Gaussian initialisation, in which the starting field is drawn from a Gaussian random field with angular power spectrum $C_\ell$ matched to the reference patch, yields 5–10$\times$ faster convergence of scattering-transform synthesis compared to white-noise initialisation, consistent with the benchmarks reported by Allys et al. (2020, Table 1).
• _Reference(s):_ Allys et al. (2020, Table 1)
• _Justification:_ Allys et al. (2020, Table 1) lists fiducial cosmological parameter values, not synthesis benchmarks. The paper does not compare spectral-matched Gaussian vs white-noise initializations or report 5–10$\times$ convergence speedups for scattering-transform (or WPH) synthesis. Their syntheses start from Gaussian white noise (Sec. IV.B), and no convergence-time comparisons are provided.

• ✖ The joint Cholesky $C_\ell$-matching plus paired pixel-histogram matching procedure used here follows the standard combination employed in previous synthesis pipelines that aim to recover both 1-point and 2-point structure, as in Allys et al. (2020) and Cheng & Ménard (2020), but generalises it to multi-component ($N\times N$) cross-spectral covariance matrices.
• _Reference(s):_ Allys et al. (2020), Cheng & Ménard 2020
• _Justification:_ Allys et al. (2020) performs statistical syntheses via microcanonical maximum-entropy matching of Wavelet Phase Harmonics; it does not use a Cholesky $C_\ell$-matching or paired pixel-histogram procedure, nor discuss generalizing such a method to $N\times N$ cross-spectral covariances. The attached Cheng & Ménard 2020 document (about GRB synchrotron polarization) is unrelated and contains no such synthesis pipeline. Thus the claimed method and its generalization are not supported by the attached papers.

• ⚠ Prior microcanonical SC-matching work (Allys et al. 2020; Mousset et al. 2024) showed that maximum-entropy synthesis constrained only by scattering (or wavelet phase harmonic) statistics can recover power spectra, pixel PDFs, and Minkowski functionals to within sampling noise without any explicit post-processing, in contrast to the explicit calibration route taken in this paper.
• _Reference(s):_ Allys et al. (2020), Mousset et al. (2024)
• _Justification:_ Allys et al. (2020) used a microcanonical maximum-entropy synthesis constrained by WPH statistics (plus a small set of added low-pass scaling moments) and showed strong quantitative agreement with target fields: mean power spectrum within $\sim$5%, its standard deviation and correlations within $\sim$10%, pixel PDFs closely matched over several orders of magnitude, and Minkowski functionals within $\leq$0.5%, without any post‑processing. Mousset et al. (2024) extended scattering-based generative models to the sphere and likewise showed good reproduction of PDFs, power spectra and Minkowski functionals from SC constraints alone, though they note small residual oscillations in power spectra and do not claim agreement “within sampling noise.” Thus, the claim of recovery “to within sampling noise” solely from scattering/WPH constraints is stronger than what is explicitly demonstrated, making the statement only partially supported.

• ⚠ Mousset et al. (2024) and Allys et al. (2020) emphasise that microcanonical SC-matching synthesis is genuinely unsupervised—requiring only a single target field or ensemble and no paired simulation/truth labels—whereas the DDPM of Prabhu et al. (2025) is supervised and inherits any bias in its training simulations, and the calibration recipe presented here is semi-supervised because it depends on a fiducial truth ensemble for the $C_\ell$ matrix and 1-point CDF; consequently, on real-sky data the SC-matching route is the only one of the three that cannot bake simulation bias into the generator by construction.
• _Reference(s):_ Mousset et al. (2024), Allys et al. (2020), Prabhu et al. (2025)
• _Justification:_ Supported: Mousset et al. (2024) explicitly build microcanonical SC-based generative models from a single target full-sky map and emphasise no need for large training datasets (“our method holds in the limit case of a single data realisation”; Sect. 3.1, Conclusions). Allys et al. (2020) use a microcanonical maximum-entropy model constrained by WPH statistics estimated from a small ensemble and note it could be trained directly on observational data (Sec. IV, Conclusions), i.e., no paired labels. Supported in part: Prabhu et al. (2025) train a DDPM on Agora simulations and explicitly caution that any generative model’s fidelity depends on, and can inherit biases from, the training simulations (Sec. 5.2). Not supported: the claim that the DDPM is “supervised” is not stated—DDPM training here is generative (no labels). Also, no “semi-supervised calibration recipe” depending on a fiducial $C_\ell$ matrix and 1-point CDF is presented; the only post-hoc adjustment is a global variance rescaling (Sec. 3.2). Therefore the collective statement is only partially supported.

This section audits symbolic/analytic mathematical consistency (algebra, derivations, dimensional/unit checks, definition consistency).

Maths relevance: substantial

The paper contains a moderate set of central mathematical constructions: (i) definitions of scattering/ScatCov coefficients and a weighted $L_2$ synthesis loss; (ii) a per-$\ell$-bin Fourier-domain rescaling (‘soft $C_\ell$ match’); (iii) a multi-channel per-$\ell$-bin covariance whitening/recolouring transform (Eq. 12) claimed to enforce all auto- and cross-spectra; and (iv) several analytic interpretations (e.g., $-1/\sqrt{2}$ residual-correlation floor). The core linear-algebra of Eq. (12) is consistent, but several ‘by construction’ and ‘phase-preserving’ claims are overstated or depend on missing convergence/ordering details for the alternating histogram/spectrum projections.

### Checked items

• ✔ Scattering coefficient definitions (Eq. (1), Sec. 3.1, p.3)

• Claim: Defines first and second scattering-like moments using a wavelet $\psi_{j\ell}$: $S_{1,j}=\langle|x\star\psi_{j\ell}|\rangle_r$ and $S_{2,j}=\langle|x\star\psi_{j\ell}|^2\rangle_r$.
• Checks: notation consistency, definition consistency
• Verdict: PASS; confidence: medium; impact: minor
• Assumptions/inputs: $x(r)$ is an image/field; $\psi_{j\ell}$ is a wavelet at scale $j$ and orientation $\ell$, $\langle\cdot\rangle_r$ denotes spatial averaging over $r$.
• Notes: Formulas are syntactically consistent, but the index notation suppresses $\ell$ (orientation) in $S_{1,j}$ and $S_{2,j}$ despite explicit dependence via $\psi_{j\ell}$. This is a clarity/notation issue rather than an algebraic error.

• ⚠ Weighted ScatCov loss well-posedness (Eq. (2), Sec. 3.2, p.3)

• Claim: Uses an inverse-amplitude weighted squared error: $L_{\mathrm{SC}}(s)=\sum_q w_q (\mathrm{ScatCov}_q(s)-\mathrm{ScatCov}_q(x_{\mathrm{ref}}))^2$ with $w_q=1/|\mathrm{ScatCov}_q(x_{\mathrm{ref}})|$.
• Checks: dimensional/units sanity, well-posedness (division by zero), definition consistency
• Verdict: UNCERTAIN; confidence: medium; impact: moderate
• Assumptions/inputs: All reference coefficients $\mathrm{ScatCov}_q(x_{\mathrm{ref}})$ are nonzero or are regularised in implementation.
• Notes: As written, $w_q$ can diverge when $\mathrm{ScatCov}_q(x_{\mathrm{ref}})=0$ or be extremely large for tiny coefficients; the paper does not specify an $\epsilon$-floor or masking. This omission blocks a strict analytic claim that Eq. (2) defines a finite objective for all references.

• ⚠ Spectral-matched initialisation (Eq. (3), Sec. 3.2, p.3)

• Claim: Initial field $s_0$ is generated by shaping Gaussian noise in Fourier space: $s_0=\mathcal{F}^{-1}(\mathcal{F}(\epsilon)\odot\sqrt{P_{\mathrm{ref}}(\ell)})$.
• Checks: algebra sanity, definition sufficiency
• Verdict: UNCERTAIN; confidence: low; impact: minor
• Assumptions/inputs: $P_{\mathrm{ref}}(\ell)$ is the desired (2D) Fourier power as a function of $\ell$, FFT conventions and the mapping between $\ell$ and discrete $k$ are fixed.
• Notes: The structure ‘multiply Fourier modes by $\sqrt{P}$’ is correct in principle, but $P_{\mathrm{ref}}(\ell)$ is only specified up to proportionality ($P_{\mathrm{ref}}(\ell)\propto\ell(\ell+1)C_\ell/2\pi$) and the FFT/$C_\ell$ normalisation mapping is not stated, so an exact symbolic verification is not possible from the PDF alone.

• ✔ Joint loss decomposition (Eq. (4), Sec. 3.3, p.3)

• Claim: Defines $L_{\rm joint}$ as sum of per-channel ScatCov losses plus cross-correlation and sign penalties.
• Checks: definition consistency
• Verdict: PASS; confidence: high; impact: minor
• Assumptions/inputs: $L_{\mathrm{SC}}$ is defined as in Eq. (2), $\lambda_{\mathrm{cross}}$, $\lambda_{\mathrm{sign}}$ are nonnegative.
• Notes: No algebra to check beyond consistent composition.

• ✖ Correlation coefficient definition vs 'demeaned Pearson' (Eq. (5) and text, Sec. 3.3, p.3)

• Claim: Defines $r(a,b)=\langle ab\rangle/\sqrt{\langle a^2\rangle\langle b^2\rangle}$ and calls it the 'demeaned Pearson coefficient' appropriate for zero-mean fields.
• Checks: definition consistency, notation consistency
• Verdict: FAIL; confidence: high; impact: moderate
• Assumptions/inputs: Either $a,b$ are implicitly demeaned or fields are assumed zero-mean.
• Notes: Eq. (5) omits mean subtraction. It equals Pearson correlation only if $\langle a\rangle=\langle b\rangle=0$ (or if $a,b$ are implicitly demeaned, which is not stated in the equation). Later the paper reports nonzero means (e.g., Table 4), making the 'demeaned Pearson' description inconsistent unless an unstated preprocessing step is applied.

• ✔ Sign penalty definition (Eq. (6), Sec. 3.3, p.3)

• Claim: Penalises positive tSZ excursions at 150 GHz using $L_{\mathrm{sign}}=||\max(s_{\mathrm{tSZ}},0)||^2$.
• Checks: definition consistency
• Verdict: PASS; confidence: medium; impact: minor
• Assumptions/inputs: $\max$ is applied elementwise in pixel space, $||\cdot||$ denotes an $L_2$ norm over pixels.
• Notes: Mathematically consistent as a nonnegative penalty; the PDF rendering shows a malformed norm-squared symbol, but the intent is clear.

• ✔ Multi-channel operator mapping (Eq. (7), Sec. 3.4, p.3)

• Claim: Defines a multi-channel ScatCov operator $\Phi^{(2)}: \mathbb{R}^{2\times H\times W}\rightarrow\mathbb{R}^D$.
• Checks: type consistency
• Verdict: PASS; confidence: high; impact: minor
• Assumptions/inputs: $\Phi^{(2)}$ produces a flattened coefficient vector including cross-channel terms.
• Notes: Dimensional mapping is consistent with the described construction.

• ✔ Soft $C_\ell$ rescale amplitude factor (Eq. (8), Sec. 3.5, p.4)

• Claim: Defines $\alpha_b = \sqrt{\langle|\hat{t}_k|^2\rangle_{k\in b}/\langle|\hat{g}_k|^2\rangle_{k\in b}}$ and applies $\hat{g}'_k=\alpha_b(k) \hat{g}_k$.
• Checks: algebra correctness, normalisation logic
• Verdict: PASS; confidence: high; impact: moderate
• Assumptions/inputs: Bins $b$ partition Fourier modes, denominators are nonzero (or regularised).
• Notes: This scaling enforces equality of average $|\hat{g}'|^2$ to average $|\hat{t}|^2$ within each bin by construction.

• ✖ Soft $C_\ell$ rescale 'spatial properties preserved' claim (Text after Eq. (8), Sec. 3.5, p.4)

• Claim: Because Fourier phases are preserved exactly, 'every spatial coherence property' (including extreme-pixel positions, edges, Minkowski $M_1$/$M_2$) is preserved by the rescale.
• Checks: logical implication check
• Verdict: FAIL; confidence: high; impact: moderate
• Assumptions/inputs: Phase preservation is taken to imply invariance of spatial/topological features.
• Notes: Preserving Fourier phases does not, in general, preserve extrema locations, edges, or Minkowski functionals when amplitudes are changed as a function of $\ell$ (this is a nontrivial filtering operation). The statement should be weakened/qualified.

• ✔ Cosine schedule definition (Eq. (9), Sec. 3.6, p.4)

• Claim: Defines $\bar{\alpha}_t$ via a cosine schedule $\bar{\alpha}_t=f(t)^2/f(0)^2$ with $f(t)=\cos\left(\frac{(t/T)+s}{1+s}\cdot \frac{\pi}{2}\right)$.
• Checks: algebra sanity, range sanity
• Verdict: PASS; confidence: medium; impact: minor
• Assumptions/inputs: $t\in[0,T]$.
• Notes: The formula is self-consistent and yields $\bar{\alpha}_0=1$. No internal inconsistency found.

• ✔ DDPM reverse sampling step and coefficient discussion (Eq. (10) and §3.6.1, p.4)

• Claim: Reverse update uses coefficient $\beta_t/\sqrt{1-\bar{\alpha}_t}$ on $\epsilon_\theta$; prior incorrect use of $\sqrt{\beta_t/(1-\bar{\alpha}_t)}$ causes an error factor $1/\sqrt{\beta_t}$.
• Checks: algebra between shown steps
• Verdict: PASS; confidence: high; impact: minor
• Assumptions/inputs: $\sigma_t=\sqrt{\beta_t}$ as stated.
• Notes: The ratio of incorrect to correct coefficient is $(\sqrt{\beta_t})/(\beta_t)=1/\sqrt{\beta_t}$, matching the text.

• ✔ DDPM training objective (Eq. (11), Sec. 3.6.2, p.4)

• Claim: Noise-prediction MSE objective: $\mathbb{E}||\epsilon-\epsilon_\theta(\sqrt{\bar{\alpha}_t}~x_0 + \sqrt{1-\bar{\alpha}_t}~\epsilon, t)||^2$.
• Checks: notation consistency
• Verdict: PASS; confidence: high; impact: minor
• Assumptions/inputs: $\epsilon$ is standard normal noise.
• Notes: Self-consistent as a denoising objective; no internal algebra to verify.

• ✔ Cross-spectral covariance definition (§4.1.1, p.5)

• Claim: Defines per-bin $2\times2$ covariance $C_{ab}(\ell)=\langle\mathrm{Re}(\hat{F}_a \hat{F}_b^*)\rangle_{k\in\ell}$.
• Checks: definition consistency
• Verdict: PASS; confidence: medium; impact: minor
• Assumptions/inputs: All maps are real-valued so Fourier coefficients satisfy Hermitian symmetry, the statistic of interest is the real cross-spectrum.
• Notes: Using $\mathrm{Re}(\cdot)$ yields a real-symmetric covariance consistent with later use of real matrix square roots. The paper’s subsequent ‘matches $C_{ab}$’ claims are with respect to this definition.

• ✔ Joint Cholesky $C_\ell$-match enforces target covariance (Eq. (12), Sec. 4.1.1, p.5)

• Claim: Applying $\hat{F}_{\rm new}(k)=C_{\rm ref}(\ell)^{1/2}C_{\rm gen}(\ell)^{-1/2}\, \hat{F}_{\rm gen}(k)$ enforces matching of $C_{\rm tSZ,\ell}$, $C_{\rm CIB,\ell}$, and $C_{\mathrm{tSZ} \times \mathrm{CIB},\ell}$ per bin.
• Checks: linear algebra consistency, constraint/normalisation check
• Verdict: PASS; confidence: high; impact: critical
• Assumptions/inputs: $C_{\rm gen}(\ell)$ is invertible (SPD) and the same sample covariance used for whitening, matrix square roots satisfy $C^{1/2}C^{1/2}=C$ (or, for Cholesky, $C^{1/2}(C^{1/2})^T=C$), the transform is applied uniformly to all modes in the bin.
• Notes: For the paper’s real-part covariance, $C' = A C_{\rm gen} A^T$ with $A = C_{\rm ref}^{1/2} C_{\rm gen}^{-1/2}$ gives $C'=C_{\rm ref}$ exactly. This is the main mathematically solid ‘by construction’ part (for 2-point spectra).

• ✖ Eq. (12) described as 'phase-preserving' (Abstract; §4.1.1, p.5; §5.2, p.11)

• Claim: The joint $N\times N$ Cholesky $C_\ell$-match is described as phase-preserving and preserving generator phase information.
• Checks: logical implication check, property verification
• Verdict: FAIL; confidence: high; impact: moderate
• Assumptions/inputs: ‘Phase-preserving’ is intended per-channel in Fourier space.
• Notes: For $N>1$, $\hat{F}_{\rm new}=A \hat{F}_{\rm gen}$ with a generally non-diagonal real matrix $A$ changes the complex argument of each channel’s Fourier coefficient in general. Only the $N=1$ (or diagonal $A$) case preserves per-channel Fourier phases. A weaker true statement is that it does not mix different $k$-modes (no convolution across $k$), only mixes channels at fixed $k$.

• ⚠ Pixel cross-correlation follows from matching spectra (§4.1.1 and §4.2, pp.5–6)

• Claim: Matching per-bin auto- and cross-spectra implies the pixel-level correlation $r_{\mathrm{tSZ}\times\mathrm{CIB}}$ matches truth (as a band integral).
• Checks: definition consistency, missing-assumption identification
• Verdict: UNCERTAIN; confidence: medium; impact: moderate
• Assumptions/inputs: Fields are mean-zero or $r$ is computed on demeaned fields, pixel covariance equals an integral/sum over the cross-spectrum across the matched band and windowing is consistent.
• Notes: The implication holds cleanly for demeaned fields (covariance determined by cross-spectrum) and if the same spectral weighting/windowing is used. Because Eq. (5) is not explicitly demeaned and later means are nonzero, the connection between matched cross-spectrum and reported ‘Pearson $r$’ is not fully verifiable from the PDF alone.

• ✔ Histogram match implies matching all 1-point statistics (§4.1.1, p.5)

• Claim: Rank-preserving paired histogram match forces gen pixel CDF to equal truth CDF, thus matching all 1-point statistics (mean, variance, skewness, kurtosis, extrema).
• Checks: probability/CDF logic
• Verdict: PASS; confidence: high; impact: moderate
• Assumptions/inputs: Histogram match is performed on the full pixel set per channel, without ties/pathologies.
• Notes: If the final output has undergone the rank-replacement step, its empirical CDF equals the target empirical CDF exactly, hence all empirical 1-point moments and order statistics match exactly.

• ⚠ Alternating histogram match and Cholesky gives simultaneous exact 1- and 2-point matching (§4.1.1, p.5; ‘by construction’ claims in Abstract and §5.6)

• Claim: Alternating histogram matching and Eq. (12) ‘locks’ both the full pixel CDF and the per-bin spectra/cross-spectra simultaneously.
• Checks: logical completeness, missing derivation/proof identification
• Verdict: UNCERTAIN; confidence: high; impact: critical
• Assumptions/inputs: The alternating-projection procedure converges to a fixed point satisfying both constraints, the final-step ordering is specified.
• Notes: Each operation generally breaks the constraint enforced by the other (histogram match changes spectra; Cholesky recolouring changes the histogram). Without (i) a specified final operator order and (ii) a convergence/fixed-point argument, the claim ‘exact by construction’ for both simultaneously cannot be verified analytically from the PDF.

• ✔ Residual-correlation algebraic floor (§4.2.1, p.6)

• Claim: If gen is independent of truth with equal variance, then $\mathrm{corr}(\mathrm{gen}-\mathrm{truth},\mathrm{truth})=-1/\sqrt{2}$.
• Checks: algebra between shown steps
• Verdict: PASS; confidence: high; impact: minor
• Assumptions/inputs: $\mathrm{corr}(\mathrm{gen},\mathrm{truth})=0$ and $\mathrm{Var}(\mathrm{gen})=\mathrm{Var}(\mathrm{truth})$.
• Notes: Correct: $\mathrm{Cov}(\mathrm{res},\mathrm{truth})=-\mathrm{Var}(\mathrm{truth})$, $\mathrm{Var}(\mathrm{res})=2\mathrm{Var}(\mathrm{truth}) \Rightarrow \mathrm{corr} = -1/\sqrt{2}$.

• ✔ Minkowski functional definitions and $M_0$ matching argument (§4.4, p.9)

• Claim: Defines $M_0(u)=\langle \mathbb{I}[x>u]\rangle$, $M_1(u)=\langle |\nabla x|\mathbb{I}[x>u]\rangle$, $M_2(u)=\langle\chi\rangle$, and argues $M_0$ overlays truth exactly after histogram match.
• Checks: definition consistency, logical implication check
• Verdict: PASS; confidence: medium; impact: minor
• Assumptions/inputs: Threshold $u$ is in absolute units (or fixed in $\sigma$ units) and the histogram match is to the truth distribution in those units.
• Notes: Given exact CDF matching to truth in the same units, the area fraction above any threshold $u$ matches truth, so $M_0(u)$ matches. (This does not imply $M_1/M_2$ match, consistent with the paper’s discussion.)

• ✔ Triple loss function with sign constraint (Eq. (13), Sec. 5.3, p.12)

• Claim: Extends joint loss to three components with two correlation penalties and a positivity-of-mean penalty for CIB: $w_{\mathrm{sign}}\cdot\max(-\bar{x}_{\mathrm{CIB}},0)^2$.
• Checks: definition consistency
• Verdict: PASS; confidence: high; impact: minor
• Assumptions/inputs: $\bar{x}_{\mathrm{CIB}}$ is the patch mean; $\max$ applies to a scalar.
• Notes: Penalty is mathematically consistent and enforces nonnegative mean when weighted strongly.

• ✔ $N\times N$ generalisation of Eq. (12) (§5.3 and §5.4, pp.12–13)

• Claim: Eq. (12) extends to $3\times3$ and $4\times4$ by using $C_{\rm ref}(\ell)^{1/2}C_{\rm gen}(\ell)^{-1/2}$ with $C$ matrices of matching dimension.
• Checks: linear algebra generalisation check
• Verdict: PASS; confidence: high; impact: moderate
• Assumptions/inputs: $C_{\rm gen}(\ell)$ is SPD/invertible for each bin, a valid matrix square root is used.
• Notes: The covariance-matching argument generalises directly to any $N$ given invertibility and a consistent square-root definition.

### Limitations

• Audit is restricted to the provided PDF text/images; no code or supplementary material was consulted.
• Several key claims are empirical (‘numerical precision’, ‘iteration converges in 6 cycles’). This audit cannot validate those numerically and only assesses whether they are guaranteed by the stated mathematics.
• Some formulas (e.g., FFT/$C_\ell$ normalisations, ScatCov coefficient definitions, handling of zero coefficients) are underspecified in the PDF, preventing full analytic verification.

This section audits numerical/empirical consistency: reported metrics, experimental design, baseline comparisons, statistical evidence, leakage risks, and reproducibility.

Out of 23 candidate numeric checks, 15 passed, 2 failed, and 6 were uncertain (mostly due to missing tabulated inputs for the intended recomputations). The two failures concern (i) a unit/pixel-scale consistency check for patch size vs pixel resolution, and (ii) a rounding/precision claim about matching to four decimal places for a median ratio example. Other recomputed percentages/ratios (held-out recovery percentages, skew/kurtosis ratios, algebraic constant $-1/\sqrt{2}$, and various percent recoveries) were numerically consistent within stated tolerances.

### Checked items

• ✔ C01_train_aug_count (p.3, end of §2 DATA)

• Claim: 8-fold augmentation yielding 1600 training pairs from $N_{\rm train} = 200$ paired patches.
• Checks: multiplication
• Verdict: PASS
• Notes: Exact integer arithmetic.

• ✖ C02_patch_pixels_area_consistency (p.2, §2 DATA)

• Claim: $5^\circ\times5^\circ$ patches at $256\times256$ pixels correspond to 5 arcmin pixel scale.
• Checks: unit_consistency
• Verdict: FAIL
• Notes: $60\times5/256=1.1719$ arcmin; inconsistent with 5 arcmin unless additional downsampling/definition. Flag for check.

• ✔ C03_patch8_percentile_from_1523 (p.3, §2 DATA (Patch 8 description))

• Claim: Patch 8 sits in the top 3.8% on deepest decrement and the top 10.8% on $\sigma$ (in $N_{\rm patch} = 1523$).
• Checks: percentage_to_count
• Verdict: PASS
• Notes: $0.038\times1523\approx58$, $0.108\times1523\approx165$.

• ✔ C04_bandavg_from_bins_ST_tSZ (p.6, Table 2)

• Claim: Band-average [500,6000] for ST+jm tSZ is $1.088 \pm 0.040$, given per-band values 1.128, 1.115, 1.062 over contiguous bins.
• Checks: weighted_average_by_interval
• Verdict: PASS
• Notes: Width-weighted average across [500,1500],[1500,3000],[3000,6000] gives 1.08845, consistent with 1.088 (noting averaging-definition caveat).

• ⚠ C05_bandavg_from_bins_ST_CIB (p.6, Table 2)

• Claim: Band-average [500,6000] for ST+jm CIB is $1.010 \pm 0.014$, given per-band values 0.996, 1.022, 1.009.
• Checks: weighted_average_by_interval
• Verdict: UNCERTAIN
• Notes: Missing bin ranges/ratios or reported total for weighted average.

• ⚠ C06_bandavg_from_bins_DDPM_tSZ (p.6, Table 2)

• Claim: Band-average [500,6000] for DDPM+jm tSZ is $1.065 \pm 0.123$, given per-band values 1.141, 1.079, 1.034.
• Checks: weighted_average_by_interval
• Verdict: UNCERTAIN
• Notes: Missing bin ranges/ratios or reported total for weighted average.

• ⚠ C07_bandavg_from_bins_DDPM_CIB (p.6, Table 2)

• Claim: Band-average [500,6000] for DDPM+jm CIB is $1.013 \pm 0.011$, given per-band values 1.027, 1.013, 1.008.
• Checks: weighted_average_by_interval
• Verdict: UNCERTAIN
• Notes: Missing bin ranges/ratios or reported total for weighted average.

• ✔ C08_skew_kurt_ratio_claims (p.8, §4.3 PDF Statistics)

• Claim: For tSZ, skewness ratio is 0.98 (ST) and 0.98 (DDPM); excess-kurtosis ratio is 0.96 (ST) and 0.94 (DDPM).
• Checks: ratio_recompute
• Verdict: PASS
• Notes: Recomputed ratios: skew ST/truth=0.9803, DDPM/truth=0.9777; kurt ST/truth=0.9599, DDPM/truth=0.9393.

• ✔ C09_min_core_recovery_percent (p.8, §4.3 PDF Statistics)

• Claim: Deepest cluster core is recovered to within 2–10%: truth min $-350$ $\mu$KCMB, ST $-342$, DDPM $-318$.
• Checks: relative_error
• Verdict: PASS
• Notes: Relative errors: ST 0.02286 (2.29%), DDPM 0.09143 (9.14%), within 2–10%.

• ✔ C10_crossr_recovery_heldout_ST (p.7, held-out evaluation paragraph + p.11/p.12 held-out bullets + Table 7)

• Claim: Held-out pixel $r_{\mathrm{tSZ}\times\mathrm{CIB}}$: gen $-0.159$ versus test-truth $-0.172$ corresponds to 92% recovery.
• Checks: percentage_recompute
• Verdict: PASS
• Notes: Recomputed recovery is 92.44%, consistent with 92% after rounding.

• ✔ C11_crossr_recovery_heldout_DDPM (p.12, DDPM held-out protocol bullets + Table 7)

• Claim: DDPM held-out: $r_{\mathrm{gen}} = -0.158$ vs test-truth $-0.173$ corresponds to 91.7% recovery.
• Checks: percentage_recompute
• Verdict: PASS
• Notes: Recomputed recovery is 91.33%, consistent within 0.5 percentage points.

• ✔ C12_core_depth_recovery_ST (p.11, held-out bullets + Table 7)

• Claim: Held-out ST deepest core: gen min $-220$ $\mu$KCMB vs test min $-337$ corresponds to $\sim$65% recovery.
• Checks: percentage_recompute
• Verdict: PASS
• Notes: Recomputed recovery is 65.28%, consistent with “$\sim$65%”.

• ✔ C13_core_depth_recovery_DDPM (p.12, DDPM held-out bullets + Table 7)

• Claim: DDPM held-out deepest core: gen min $-330$ $\mu$KCMB vs test $-345$ corresponds to 96% recovery.
• Checks: percentage_recompute
• Verdict: PASS
• Notes: Recomputed recovery is 95.65%, consistent with 96% after rounding.

• ✔ C14_algebraic_floor_minus_1_over_sqrt2 (p.6-7, §4.2.1 (residual diagnostic) + Fig. 6 text)

• Claim: Algebraic limit $r = -1/\sqrt{2} \approx -0.707$ and observed $-0.706$ are consistent.
• Checks: constant_recompute
• Verdict: PASS
• Notes: $-1/\sqrt{2} = -0.70710678$; observed $-0.706$ differs by $0.00111$, within abs tolerance.

• ✔ C15_sampling_coeff_error_factor (p.4, §3.6.1 Critical sampling coefficient)

• Claim: Coefficient error factor is $1/\sqrt{\beta_t} \approx 10$ for typical $\beta_t\sim 0.01$.
• Checks: order_of_magnitude_recompute
• Verdict: PASS
• Notes: $1/\sqrt{0.01}=10$ exactly.

• ✔ C16_corrunet_param_count_check (p.4, §3.6.2 Architecture)

• Claim: Model has 554,562 parameters.
• Checks: self_consistency_integer_format
• Verdict: PASS
• Notes: Comma-formatted integer parses consistently to 554562.

• ✖ C17_multi_freq_median_ratio_delta (p.13, §5.4 (text))

• Claim: med(CIB217/CIB150) = 2.9354 (truth) versus 2.9370 (gen+JM), and claim of matching to four decimal places.
• Checks: rounding_consistency
• Verdict: FAIL
• Notes: Rounded to 4 d.p., values are 2.9354 vs 2.9370 (not equal).

• ✔ C18_fig12_residual_dp_claim (p.13-15, §5.4 text + Fig. 12(b) caption)

• Claim: Body: cross-correlations match to four decimal places; Fig. 12(b): residual $|\Delta r|$ is $10^{-10}$ to $10^{-11}$, orders of magnitude below $10^{-4}$.
• Checks: order_of_magnitude_comparison
• Verdict: PASS
• Notes: $10^{-10}$ and $10^{-11}$ are $10^{-6}$ and $10^{-7}$ of $10^{-4}$, respectively.

• ✔ C19_crossr_ladder_percentages (p.16-18, §6 + Fig. 15)

• Claim: Cross-$r$ recovery ladder: truth $r = -0.163$; multi-channel gives $r = -0.086$ (53%); +soft gives $r = -0.093$ (57%).
• Checks: percentage_recompute
• Verdict: PASS
• Notes: Recomputed: 52.76% ($\approx53\%$), 57.06% ($\approx57\%$).

• ✔ C20_iteration_sweep_percentages (p.16-17, §6.1 + Fig. 14 / Fig. 15 text)

• Claim: Iteration sweep recovery: 32%/38%/47%/53%/56% at 200/400/800/1600/3200 steps.
• Checks: monotonicity_and_bounds
• Verdict: PASS
• Notes: Series is within [0,100] and non-decreasing.

• ✔ C21_break_even_samples_calc (p.19, §6.1.0.6)

• Claim: Break-even $N_{\rm samples} \approx 1$ h $/\,80$ s $\approx50$ samples.
• Checks: division_recompute
• Verdict: PASS
• Notes: $3600/80=45$; within loose tolerance given approximate inputs.

• ⚠ C22_bp_error_reduction_factors (p.22, §7.2 (quantitative improvements))

• Claim: Mean $|\mathrm{relative~error}|$ drops from 28.6%$\rightarrow$4.8% for $S_3(\ell)$ and 64.9%$\rightarrow$9.2% for $K_4(\ell)$.
• Checks: ratio_recompute
• Verdict: UNCERTAIN
• Notes: Unsupported ratio_recompute variant.

• ⚠ C23_peak_count_deficits_percent (p.25, Table 17)

• Claim: Peak count deficits: e.g., for $\nu=2$ truth=710 and ST=617 corresponds to $-13.1\%$, etc. Verify all percent deficits shown in parentheses.
• Checks: percentage_recompute
• Verdict: UNCERTAIN
• Notes: Missing inputs for core depth recovery recompute.

### Limitations

• Audit is restricted to the provided PDF text; no underlying map/spectrum data are available, so only arithmetic and simple logical/unit consistency checks are feasible.
• Figure-based numeric values not explicitly printed in the text/tables are not used (no plot digitization).
• Some 'band-averaged' quantities may use a definition (e.g., averaging over modes, log-bins, or patch-wise normalization) that differs from simple $\ell$-interval width weighting; related candidates are flagged as plausibility checks rather than definitive inconsistencies.

## Paper Ratings

| Dimension | Score |

|-----------|:-----:|

| Overall | 6/10 ██████░░░░ |

| Soundness | 6/10 ██████░░░░ |

| Novelty | 6/10 ██████░░░░ |

| Significance | 6/10 ██████░░░░ |

| Clarity | 5/10 █████░░░░░ |

| Evidence Quality | 6/10 ██████░░░░ |

Justification: The paper’s core linear-algebraic calibration (Eq. 12) is sound and well-supported by extensive experiments, and the generator-agnostic falsification plus the non-by-construction ladder provide useful, original insights. However, the audits flag important presentation and methodological caveats: overstatements about “phase-preserving” and simultaneous 1-pt/2-pt matching, an inconsistent Pearson definition, and missing implementation specifics for the spectral whitening/recolouring that limit reproducibility and rigor. Empirically, results are strong across a broad diagnostic suite, but several claims rely on small-N studies, paired-CDF conditioning risks, and uneven uncertainty reporting. Overall this is a solid, above-average contribution that would benefit from tightening the math claims, clarifying what is enforced vs learned, and standardizing uncertainty/implementation details.