{"id":"f3b702d7-8dea-4073-be9f-3f7f068f424e","arxiv_id":"1908.02016","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A curved-sky position-space framework derives analytic responses, Gaussian noise biases, and optimal joint gradient-curl quadratic estimators for arbitrary-spin CMB anisotropy.","lead":"This paper presents a unified position-space formalism for quadratic estimators that reconstruct anisotropy in the cosmic microwave background, including lensing, patchy reionization, and cosmic birefringence. It ships the analytic response and noise formulas used by the public Planck 2018 lensing pipeline and derives optimal joint gradient-curl estimators for arbitrary spin.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (4.4) as printed has a spin-index sign inconsistency with Eq. (4.3): the second-leg output spin is -s+r instead of s+r, breaking the spin-r estimate.","rationale":"The reader's verdict was CONDITIONAL, with the condition that the author supply derivations or numerical validation for the central formulas. The spin-index inconsistency between Eq. (4.3) and Eq. (4.4) is a concrete instance of why that condition matters: it is an internal consistency error, not a disagreement with external consensus. The reader's weakest_assumption focused on Gaussianity and the separability of the covariance response; my concern is distinct but compatible. If the sign error is real, the central claim as stated is not directly usable, but the fix is likely a one-line correction, so the appropriate disposition remains conditional acceptance pending verification. The proposed check settles whether the printed Eq. (4.4) can be trusted as a recipe for the optimal estimator.","tokens_in":7670,"tokens_out":36100,"duration_ms":357745,"concrete_test":"Set r=0 and s=2 in Eqs. (4.3) and (4.4). Eq. (4.3) gives a term with first leg -2 \\bar X (spin -2) and second harmonic _2 Y_lm (spin +2), yielding spin 0. Eq. (4.4) as printed gives second-leg output spin -2, yielding spin -4. Implement this one term in a small harmonic-space code, or apply the spin product rule in Eq. (1.7), and check the resulting spin. If the spin is not 0, Eq. (4.4) is misprinted. Then compare with the polarization-rotation estimator in Section V to determine the correct sign for the second-leg output spin.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim tells readers to construct the optimal estimator with the weights of Eq. (4.4). As printed, the second-leg weight is w^{-s+r,t} (output spin -s+r), and the W superscript is W^{-r,-st}. But Eq. (4.3), which is the derivative result being summarized, has the second-factor harmonic _s+r Y_lm, i.e. output spin s+r. For the product to have the stated spin r, the second leg must have output spin s+r. Taking r=0, s=2, Eq. (4.4) would make the estimator a spin -4 object, contradicting the scalar examples in Section V (modulation, polarization rotation, point sources). The sign flip on the second index of W between Eq. (4.3) (W^{-r,st}) and Eq. (4.4) (W^{-r,-st}) compounds the inconsistency. Unless there is an unstated index convention, the printed weight formula is internally inconsistent with the derivation it is meant to encode, so the central recipe cannot be used as-is.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a position-space, spin-weight formalism for quadratic estimators (QEs) of statistical anisotropy of the CMB on the curved sky. It derives analytic expressions for Gaussian noise biases (Section II), estimator responses (Section III), and optimal joint gradient/curl weights from a general covariance response W (Section IV). The document is a supplement to the public Planck 2018 lensing pipeline and applies the formalism to lensing, modulation, patchy reionization, polarization rotation, point sources, and noise inhomogeneities.","tokens_in":7870,"tokens_out":9138,"duration_ms":92944,"significance":"If the formulas are correct, the paper is a useful reference that unifies several anisotropy estimators into one separable QE framework and reduces response and noise computations to one-dimensional Wigner small-d transforms. It also makes concrete contact with known Okamoto-Hu lensing estimators. However, the printed weight formula in Section IV contains a spin-index inconsistency that prevents the central recipe from being used as written; this significantly limits the immediate utility of the paper until corrected.","major_comments":[{"comment":"The printed weight formula is internally inconsistent with the expression it is meant to encode. In Eq. (4.3), the second leg has the spherical harmonic s+r Y_lm, so in the notation of Eq. (1.7) its output spin is t_o = s+r, and the factor is 2 W^{-r,st}_l / t_2. Equation (4.4), however, gives w^{-s+r,t}_l = 2/t_2 W^{-r,-st}_l, i.e. output spin -s+r and a second index -st on W. Since the first leg has output spin -s, the total output spin would be -2s+r rather than r. For r=0, s=2, this would produce a spin -4 object, contradicting the scalar modulation and polarization-rotation examples in Section V. Unless there is an unstated index convention, Eq. (4.4) must be corrected to w^{s+r,t}_l = 2/t_2 W^{-r,st}_l (or the equivalent with clearly defined index symmetries), and the derivation should be checked against the examples.","section":"Section IV, Eqs. (4.3)-(4.4)"},{"comment":"The transition from the functional derivative in Eq. (4.2) to the explicit weight expression in Eq. (4.3) is not shown; the text states only that performing the derivative using representation (3.3) gives the result. Because the resulting weight formula Eq. (4.4) contains a sign/spin inconsistency, the omitted algebra is not merely a presentation issue. The authors should provide the intermediate spin-weight manipulation for at least one case, or otherwise verify that the printed equations satisfy the spin-sum rule s_o + t_o = r.","section":"Section IV, between Eqs. (4.2) and (4.3)"}],"minor_comments":[{"comment":"The symbols s2 and t2 are defined in the text as \"1(s=0) or 2(s≠0)\", but this notation is easy to misread as s^2 and t^2; using s_2 and t_2 would be clearer.","section":"Section IV, Eq. (4.4)"},{"comment":"The same symbol w is used for both legs even though the weights may differ; introducing separate symbols, e.g. w^{(1)} and w^{(2)}, would reduce ambiguity in equations such as Eq. (4.4).","section":"Section I.B, Eq. (1.7)"},{"comment":"The paper states that most formulas follow by applying Eq. (1.10), but no representative derivation is shown. Since this is a methods supplement, adding a short worked derivation of one of the N(0) or response formulas would greatly help readers verify the sign conventions.","section":"Sections II and III"}],"recommendation":"major_revision","confidential_remarks":"This manuscript is a code-release supplement rather than a standalone research article. The index error in Eq. (4.4) is load-bearing because the document is explicitly meant as a reference for users of the plancklens pipeline. I recommend the editor ask the authors to correct the formula and to add a consistency check, for example by explicitly deriving the polarization-rotation estimator (r=0, s=2) from Eq. (4.3). If the journal publishes technical methods notes, the paper has value after revision; otherwise its scope should be clarified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a useful reference for the position-space quadratic-estimator formalism, but the printed optimal-weight recipe is not usable as-is. The N(0) and response formulas in Secs. II-III are a clean generalization of Dvorkin-Hu-Smith and reduce to the Okamoto-Hu limits, and the public code release is a real asset. The author clearly knows the material.\n\nThe problem is in Sec. IV. Eq. (4.3) gives the derivative with a second-leg harmonic of spin s+r and a W^{-r,st} kernel. Eq. (4.4) then writes the second-leg weight as w^{-s+r,t} = (2/t2) W^{-r,-st}. For r=0, s=2 that makes the estimator spin -4 instead of scalar. The W superscript also flips sign on the first index. So the advertised optimal estimator cannot be constructed from the text without re-derivation. It may be a simple typo, but it is load-bearing: the abstract promises this exact recipe.\n\nThe rest is in better shape. The Gaussian covariance and cross-response formulas are stated as one-dimensional Wigner-d integrals and are plausible, and the examples (lensing, modulation, rotation, point sources) are useful. The main methodological gap, as the reader notes, is that the intermediate algebra from Eq. (1.10) to Eqs. (2.4-2.5) and (3.5-3.7) is not shown, which makes independent verification slow. A short derivation appendix or a numerical cross-check against the code would fix that. The circularity burden is low: responses are derived from a forward model, not fitted.\n\nWho should read it? Anyone needing explicit N(0) and response kernels for arbitrary-spin anisotropy, and anyone using the plancklens code. It deserves a serious referee, but only after the Eq. (4.4) signs are corrected and the derivations are at least sketched. I would not cite the optimal-weight formula from v1 as-is.","headline":"Useful methods supplement, but the printed optimal-weight recipe in Eq. (4.4) has a spin-index sign error that makes it unusable as-is.","tokens_in":8386,"tokens_out":16897,"would_cite":false,"duration_ms":162431,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Optimal curved-sky quadratic estimators for CMB anisotropies follow from one covariance-response kernel and one weight formula.","keywords":["cosmic microwave background","quadratic estimators","CMB lensing","statistical anisotropy","spin-weight correlation functions","gradient and curl modes","curved sky","Wigner small-d transforms"],"falsifier":"Simulate Gaussian CMB skies containing a known anisotropy whose covariance response is $W$, apply the optimal weights of Eq. (4.4), and compare the estimator's variance to the inverse of the corresponding Fisher matrix; any significant excess variance would show that the analytic optimality argument misses something. On the noise side, averaging the quadratic estimator over isotropic simulations should reproduce the analytic $N^{(0)}_{\\ell}$ of Eqs. (2.4)-(2.5) to numerical precision, so a mismatch would falsify the Gaussian covariance calculations.","tokens_in":7455,"feed_emoji":"🌌","tokens_out":8600,"duration_ms":83727,"temperature":0.7,"pith_summary":"This paper establishes a general recipe for building optimal quadratic estimators of statistical anisotropies in the cosmic microwave background on the curved sky. Its central claim is that, once the data covariance response $W$ of Eq. (3.3) is known, the optimal joint gradient and curl estimator is a separable product of two position-space, spin-weighted filtered maps with explicit weights, and that responses and Gaussian noise biases follow from compact one-dimensional Wigner $d$-function integrals. This matters because lensing, patchy reionization, cosmic birefringence, point sources, and noise inhomogeneities are all anisotropy probes that can be treated as special cases of the same construction, with no separate derivations needed. The document also shows that the standard temperature, polarization, and minimum-variance lensing estimators are recovered as gradient-mode special cases.","feed_headline":"One covariance response gives optimal CMB anisotropy estimators","feed_subtitle":"The same compact formula covers lensing, patchy reionization, and polarization rotation.","key_machinery":"The load-bearing object is the covariance response kernel $W^{a,st}_{\\ell}$ of Eq. (3.3), which encodes how the data covariance changes under a spin-$r$ anisotropy source. The estimator itself is the separable quadratic form of Eq. (1.7), a product of two spin-weighted filtered maps, whose weights are set by Eq. (4.4) as the first Newton-Raphson step of the Gaussian likelihood evaluated at zero anisotropy. Responses and noise covariances are then computed by projecting these position-space products onto gradient and curl modes, which reduces every integral to a one-dimensional Wigner $d$-function transform. This combination of a position-space correlation-function representation and the covariance response is what turns each new anisotropy source into a known optimal estimator.","core_discovery":"The core claim is that for a Gaussian CMB whose covariance responds to a spin-$r$ anisotropy through the separable kernel $W$ of Eq. (3.3), the minimum-variance quadratic estimator of the anisotropy's gradient and curl modes is given by Eq. (1.7) with the weights of Eq. (4.4): one leg of the estimator is a delta function in spin, the other leg is proportional to $W$, and both legs act on inverse-variance-filtered maps. The same formalism supplies analytic responses, Eqs. (3.5)-(3.7), and Gaussian noise biases between arbitrary pairs of estimators, Eqs. (2.4)-(2.5), all reduced to one-dimensional Wigner small-$d$ integrals. In the lensing case the construction reproduces the known full-sky lensing estimators, including temperature, polarization, and minimum-variance versions, and extends them to curl modes and to anisotropy sources of arbitrary spin.","pith_inferences":["A direct test of the optimality claim would be to feed a numerical covariance response $W$ measured from simulated anisotropic skies, compare the resulting estimator variance with the inverse Fisher matrix, and see whether any excess variance reveals terms beyond Eq. (3.3).","The same Newton-Raphson logic could be applied to non-separable covariance responses numerically, which would quantify how much optimality is lost when the separable form is violated.","Because the method only assumes Gaussianity under the anisotropy, the estimator construction could be reused for foregrounds or secondary anisotropies whose covariance response is known, not just the CMB examples listed in the paper."],"forward_implications":["Any anisotropy whose covariance response has the separable form of Eq. (3.3) immediately gets an optimal joint gradient and curl estimator: supplying $W$ fixes all weights, responses, and leading Gaussian noise biases.","The analytic responses of Eqs. (3.5)-(3.7) allow gradient and curl estimates to be normalized independently, so one pipeline can produce unbiased maps for lensing, birefringence, or patchy reionization.","The Gaussian noise-bias formulae of Eqs. (2.4)-(2.5) provide the leading $N^{(0)}_{\\ell}$ terms between arbitrary pairs of estimators, the quantity needed to debias anisotropy power spectra and cross-spectra.","The known temperature, polarization, and minimum-variance lensing estimators are recovered as gradient-mode special cases of the single weight rule of Eq. (4.4), confirming the construction against established results."],"supporting_citations":[{"why":"It supplies the CMB lensing analysis pipeline that this document supplements and whose public release motivates the calculations.","marker":"[1]"},{"why":"It introduces the position-space spin-weight correlation function formalism for patchy-screening anisotropies that the paper builds on.","marker":"[2]"},{"why":"It provides the curved-sky quadratic estimator framework and the use of Wigner small-$d$ transforms for responses and noise.","marker":"[3]"},{"why":"It supplies the Gaussian-likelihood Newton-Raphson estimator whose first step defines the optimal anisotropy weights.","marker":"[10]"},{"why":"It provides the standard full-sky lensing estimators that the derived weight rule recovers as special cases.","marker":"[11]"},{"why":"It supplies the lensing response kernel used in the worked lensing example.","marker":"[15]"}],"fun_headline_variants":["One covariance response optimizes all CMB anisotropy estimators","Single formula unifies optimal CMB gradient and curl estimators","Compact kernel yields optimal CMB probes for any spin anisotropy","One integral formula gives minimum-variance CMB anisotropy estimators","Curved-sky quadratic estimators: one response does it all"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The optimality argument assumes that the anisotropy enters only through the separable covariance response $W$ of Eq. (3.3), that the CMB remains Gaussian under the anisotropy, and that the Newton-Raphson step is taken at zero anisotropy; if any of these fails, the proposed weights are approximate rather than optimal.","fun_headline_variants_meta":{"raw":{"variants":["One covariance response optimizes all CMB anisotropy estimators","Single formula unifies optimal CMB gradient and curl estimators","Compact kernel yields optimal CMB probes for any spin anisotropy","One integral formula gives minimum-variance CMB anisotropy estimators","Curved-sky quadratic estimators: one response does it all"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000435,"raw_usage":{"total_tokens":2134,"prompt_tokens":782,"completion_tokens":1352,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":398,"completion_tokens_details":{"reasoning_tokens":1271}},"tokens_in":398,"tokens_out":1352,"duration_ms":10745,"temperature":1.0,"reasoning_tokens":1271,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:55:49.943631+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate Gaussian CMB skies containing a known anisotropy whose covariance response is $W$, apply the optimal weights of Eq. (4.4), and compare the estimator's variance to the inverse of the corresponding Fisher matrix; any significant excess variance would show that the analytic optimality argument misses something. On the noise side, averaging the quadratic estimator over isotropic simulations should reproduce the analytic $N^{(0)}_{\\ell}$ of Eqs. (2.4)-(2.5) to numerical precision, so a mismatch would falsify the Gaussian covariance calculations.","supporting_citations":[],"review_version":1}