{"id":"1d377640-2d4e-4dc7-a647-677d523fa726","arxiv_id":"2508.17558","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"The BUQEYE truncation-error model for chiral EFT nucleon-nucleon observables requires a momentum-dependent angular length scale, and after warping the angular variable it works for several potentials, though breakdown-scale estimates only stay consistent across orders for the SMS 450/500 MeV…","lead":"This paper tests the BUQEYE statistical model for estimating uncertainties in chiral effective field theory predictions of nucleon-nucleon scattering, across several modern potential families. It finds that the model's key stationarity assumption fails because angular correlation lengths shrink with momentum, and that a warping correction improves the model's diagnostics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Diagnostics comparison conflates warping with downsampling; improvement cannot be attributed to the ℓθ ~ 1/p remediation without a controlled test.","rationale":"The reader's verdict identifies the fixed power-law form b=1 as the weakest assumption. I agree that this is a vulnerability, but I see a more immediate confound in the evidence adduced for the claim. The abstract asserts that diagnostics improve significantly after allowing ℓθ to vary with momentum, 'validating' the modification. The only quantitative-looking evidence is a visual contrast between Figs. 6 and 7 (and 8 vs 9). Those figures differ in two ways at once: the presence of warping and the number/placement of training and testing points. Because Fig. 7 uses fewer than half the training points and fewer test points, and because the dropped points are concentrated at low momentum where the stationary GP's long ℓθ makes predictions least informative, the improved coverage and Mahalanobis distance may not indicate a corrected covariance structure. The paper even remarks in Sec. IVC that 'the choice of train-test split ... is a more important decision than the specific method of remediating nonstationarity,' which undermines the attribution of the improvement to the warping. The b=0 re-fit after warping is suggestive but circular: the warp was constructed from the fitted 1/p law, so finding b≈0 after warping is expected. A proper validation would hold out potentials or observables, or at least fix the test set while varying the warp. The proposed check—applying the exact downsampling without warping—would settle whether the improvement is due to the ℓθ(p) model or merely to discarding hard-to-fit low-momentum points. This does not invalidate the empirical ℓθ~1/p finding, which is well-documented in Fig. 3, but it does call into question the strong 'validating' language in the abstract. The reader's conditional verdict remains appropriate: the paper should be accepted with major revisions or conditions requiring a controlled diagnostic comparison and, ideally, a quantitative significance statement (e.g., posterior predictive p-values for D2_MD). I therefore keep verdict_should_be as UNCHANGED relative to the reader's CONDITIONAL, while noting that the concern sharpens the conditions.","tokens_in":20135,"tokens_out":10968,"duration_ms":101013,"concrete_test":"Using the public notebook, rerun the SMS 500 MeV dσ/dΩ diagnostics with the same downsampled 14 training and 33 testing points (mapped back to the original (p_rel, -cosθ) coordinates) but with the stationary RBF kernel and no warping. Compare D2_MD (relative to its expected degrees of freedom), DCI coverage, and DPC to Figs. 6 and 7. If the no-warp/downsampled diagnostics are as good as Fig. 7, the improvement is due to downsampling and the claim that ℓθ ∝ 1/p is validated fails; if they resemble Fig. 6, the warping is the operative ingredient.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that ℓθ is approximately ∝1/p_rel and that remediating this nonstationarity via warping validates the modified BUQEYE model—rests on the diagnostic improvement shown in Sec. IVD. The comparison is not controlled: Fig. 6 (no warping, no downsampling) uses 30 training and 69 testing points, while Fig. 7 (warping + downsampling) uses 14 training and 33 testing points. Warping is applied together with a nonuniform re-gridding that removes many low-momentum points, precisely the regime where the stationary GP performed worst. The observed improvement in DCI, D2_MD, and DPC could therefore result from the downsampling or the changed test-point set rather than from the warped space's restored stationarity. The subsequent b=0 re-fit after warping (Sec. IVD) is a consistency check on the same data used to fix B=1; it confirms that the warp removed the fitted trend but does not independently validate the 1/p law. Without a comparison that isolates warping from downsampling—or a quantitative significance test of the diagnostics—the abstract's 'validating' conclusion is not supported by the evidence shown.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper assesses the BUQEYE correlated-truncation-error model across multiple modern chiral EFT nucleon-nucleon potentials. Using six scattering observables as functions of relative momentum and scattering angle, the authors first identify an even-odd staggering in coefficient sizes for soft potentials and exclude the softest cases from further analysis. They then report evidence that the Gaussian-process angular length scale ℓ_θ is approximately inversely proportional to p_rel, fit this as a power law, and introduce a warping transformation with fixed exponent B=1 to restore stationarity. After warping and low-momentum downsampling they report improved diagnostics (DCI, D2_MD, DPC) for SMS 500 MeV and EMN 500 MeV, and then extract posterior distributions for the breakdown scale Λ_b at fixed meff and with meff variable. They find statistically consistent Λ_b posteriors across orders only for SMS 450 and 500 MeV, and they interpret the variable-meff results as an overfitting problem. The paper includes a publicly available Jupyter notebook for reproduction.","tokens_in":20418,"tokens_out":3116,"duration_ms":35365,"significance":"If the central claim is established, the paper would provide an important practical modification of the BUQEYE model for NN scattering, with a physically motivated treatment of nonstationarity in the angular correlation structure and a systematic survey of regulator schemes and scales. The paper is commendably honest about the limitations of cross-order Λ_b consistency, and it ships reproducible code. However, the validation of the proposed warping is currently under-supported: the diagnostics comparison in Sec. IVD changes two things at once (warping and downsampling), and the b=0 result after warping is a consistency check on data used to fix B=1 rather than an independent test. The central claim therefore needs a controlled validation before the abstract's 'validating' conclusion can be accepted.","major_comments":[{"comment":"The comparison that is claimed to validate the warping is not controlled. Figure 6 uses 30 training and 69 testing points with no warping and no downsampling, while Figure 7 uses 14 training and 33 testing points with warping and downsampling; the same is true of Figs. 8 and 9 for EMN 500 MeV. The improvement in DCI, D2_MD, and DPC could therefore be due to the removal of low-momentum test points or to the smaller training set rather than to the restoration of stationarity in the warped space. To support the abstract's claim that the diagnostics 'validate' the modification, the authors should provide a controlled comparison that isolates warping from downsampling, for example by applying the warp without changing the training/testing points, by applying the downsampling without warping, or by testing on a fixed held-out set. A quantitative significance statement for the diagnostics would also strengthen the claim.","section":"Sec. IVD, Figs. 6-9"},{"comment":"The validation logic is partially circular. The power law ℓ_θ = a (p_rel/450 MeV)^{-b} is extracted from the data, the warping in Eq. (16) is chosen as its inverse with B=1, and the re-fit in Sec. IVD then reports that b is consistent with 0 after warping. This confirms that the warp removes the fitted trend, but it is not independent evidence that the 1/p_rel law is the correct description of the nonstationarity. The authors should test the warping on held-out data or observables not used to fit Eq. (14), or compare the warped-space GP against alternative nonstationarity models (e.g., other exponents or a nonlinear ℓ_θ(p_rel)) using predictive diagnostics. Without such a test, the claim that the modification is 'validated' goes beyond what the current evidence supports.","section":"Sec. IIIB1, Eq. (14); Sec. IVA, Eq. (16); Sec. IVD"},{"comment":"All Λ_b and meff results in Sec. V are obtained with the fixed warp exponent B=1 and the accompanying downsampling. If the assumed power-law form or the fixed exponent is misspecified for one of the potentials, the subsequent Λ_b posteriors inherit that misspecification, so the finding that only SMS 450 and 500 MeV show cross-order consistency is conditional on the B=1 assumption. The paper would be stronger if it reported a sensitivity analysis of the Λ_b posteriors to B (or to the choice of downsampling scheme), at least for one potential, so that the robustness of the central 'statistically consistent only for SMS' conclusion can be assessed.","section":"Sec. VA and Sec. VB, Table II"}],"minor_comments":[{"comment":"The caption contains typographical errors: 'Note thesimilarvaluesacrossforSMS500MeVandandacrossSCS 0.9 fm' should be cleaned up, and the sentence about even-order and odd-order values is grammatically incomplete.","section":"Table I caption"},{"comment":"The phrase 'There are are notable jumps between orders' contains a duplicated word, and the sentence beginning 'It may also be the case that how potentials are fitted matters here' would benefit from a smoother transition.","section":"Sec. VA, text after Fig. 13"},{"comment":"The phrase 'the sy-setmatic separation of the curves' contains a typo, and the qualitative statement about the DCI trend could be complemented by a numerical measure, such as the coverage deficit or the slope of the DCI curves.","section":"Sec. IVD, Fig. 10 discussion"},{"comment":"The presentation states that 'the values of b are not far from 1' and then fixes B=1; it would be helpful to report the fitted b values and their uncertainties numerically, at least in a table, rather than only showing them graphically in Fig. 3.","section":"Sec. IIIB1"},{"comment":"The discussion of the equivalence between warping and the nonstationary kernel is clear, but it should be explicitly noted that the practical claim of equivalence in the paper applies only to the diagnostics computed with the specific downsampled training and testing sets, since the paper does not present a head-to-head comparison with identical point sets for both methods.","section":"Sec. IVC"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope and the authors have been appropriately cautious about the cross-order Λ_b inconsistency. My main concern is the validation of the warping step: the evidence in Sec. IVD conflates warping with downsampling, and the b=0 refit is close to a consistency check of the fitted power law. I recommend major revision rather than rejection because the underlying idea is plausible and the paper explicitly provides reproducible code; a controlled validation, a sensitivity check on B, and a softened or more carefully qualified abstract would address my concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe punchline: the paper's central empirical claim—that the BUQEYE angular length scale ℓ_θ falls like 1/p_rel—is well supported and worth knowing. The validation of the warping fix, however, is weaker than the abstract suggests, because the improvement shown in Figs. 6–9 conflates two changes at once.\n\nWhat's genuinely new: the authors extend their earlier single-potential BUQEYE analysis to six potential families, document the even-odd coefficient staggering in soft potentials, and make a clean, well-illustrated case for a specific power-law nonstationarity. The reproducibility notebook is a real plus, and the paper is honest about where the model fails—cross-order Λ_b consistency only for SMS 450/500, and overfitting when both Λ_b and m_eff are free. That candor counts.\n\nThe soft spots, in order of seriousness. First, the diagnostics comparison is not controlled: Fig. 6 uses 30 training points and no warping; Fig. 7 uses 14 training points after warping and downsampling. You can't attribute the improvement to the warping alone. A proper test would compare warped vs. unwarped kernels on the same grid, or at least report a significance test. Second, the b=0 re-fit in Sec. IVD is a consistency check on the same data that fixed B=1—it confirms the warp removed the fitted trend, but it doesn't independently validate the 1/p law. The semi-classical argument helps, but the validation loop is partly circular. Third, the exclusion of soft potentials without the promised scan is a minor weakness; the paper's justification is plausible but not a substitute for the analysis.\n\nNone of this undermines the empirical core. The 1/p scaling is consistent across six observables with small errors, and the physical rationale is sensible. The paper overstates its validation in the abstract, but the underlying work is careful and reproducible.\n\nWho is this for? Anyone using BUQEYE truncation errors for chiral potentials, and anyone building GP surrogates for scattering observables. It deserves a serious referee; the confounds should be addressed in revision, but the paper has enough new, honestly reported content to justify the time.\n\nMy recommendation: send it to review, with a request for a controlled warping comparison or a clearly stated limitation.","headline":"ℓ_θ ~ 1/p is likely right; the warping validation is confounded and the abstract oversells it, but the paper is honest, reproducible, and deserves refereeing.","tokens_in":20999,"tokens_out":3238,"would_cite":true,"duration_ms":32100,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["13.75.Cs","21.30.-x"],"model":"deepseek-v4-flash","headline":"The angular correlation length of chiral-EFT expansion coefficients falls as one over the relative momentum, and warping the angle axis by momentum restores the BUQEYE truncation-error model's statistical consistency.","keywords":["chiral effective field theory","nucleon-nucleon potentials","EFT truncation errors","Gaussian process","BUQEYE model","nonstationarity","input-space warping","breakdown scale"],"falsifier":"Extract the angular length scale $\\ell_\\theta$ in fine momentum bins from order-by-order predictions of a chiral potential outside the fitted set — for example a $\\Delta$-full interaction, or the same potentials extended beyond $p_{\\rm rel} = 405$ MeV. If the fitted exponent in $\\ell_\\theta = a\\,(p_{\\rm rel}/450\\,\\mathrm{MeV})^{-b}$ departs significantly from $b = 1$, or if the warped-model diagnostics degrade once the low-momentum region is sampled without downsampling, the proposed warping is the wrong remedy rather than a validated one.","tokens_in":19883,"feed_emoji":"⚛️","tokens_out":16720,"duration_ms":140690,"temperature":0.7,"pith_summary":"The paper asks whether a Bayesian model for correlated effective field theory truncation errors — the BUQEYE model, which treats the dimensionless coefficients of an order-by-order observable expansion as draws from one Gaussian process — survives contact with sixteen modern chiral nucleon-nucleon potentials from four regulator families. It does, but only after one modification: the correlation length in the scattering-angle direction is not constant but falls roughly as $1/p_{\\rm rel}$, the inverse of the relative momentum, so a stationary Gaussian process is misspecified. Warping the angle coordinate by a single power of $p_{\\rm rel}$ restores stationarity and makes the model's coverage and consistency diagnostics pass for the potentials that behave. The authors also show that soft regulators produce an even-odd staggering in coefficient sizes that no choice of the breakdown scale $\\Lambda_b$ can repair, and that order-stable values for $\\Lambda_b$ emerge only for the SMS potential at 450 or 500 MeV. Why this matters: if the modified model is right, truncation-error bars on chiral EFT predictions can be computed from lower orders alone, for the potentials where the convergence pattern is regular.","feed_headline":"One power law fixes the error bars on chiral nuclear forces","feed_subtitle":"Falling as 1 over momentum, the angular correlation length turns stationary once the angle axis is warped.","key_machinery":"The load-bearing object is the BUQEYE model itself: a zero-mean Gaussian process with a squared-exponential kernel, $\\kappa(x,x') = \\bar{c}^2 \\exp[-(x_E-x'_E)^2/(2\\ell_E^2) - (x_\\theta-x'_\\theta)^2/(2\\ell_\\theta^2)]$, in the input space $x_E = p_{\\rm rel}$, $x_\\theta = -\\cos\\theta$, whose draws model the coefficient functions $c_n(x)$. The mechanism that carries the argument is the warping transformation $w_\\theta(x_E,x_\\theta) = x_\\theta\\,(p_{\\rm rel}/405\\,\\mathrm{MeV})^B$ with $B = 1$, which shrinks the angular input at low momentum so that a stationary length scale becomes an accurate description; it is exactly equivalent to a nonstationary kernel whose angular length scale depends on momentum, and it guides the downsampling of low-momentum training points, where the long correlation length makes each point carry less information.","core_discovery":"On the paper's own terms: the assumption that the dimensionless coefficients $c_n(x)$ extracted from order-by-order observable predictions are draws from a single Gaussian process fails for modern chiral NN potentials as currently implemented, and the specific failure is identified. The GP's angular length scale obeys $\\ell_\\theta = a\\,(p_{\\rm rel}/450\\,\\mathrm{MeV})^{-b}$ with $b \\approx 1$ for all six observables tested, a semi-classical reflection of the number of partial waves accessible at momentum $p_{\\rm rel}$; the variance $\\bar{c}^2$ and the momentum-direction length scale $\\ell_E$, by contrast, show no systematic trend and can be treated as stationary. Making $\\ell_\\theta$ momentum-dependent restores the model: the authors implement this as a warping of the angular input, $w_\\theta = x_\\theta\\,(p_{\\rm rel}/405\\,\\mathrm{MeV})^B$ with $B = 1$, show it is equivalent to a nonstationary kernel with a momentum-dependent length scale, and find that after warping with low-momentum downsampling the re-extracted exponent is consistent with $b = 0$ and all three diagnostics (credible-interval coverage, Mahalanobis distance, pivoted Cholesky decomposition) improve substantially. The modified model then yields statistically consistent truncation errors for the regular potentials, provided $\\Lambda_b$ is set near 600–750 MeV; cross-order consistency of the $\\Lambda_b$ posterior is the exception, holding only for SMS 450 and 500 MeV.","pith_inferences":["The $1/p_{\\rm rel}$ law for the angular correlation length is a partial-wave counting argument, so the same warping with $B = 1$ could plausibly serve as a default prior for other scattering observables — pp with Coulomb, $\\Delta$-full potentials, and 3N observables — without re-deriving the exponent, an extension the paper leaves to future work.","A testable stress point: the semi-classical argument predicts the exponent should fail in regions where a single partial wave dominates, so measuring $\\ell_\\theta$ near thresholds or in kinematically constrained configurations would probe whether $B = 1$ is a universal feature or a fitted coincidence.","The even-odd staggering pattern gives a cheap quantitative diagnostic of regulator artifacts: alternating rms coefficient sizes beyond roughly a factor of two between consecutive orders signal that pion exchange has been pushed into even-order contact terms, connecting directly to pionless-EFT expectations for these potentials.","The correlation between the warping exponent $B$ and the extracted $\\Lambda_b$ is not explored in the paper; fitting both together would reveal whether the quoted 600–750 MeV range carries an additional systematic error from fixing $B = 1$."],"forward_implications":["For potentials with regular convergence (SMS 450–550 MeV, SCS 0.9–1.0 fm, EMN 500 MeV), order-by-order error bars computed with the warped model cover the next order at the claimed rate, so those truncation errors can be propagated into nuclear-structure predictions.","Soft potentials (SMS 400 MeV, SCS 1.1 and 1.2 fm) are diagnosably irregular: the even-odd coefficient staggering persists for any choice of $\\Lambda_b$ or $m_{\\rm eff}$, so they should be excluded from this kind of uncertainty quantification.","Reported values of the chiral breakdown scale depend on the truncation order used to extract them; only SMS 450 and 500 MeV give $\\Lambda_b$ posteriors that are stable across orders, so single-order $\\Lambda_b$ claims for other potentials carry unquantified order dependence.","Treating both $\\Lambda_b$ and $m_{\\rm eff}$ as free parameters overfits: the joint posteriors do not narrow as orders are added, which points to the parametrization of $Q$ rather than the GP kernel as the model's limiting assumption.","Because warping and a nonstationary kernel give identical results under an equivalent train-test split, the practical lesson transfers: the split, and specifically low-momentum downsampling, matters more than which implementation is chosen."],"supporting_citations":[{"why":"The prior single-potential BUQEYE analysis of SMS 500 MeV that supplies the Q-sum parametrization, reference scales, and the three validation diagnostics this paper adopts and extends.","marker":"[2]"},{"why":"The paper that launched correlated EFT truncation-error estimation, whose framework the BUQEYE model formalizes.","marker":"[3]"},{"why":"Derives the Student-t process for coefficient functions, the squared-exponential kernel, and the hyperparameter update formulas the whole analysis runs on.","marker":"[4]"},{"why":"Source of the SMS potential family at regulators 400–550 MeV, including the flagship SMS 500 MeV case.","marker":"[5]"},{"why":"Source of the SCS coordinate-space potential family, including the soft 1.1 and 1.2 fm cases that fail the convergence diagnostics.","marker":"[6]"},{"why":"Source of the EMN potential family, represented by EMN 500 MeV in the modified analysis.","marker":"[7]"},{"why":"Source of the GT+ local potential family, used to show that N2LO-only potentials cannot pin down the breakdown scale.","marker":"[8]"},{"why":"The earlier uncorrelated-error analysis that first flagged irregular convergence in the softest SCS potentials, which the new diagnostics corroborate and extend.","marker":"[13]"},{"why":"Supplies the alternative value $m_{\\rm eff} \\approx 200$ MeV used when the soft scale is fixed during breakdown-scale extraction.","marker":"[24]"}],"fun_headline_variants":["Warp angle axis by momentum to fix chiral force error bars","Nuclear force errors: angular correlation falls as 1 over p","Nonstationary GP in chiral EFT fixed by angular scaling","Chiral NN convergence: one inverse-power law for angle","Momentum-dependent angular warp validates BUQEYE model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on one power-law assumption: that the angular correlation length follows $\\ell_\\theta = a\\,(p_{\\rm rel}/450\\,\\mathrm{MeV})^{-b}$ with the exponent fixed at $b = 1$ for all observables and potentials; if the true momentum dependence is not this single-exponent form, the warped Gaussian process is misspecified and the improved diagnostics do not validate the model.","fun_headline_variants_meta":{"raw":{"variants":["Warp angle axis by momentum to fix chiral force error bars","Nuclear force errors: angular correlation falls as 1 over p","Nonstationary GP in chiral EFT fixed by angular scaling","Chiral NN convergence: one inverse-power law for angle","Momentum-dependent angular warp validates BUQEYE model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000286,"raw_usage":{"total_tokens":1816,"prompt_tokens":1215,"completion_tokens":601,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":831,"completion_tokens_details":{"reasoning_tokens":516}},"tokens_in":831,"tokens_out":601,"duration_ms":7326,"temperature":1.0,"reasoning_tokens":516,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:04:25.169442+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Extract the angular length scale $\\ell_\\theta$ in fine momentum bins from order-by-order predictions of a chiral potential outside the fitted set — for example a $\\Delta$-full interaction, or the same potentials extended beyond $p_{\\rm rel} = 405$ MeV. If the fitted exponent in $\\ell_\\theta = a\\,(p_{\\rm rel}/450\\,\\mathrm{MeV})^{-b}$ departs significantly from $b = 1$, or if the warped-model diagnostics degrade once the low-momentum region is sampled without downsampling, the proposed warping is the wrong remedy rather than a validated one.","supporting_citations":[{"cited_title":"low, high, low","cited_arxiv_id":null,"evidence_quote":"The prior single-potential BUQEYE analysis of SMS 500 MeV that supplies the Q-sum parametrization, reference scales, and the three validation diagnostics this paper adopts and extends."},{"cited_title":"deformed,","cited_arxiv_id":null,"evidence_quote":"The paper that launched correlated EFT truncation-error estimation, whose framework the BUQEYE model formalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the alternative value $m_{\\rm eff} \\approx 200$ MeV used when the soft scale is fixed during breakdown-scale extraction."}],"review_version":2}