{"id":"3c7bed1d-faef-4b9e-afb1-20f64c8edcf2","arxiv_id":"2506.24037","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper defends the inverse-problem view of LaMET reconstructions and argues that rigid parametric extrapolations underestimate PDF uncertainties when lattice data are noisy.","lead":"This comment defends the claim that extracting parton distributions from noisy lattice data is an ill-posed inverse problem. It argues that simple exponential-tail fits underestimate uncertainties and that the smoothness of the PDF, not the asymptotic behavior of correlators, controls the reconstruction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claim that the exact asymptotic behavior matters little rests on an unverified smoothness scale; for realistic large-x falloffs or moderate-x structures, exponential smearing can become the controlling regulator.","rationale":"The reader's conditional verdict is appropriate. The paper's argument is reputable and conceptually clear, but its strongest claim—that the asymptotic behavior matters little—is established only for smooth, well-behaved PDFs. The authors explicitly rely on the smoothness of usual parton distributions, and they do not provide a quantitative criterion for what counts as smooth relative to the smearing scale meff/Pz. Their own Fig. 4 shows that a sharp moderate-x bump is erased by the exponential tail, which means the exact tail behavior is not irrelevant for realistic features; it is exactly the mechanism that limits resolution. This does not refute the paper's secondary claim that LaMET cannot reliably reconstruct point-by-point x-dependence, but it does weaken the statement that the asymptotic behavior is a minor source of uncertainty. A concrete test using global PDFs or a one-parameter family of large-x exponents would settle whether the claimed insensitivity extends to realistic functions. Since the reader already assigned CONDITIONAL status based partly on this assumption, my read does not change the verdict; it sharpens the reason for the condition.","tokens_in":12541,"tokens_out":12029,"duration_ms":145139,"concrete_test":"Take a set of global PDFs (e.g., CT18, MSHT20, NNPDF4.0) at a scale of about 2 GeV. For each PDF f(x), compute the Fourier harmonic h(λ) = ∫_0^1 f(x) cos(xλ) dx, apply the same exponential damping used in Fig. 4 (h(λ) → h(λ) exp(1 - λ meff/Pz) for λ > Pz/meff with Pz = 2 GeV and meff = 0.3 GeV), invert the Fourier transform, and record the relative change in the reconstructed x-space function for x in [0.3, 0.8]. If the change exceeds 10% for any realistic PDF, the claim that the exact asymptotic behavior matters little in the moderate-x region is not robust. As a sensitivity check, repeat the exercise with the toy model f(x) = (1-x)^β for β = 1, 2, 3 to map how the result depends on the large-x exponent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim that the exact asymptotic behavior of space-like lattice matrix elements matters little in the moderate-x region is demonstrated only with the smooth model of Eq. (3), whose Fourier transform decays as 1/lambda^2. Figure 4 shows that enforcing an exponential decay with meff/Pz = 0.15 barely changes the x-reconstruction for x > 0.3, but this conclusion depends on the assumption that x-space features are wider than the smearing scale meff/Pz. The authors state that usual parton distributions are smooth and therefore high Fourier harmonics are suppressed, but they do not test this assumption against realistic PDFs. If a real PDF has a sharper large-x threshold, for example behavior like (1-x)^beta with beta <~ 2, or has a moderate-x bump or oscillation, then the high harmonics are not negligible. In that case the exponential damping—with its uncertain, dataset-dependent meff—becomes the dominant regulator, and the exact asymptotic behavior matters greatly, contrary to the paper's headline claim. The authors' treatment of the bump case in Fig. 4 actually confirms that the exponential tail erases genuine features, so their assertion that the tail 'matters little' is only valid under an unquantified smoothness assumption. The reader's weakest-assumption diagnosis is correct: this smoothness prior is load-bearing, and the paper does not bound the range of 'usual' PDFs over which its conclusion holds.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This is a reply/comment to Chen et al. (arXiv:2505.14619) defending the authors' earlier critique of rigid parametric extrapolations in LaMET. The paper argues that reconstructing quasi-PDFs from finite, noisy lattice matrix elements is an ill-posed inverse problem whose defining symptom is strong sensitivity to the choice of regularization. It criticizes the rigid fits of Refs. [5,6] for underestimating uncertainty, contends that the exact asymptotic behavior of space-like correlators matters little in the moderate-x region when parton distributions are smooth, and claims that LaMET smears point-like x-space features over a width of order meff/Pz, making point-by-point PDF reconstruction dubious. The paper includes two toy-model figures (Figs. 3 and 4), visual re-examinations of published extrapolations (Figs. 1 and 2), and a correction of what the authors see as misunderstandings in Ref. [2].","tokens_in":12854,"tokens_out":6467,"duration_ms":75308,"significance":"If the central claims are correct, the paper provides a useful caution about the reliability of uncertainty bands from rigid few-parameter fits in current LaMET analyses and argues for more conservative, regularization-aware uncertainty quantification. The paper's strengths include explicit toy-model demonstrations, a clearly stated criterion (regularization sensitivity) for identifying ill-posedness, and a falsifiable expectation for smearing of x-space features. It also openly acknowledges that the Gaussian-process priors depend on the dataset (footnote [13]), which is an important limitation. However, the quantitative support for the main claims largely resides in the authors' previous papers (Refs. [1,8]), and the present manuscript does not deliver a systematic robustness test across plausible PDF shapes or hyperparameter choices. The smoothness assumption on which the 'asymptotic behavior matters little' claim rests is explicit but unquantified.","major_comments":[{"comment":"The claim that the exact asymptotic behavior of space-like correlators 'matters little' in the moderate-x region is demonstrated only for the smooth family of Eq. (3), whose Fourier harmonics decay as 1/lambda^2. The bump example in the same figure shows that if a PDF contains features narrower than the smearing scale meff/Pz, then exponential damping erases those features and the asymptotic behavior becomes the controlling regulator. The paper calls such bump models 'unphysical' but does not justify the implicit assumption that realistic PDFs are smooth on all scales relevant at moderate x. Please either quantify the class of PDFs for which the claim holds (for example by testing additional shapes with (1-x)^beta for beta <= 2 or with moderate-x oscillatory components), or explicitly state the claim as conditional on an assumed minimum feature width in x.","section":"Inverse problem and upper-bound on uncertainty; Fig. 4"},{"comment":"The authors acknowledge that the Gaussian-process hyperparameters are set with respect to the dataset at hand and that the priors are designed to enforce a desired behavior on the posterior. This makes the demonstration of 'strong sensitivity to the choice of regularization' in Fig. 3 partly a statement about the hyperparameter-setting rule, rather than a purely intrinsic property of the inverse problem. The closure tests in Ref. [8] mitigate this concern, but those tests are not reproduced or summarized here. Please provide a robustness check in which the hyperparameters are varied over a systematic, physically motivated range for the same dataset, and report how the spread of reconstructions changes. Without such a check, the claim that the bands in Fig. 3 reflect the full model uncertainty remains under-supported.","section":"Footnote [13] and 'Our proposal to study uncertainty'"},{"comment":"The argument that LaMET smears x-space features over a width of order meff/Pz is made with a simple Fourier toy model, not with the actual LaMET matching formalism. The light-cone PDF is obtained from the quasi-PDF through a perturbative matching kernel, and it is not automatic that exponential damping of the hadronic matrix element directly translates into a smearing of the light-cone PDF with no compensating structure. Please either derive the smearing kernel in the matching formalism, showing how the exponential decay of h(z,Pz) limits the resolution in x after matching, or restrict the claim to the quasi-PDF itself rather than to the light-cone PDF. As written, the conclusion that 'the claim that LaMET can reconstruct a point-by-point x-dependence of light-cone PDFs ... seems dubious' is plausible but not established by the evidence presented.","section":"Point-by-point reconstruction in the LaMET formalism"}],"minor_comments":[{"comment":"The caption says 'dotted lines' twice: 'We now add an unphysical bump at x = 0.5 (dotted lines). When the exponential decay is enforced (dotted lines), the bump is erased.' The second occurrence should presumably be 'solid lines' for the enforced-exponential-decay curves; please correct this to avoid ambiguity.","section":"Fig. 4 caption"},{"comment":"The parameter a in Eq. (3) is said to be varied, but the text does not specify the range of a used or the criteria for choosing the displayed values. For reproducibility, please state the values of a and the decay parameters (meff, Pz, threshold) used in Fig. 4.","section":"Eq. (3) and Fig. 4"},{"comment":"The footnote is long and defensive; the discussion about why dataset-dependent priors are still called 'priors' is not central to the scientific reply and would be better placed in a methods appendix or moved to the main text only if it is essential to the argument.","section":"Footnote [13]"},{"comment":"The term 'unphysical PDF models' is asserted without definition. If the criterion is simply that the bump is not part of the smooth benchmark family, please say so explicitly, or specify what physical conditions (positivity, support, smoothness) are meant.","section":"Fig. 4, lower panels"},{"comment":"The discussion first states that the bound of Ref. [5] is 'not really an upper bound' because parts of the quantity are ignored, then immediately concedes that 'there must be a value of N of the order of a few units where this broad estimate is reasonable.' This reads as contradictory; please clarify the intended status of the bound (e.g., a heuristic estimate with explicit caveats rather than a rigorous upper bound).","section":"Section 'Inverse problem and upper-bound on uncertainty'"}],"recommendation":"major_revision","confidential_remarks":"The paper is a scoping comment and is largely a restatement of positions already defended in Refs. [1,8]. The new material (Figs. 3 and 4) is illustrative rather than a complete proof. The main claim about asymptotic behavior is conditional on a smoothness assumption that, as the stress-test note correctly points out, is not quantified. The authors are already aware of the caveat, but the manuscript would be strengthened by either additional tests or a clearer conditional formulation. The polemical tone is within normal bounds for a comment-reply exchange, but the reliance on dataset-dependent hyperparameters (footnote [13]) deserves a more careful treatment before the paper can be considered fully convincing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a reply-comment that restates the authors' earlier critique of rigid LaMET extrapolations, and it makes one new illustrative point. The genuinely new element is the toy model in Fig. 4: imposing an exponential decay on Fourier modes leaves a smooth (1-x)^3-ish PDF nearly unchanged for x>0.3, but completely erases a sharp bump at x=0.5. That is a clean, honest demonstration that the relevant regulator is the assumed smoothness of the PDF, not the exact asymptotic tail.\n\nThe paper does a real service by pushing back on the 'forward problem' framing. The test of an ill-posed inverse problem is sensitivity to regularization, and the authors give concrete published examples (Figs. 1 and 2) where rigid few-parameter extrapolations look dubious—an anomalously fast decay, a central value drifting against the data, uncertainties shrinking too quickly. The examples are well chosen and fairly presented. They also acknowledge in footnote 13 that their Gaussian-process 'priors' are tuned per dataset, which is honest, even if it undercuts the claim of being a fully Bayesian procedure.\n\nThe soft spots are real. The central claim that the asymptotic behavior matters little in the moderate-x region is shown only for smooth models; the authors assert that 'usual' PDFs are smooth, but they never quantify the domain of validity. If a PDF has a steeper large-x falloff or resonance-like bumps, the toy model itself shows the exponential smearing erases features. So the conclusion is conditional in exactly the way the stress-test says, and the condition is not proven. Second, most of the substance is already in Refs. [1] and [8]; this comment is not self-contained. No code or data are provided, so the closure tests are not independently checkable. The examples are illustrative, not a systematic scan.\n\nWho should read it: lattice QCD practitioners and anyone using LaMET PDFs. It deserves a serious referee, but the referee should ask for a quantitative smoothness bound and ideally public code for the GPR closure tests. The negative point—that rigid fits can under-cover uncertainty—is persuasive; the positive point—that asymptotics are irrelevant—is weaker than the authors claim.\n\nRecommendation: send to peer review as a comment, not a desk reject.","headline":"A clear, well-argued reply-comment that restates the authors' earlier critique of rigid LaMET fits; the only new element is a toy model whose lesson is explicitly conditional on PDF smoothness.","tokens_in":13396,"tokens_out":4101,"would_cite":true,"duration_ms":44996,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.Gc"],"model":"deepseek-v4-flash","headline":"This comment argues that LaMET's reconstruction of parton distributions from noisy lattice data is an ill-posed inverse problem whose uncertainty is controlled by the assumed smoothness of the PDF, not by the exponential decay of the…","keywords":["LaMET","parton distribution functions","inverse problem","regularization dependence","Bayesian priors","Gaussian process","lattice QCD","uncertainty quantification"],"falsifier":"Run a closure test with a synthetic PDF that has a narrow bump of width much smaller than $m_{\\rm eff}/P_z$ at, say, $x=0.5$: generate its Fourier harmonics, impose the expected exponential decay, add noise at the level of current lattice data, and reconstruct with the paper's Bayesian procedure. If the bump survives at the claimed point-by-point resolution, the smearing argument is wrong; if the bump is erased, the comment's central limitation is confirmed.","tokens_in":12351,"feed_emoji":"⚛️","tokens_out":8929,"duration_ms":97553,"temperature":0.7,"pith_summary":"This comment responds to a rebuttal paper that defends rigid few-parameter extrapolation of noisy lattice data in the large momentum effective theory (LaMET). The authors argue that recovering a continuous quasi-PDF from a limited, noisy set of Fourier harmonics is an ill-posed inverse problem, and its tell-tale sign is strong sensitivity to the choice of regularization, not whether the integral relation can be formally inverted. They maintain that the exact exponential asymptotic behavior of space-like matrix elements has little effect on the moderate-x region; the real regulator is the assumed smoothness of parton distributions. If they are right, uncertainty bands from rigid fits are systematically too small, and the advertised point-by-point reconstruction of light-cone PDFs is instead smeared at a width of order the effective mass over the hadron momentum. The paper is a correction of what its authors call misunderstandings of their earlier proposal, not a new computation.","feed_headline":"Smoothness, not decay, controls moderate-x PDF errors","feed_subtitle":"Rigid fits to noisy lattice data understate the true uncertainty and wash out sharp x-features.","key_machinery":"The machinery is the truncated Fourier relation $f(y,P_z)=P_z\\int_{-\\infty}^{\\infty}\\frac{dz}{2\\pi}e^{iyP_z z}h(z,P_z)$ between the quasi-PDF and the space-like matrix elements, together with a diagnostic that defines ill-posedness by sensitivity to regularization rather than by formal invertibility. The second ingredient is a scaling estimate: exponential suppression of high harmonics with a decay length $P_z/m_{\\rm eff}$ acts like a convolution in $x$-space with width of order $m_{\\rm eff}/P_z$, so any feature narrower than that width is unrecoverable. The paper's proposed alternative regularization is a Bayesian prior combining smoothness and correlation structure in $x$-space with a Fourier-space prior that enforces exponential decay only beyond the largest measured separation, with hyperparameters chosen per dataset.","core_discovery":"The central claim is that LaMET and pseudo-PDF approaches face the same ill-posed inverse problem: both reconstruct a continuous function in $x$ from a limited, noisy set of Fourier-space data. The authors' diagnostic is empirical: the problem is ill-posed when different physically reasonable regularizations of that data produce markedly different reconstructions and uncertainty estimates. They argue that the exponential decay of large-$z$ matrix elements is not the controlling regulator: many current datasets end before one decay length, and varying the decay form barely changes the moderate-$x$ reconstruction. The quantity that actually controls the answer is the prior assumption that usual parton distributions are smooth on scales larger than $m_{\\rm eff}/P_z$, with sharp features erased by the Fourier inversion. From this it follows that the advertised point-by-point reconstruction of light-cone PDFs is dubious, because the reconstruction is intrinsically smeared at that width.","pith_inferences":["The smearing argument implies a quantitative resolution scale: comparing $m_{\\rm eff}/P_z$ with the narrowest features expected in realistic PDFs would show how much of the advertised point-by-point resolution is practically real at current momenta.","The regularization-sensitivity diagnostic could be exported to other hadronic inverse problems, such as global PDF extraction from deep-inelastic structure functions, to test whether quoted uncertainty bands depend on parametrization choice in the same way.","A concrete community protocol follows: closure tests with a truth distribution containing a deliberately sharp moderate-$x$ feature. The paper's logic predicts such features are systematically washed out by the exponential tail, and the test would confirm or refute that prediction."],"forward_implications":["Uncertainty bands from rigid few-parameter LaMET fits should be read as lower bounds on model dependence, not as the full error.","The distinction between LaMET and pseudo-PDF is not one of inverse problem versus forward problem; both share the same ill-conditioned Fourier inversion and the same sensitivity to regularization.","At moderate $x$, the precise functional form chosen for the asymptotic $z$ tail is not the dominant error; the smoothness prior on the PDF is.","At finite hadron momentum, LaMET cannot resolve sharp $x$-space features narrower than $m_{\\rm eff}/P_z$, so point-by-point claims hold only up to that smearing.","For datasets with small noise in the range $\\lambda\\sim5$–$15$, the concerns are reduced; the criticism is aimed at currently noisy datasets rather than at all LaMET measurements."],"supporting_citations":[{"why":"Previous paper by the same authors that first framed LaMET reconstruction as an ill-posed inverse problem and proposed the Bayesian regularization method; the current comment defends and corrects it.","marker":"[1]"},{"why":"The target comment that defends rigid parametric extrapolation and rejects inverse-problem methods; the paper's central disagreement is with this reference.","marker":"[2]"},{"why":"Lattice determination of the pion valence PDF whose dataset and parametric extrapolations illustrate where data lose signal and where the asymptotic tail is imposed.","marker":"[5]"},{"why":"Transversity PDF lattice calculation used to show the large model dependence of rigid extrapolation in both central value and uncertainty.","marker":"[6]"},{"why":"Recent independent work confirming that LaMET has a moderately tractable ill-posed inverse problem, cited to support the existence of the inverse problem.","marker":"[7]"},{"why":"The authors' earlier nonparametric reconstruction paper supplying the Gaussian-process method and closure-test-based hyperparameter choices.","marker":"[8]"},{"why":"Closure test with Bayesian and neural-network reconstructions that reproduced PDFs within the proposed uncertainty bound without assuming exponential decay.","marker":"[16]"},{"why":"Another closure test using Bayes-Gauss-Fourier transforms that reproduced PDFs within the bound, supporting the claim that smoothness rather than exponential decay is the regulator.","marker":"[17]"}],"fun_headline_variants":["Smoothness, not decay, sets PDF error bars","Rigid lattice fits undersell PDF uncertainty","Inverse problem blurs moderate-x PDF features","Regularization choice, not decay, controls PDF errors","Lattice PDFs: ill-posed inversion smears x-features"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that physically realized parton distributions are 'usual', i.e. smooth enough that high Fourier harmonics are suppressed; if a real PDF contains a sharp moderate-$x$ feature, the exponential smearing will erase it and the asymptotic behavior of the correlator would become important after all.","fun_headline_variants_meta":{"raw":{"variants":["Smoothness, not decay, sets PDF error bars","Rigid lattice fits undersell PDF uncertainty","Inverse problem blurs moderate-x PDF features","Regularization choice, not decay, controls PDF errors","Lattice PDFs: ill-posed inversion smears x-features"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000216,"raw_usage":{"total_tokens":1397,"prompt_tokens":873,"completion_tokens":524,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":445}},"tokens_in":489,"tokens_out":524,"duration_ms":5187,"temperature":1.0,"reasoning_tokens":445,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:24:58.838653+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a closure test with a synthetic PDF that has a narrow bump of width much smaller than $m_{\\rm eff}/P_z$ at, say, $x=0.5$: generate its Fourier harmonics, impose the expected exponential decay, add noise at the level of current lattice data, and reconstruct with the paper's Bayesian procedure. If the bump survives at the claimed point-by-point resolution, the smearing argument is wrong; if the bump is erased, the comment's central limitation is confirmed.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous paper by the same authors that first framed LaMET reconstruction as an ill-posed inverse problem and proposed the Bayesian regularization method; the current comment defends and corrects it."},{"cited_title":"might be perceived as a model","cited_arxiv_id":null,"evidence_quote":"The target comment that defends rigid parametric extrapolation and rejects inverse-problem methods; the paper's central disagreement is with this reference."},{"cited_title":"Comment on \"LaMET's Asymptotic Extrapolation vs. Inverse Problem\"","cited_arxiv_id":"2506.24037","evidence_quote":"Lattice determination of the pion valence PDF whose dataset and parametric extrapolations illustrate where data lose signal and where the asymptotic tail is imposed."},{"cited_title":"The lattice data should be truncated at a point within the sub-asymptotic region where they have decreased smoothly to near zero","cited_arxiv_id":null,"evidence_quote":"Transversity PDF lattice calculation used to show the large model dependence of rigid extrapolation in both central value and uncertainty."},{"cited_title":"The extrapolation model must preserve the cor- rect asymptotic behavior of the correlation function without introducing unphysical distortions","cited_arxiv_id":null,"evidence_quote":"Recent independent work confirming that LaMET has a moderately tractable ill-posed inverse problem, cited to support the existence of the inverse problem."},{"cited_title":"The extrapolated results should continue to de- crease in amplitude even with the presence of oscil- lations, consistent with asymptotic decay","cited_arxiv_id":null,"evidence_quote":"The authors' earlier nonparametric reconstruction paper supplying the Gaussian-process method and closure-test-based hyperparameter choices."}],"review_version":1}