{"id":"cf9bef27-696b-49dd-b0e0-5cfe4cb34f46","arxiv_id":"2608.07085","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A DNN fit to pion form-factor and lattice data produces quark and gluon GPDs; quark results agree with lattice, gluon results rely on a fitted normalization factor.","lead":"A neural network trained on pion form-factor data and lattice QCD maps out the pion's quark and gluon distributions in momentum and position space. The quark results match lattice calculations, but the gluon part needs a fitted rescaling factor to agree with lattice, a limitation the authors acknowledge.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed exact forward-limit constraint is not enforced by the stated quark NN output layer; if this is not fixed or clarified, the central claim that the framework preserves essential theoretical constraints is unsupported.","rationale":"The reader's weakest assumption, the x-independent gluon rescaling factor in Sec. IV, is a serious limitation, and I partly agree with it. But the even more load-bearing issue is in the quark sector, because the abstract's headline claim is about the quark GPDs and about preserving theoretical constraints. The architecture as written cannot satisfy the forward limit exactly: a generic network with exp output at t=0 has no reason to equal 1, and no term in the loss enforces it pointwise. This is not a disagreement with current consensus; it is an internal inconsistency between Eq. (4), Sec. IV's 'by construction' claim, and Sec. VI Eq. (23). The same pattern affects the gluon moment definitions (Eq. (6) vs Eq. (19)), reinforcing the conclusion that the paper's stated constraints are not reliably implemented. These issues are concrete and fixable, for example by adding an explicit t->0 architecture constraint or a pointwise penalty, stating which Mellin-moment formula is used, and releasing the code. They therefore do not justify rejection, but they do justify the reader's CONDITIONAL verdict. I would not change that verdict, hence UNCHANGED. I also note that no code or data are provided, which makes the requested check essential rather than optional.","tokens_in":18416,"tokens_out":13232,"duration_ms":115171,"concrete_test":"Train the valence-quark network exactly as described in Sec. III with the output activation of Sec. VI Eq. (23) and the loss of Eq. (11), then evaluate H_q(x,0)/q_v(x) for x in [0.05, 0.95]. If the ratio is not 1 for every x, Eq. (4) is not satisfied by construction and the lattice comparison at t=0 in Fig. 10 is not a meaningful check; if the authors instead impose NN(x,0)=1 pointwise, that constraint must be stated and placed in the loss function.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sec. II Eq. (3) parameterizes H_q(x,t)=q_v(x)e^{c|t|}NN(x,t) and Eq. (4) claims H_q(x,0)=q_v(x). Sec. IV says 'No additional normalization penalty is required for the valence-quark sector because the NN correction is constrained to satisfy NN(x,0)=1 for both the quark and gluon GPDs,' so the forward limit is 'satisfied exactly by construction.' But the only output-layer formula for quarks in Sec. VI, Eq. (23), is NNq(x,t)=exp(NN(x,t)), which does not force NNq(x,0)=1; only the gluon activation, Eq. (24), contains the explicit factor |t|/(1+|t|) that enforces NNg(x,0)=1. Thus the valence-quark forward limit is either not exact, making the 'excellent agreement' with lattice at t=0 in Fig. 10 non-interpretable, or a pointwise constraint is being imposed somewhere that is not described or included in the loss (Eq. 11). The normalization statements are also contradictory: Eq. (15) imposes chi2_norm=[(F_pi(0)-1)/0.01]^2, while Sec. IV says no normalization penalty is needed. And the gluon sector has the same kind of inconsistency: Eq. (6) defines A_g(t)=integral_0^1 dx x H_g(x,t), while Eq. (19) uses A_g(t)=integral_0^1 dx H_g(x,t); one formula is wrong, and whichever is used changes what the gluon fit actually constrains. These are not cosmetic typos: they determine whether the extracted objects satisfy the foundational sum rules the paper claims to preserve.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents a deep neural-network (DNN) framework for extracting the pion's unpolarized quark and gluon generalized parton distributions (GPDs) at zero skewness. The GPDs are parameterized as input pion PDFs multiplied by an exponential Regge-inspired momentum-transfer factor and a trainable neural-network correction. The quark-sector network is fitted to pion electromagnetic form factor (EMFF) data, squared EMFF data, and lattice-QCD EMFF points, while the gluon-sector network is fitted to lattice-QCD determinations of the gluon gravitational form factor A_g(t). The authors use the JAM21 and xFitter PDF ensembles to propagate uncertainties and compare their extracted quark GPDs with lattice-QCD calculations. The paper claims that the framework preserves theoretical constraints such as the forward limit H(x,0)=PDF and the charge normalization F_pi(0)=1, and that the resulting quark GPDs agree with lattice QCD.","tokens_in":18862,"tokens_out":5922,"duration_ms":51418,"significance":"If the framework were fully consistent, this would be a useful contribution: it demonstrates a flexible, nonparametric way to map pion PDFs into GPDs, reports a genuinely good EMFF fit (chi^2/N about 1.4), and uses the full PDF replica ensemble to quantify uncertainties. The comparison of the extracted quark GPDs with lattice QCD is a meaningful independent check. However, several load-bearing issues currently prevent the central claims from being accepted: the quark forward limit is not enforced by the stated output layer, the gluon Mellin moment is used inconsistently between the formalism and the fit, the gluon normalization agreement is imposed by an ad hoc rescaling, and the text and loss function contradict each other on the normalization constraint. These issues affect the interpretation of the main results and require substantial revision.","major_comments":[{"comment":"The forward-limit constraint is not enforced by the stated quark output layer. Equation (3) defines H_q(x,t)=q_v(x)e^{c|t|}NN(x,t), and Eq. (23) gives NN_q(x,t)=exp(NN(x,t)) for the valence-quark sector. Consequently, H_q(x,0)=q_v(x)exp(NN(x,0)), which equals q_v(x) only if the network output happens to vanish at t=0. No such pointwise constraint is listed in the loss function, and the claim in Sec. IV that \"NN(x,0)=1\" and that the forward limit is \"satisfied exactly by construction\" is therefore incorrect. This also undermines the interpretation of the -t=0 lattice comparison in Fig. 10, since the quark GPD at t=0 is not guaranteed to reduce to the input PDF. The authors must either modify the quark output layer to enforce NN_q(x,0)=1, add an explicit penalty, or remove the exactness claim and re-interpret the lattice comparison.","section":"Sec. II / Appendix Eq. (23)"},{"comment":"The gluon gravitational form factor is defined inconsistently. Equation (6) states A_g(t)=∫ dx x H_g(x,t), which is the second Mellin moment, while Eq. (19), used in the fit, states A_g(t)=∫ dx H_g(x,t), which is the first Mellin moment. These are different quantities. Since Eq. (19) is the formula actually implemented in the loss function, the quantity constrained by the lattice A_g(t) data is the first moment of H_g, not the second moment entering the sum rule of Eq. (6). The reported values A_g(0)=0.37 (JAM21) and 0.26 (xFitter) are consistent with the PDF gluon momentum fraction ∫ xg(x)dx, not with the second Mellin moment of Eq. (6). The authors should specify which sum rule is being fitted and correct either Eq. (6) or Eq. (19) and all subsequent interpretation.","section":"Sec. II Eq. (6) / Sec. III Eq. (19)"},{"comment":"The introduction of rescaling factors 1.46 (JAM21) and 2.07 (xFitter) makes the agreement between the rescaled gluon GPD normalization and the lattice A_g(0) a construction, not a validation. A single x-independent multiplicative constant cannot undo a mismatch that the authors themselves attribute to the poorly constrained small-x gluon PDF; the x-dependence of the extracted gluon GPD remains inherited from the input PDF, and the fit only constrains the t-dependence of a single moment integral. The gluon extraction should be presented as a model-dependent estimate with a dedicated systematic uncertainty associated with the rescaling, rather than as an independent determination of the gluon GPD.","section":"Sec. IV, gluon rescaling paragraph"},{"comment":"The text and the loss function contradict each other on the charge-normalization constraint. The loss function in Eq. (11) explicitly includes chi^2_norm = [(F_pi(0)-1)/0.01]^2, and Fig. 6 shows a separate chi^2_F(0) loss component, but Sec. IV states that \"No additional normalization penalty is required for the valence-quark sector\" because the forward limit is satisfied by construction. These statements cannot both be true. The authors should state clearly whether F_pi(0)=1 is imposed by a penalty, by the network architecture, or by both, and adjust the text and the loss accordingly.","section":"Sec. III Eq. (15) / Sec. IV"}],"minor_comments":[{"comment":"Both equations use the symbol A_g(t) for different Mellin moments; please use distinct notation (e.g., A_g^{(2)}(t) and A_g^{(1)}(t)) or state explicitly which moment is being considered in each context.","section":"Eq. (6) and Eq. (19)"},{"comment":"The legend entries \"LQCD\" and \"Lattice QCD\" appear to refer to the same dataset; unify the naming to avoid confusion.","section":"Fig. 11 caption"},{"comment":"The parameter T_max is used in the cosine-annealing schedule and also identified as the total number of training epochs; if these are intended to be the same, please say so explicitly, otherwise use separate symbols.","section":"Eq. (22) and text"},{"comment":"The regularization term L_reg is defined in Eq. (16) for the quark sector, but the paragraph discussing gluon regularization mentions \"additional regularization terms\" without giving their explicit form; provide the gluon L_reg expression or remove the statement.","section":"Sec. III, text after Eq. (16)"},{"comment":"The sentence \"The exact form of the output layer have been discussed in Appendix VI\" contains a subject-verb agreement error and should refer to the appendix without a section number that duplicates the main text numbering.","section":"Sec. III, output layer sentence"}],"recommendation":"major_revision","confidential_remarks":"The paper presents a promising approach, but the internal inconsistencies enumerated in the major comments are load-bearing for the central claims. The quark forward-limit issue and the gluon Mellin-moment inconsistency mean the current manuscript does not support its stated conclusions. I would encourage the authors to fix the architecture or the claims, re-run the fits, and resubmit; with those corrections, the framework could be a valuable contribution to pion tomography. In its present form, however, the manuscript should not be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is the first NN-based pion GPD extraction, and the quark-sector part is a genuinely useful result. The gluon sector is weaker, and several claims in the paper don't survive contact with its own equations.\n\nWhat is actually new: an established DNN-extraction recipe, already used for nucleon GPDs, applied to the pion. JAM21 and xFitter PDFs supply the forward limit; EMFF data and lattice results supply the constraints. The quark fit is solid: chi2/N about 1.4 over 176 points, extracted charge radius 0.668 fm against the PDG 0.659 fm, and the comparison with the lattice quark GPDs of Ding et al., especially the RGR version, is a meaningful independent check. The paper is also candid about the gluon normalization problem.\n\nThe soft spots, in order of real weight. First, the 'exact by construction' forward-limit claim is not supported for the quark sector. Eq. (23) sets NNq(x,t)=exp(output), which does not force NNq(x,0)=1; only the gluon layer, Eq. (24), has the explicit |t|/(1+|t|) factor. Meanwhile Sec. IV says no normalization penalty is needed, while Eq. (15) and Fig. 6 show a chi2_F(0) term doing that job. The fit probably delivers Hq(x,0) approximately equal to qv(x), so the science is likely fine, but the text contradicts itself, and the t=0 lattice comparison in Fig. 10 is only interpretable once the text is corrected.\n\nSecond, the gluon Mellin-moment formulas disagree: Eq. (6) carries an x-weight, Eq. (19) does not. The quoted A_g(0) values suggest Eq. (19) was used, but a reader cannot tell. This changes what the gluon fit constrains, so it has to be fixed.\n\nThird, the gluon rescaling factors, 1.46 and 2.07, matter more than the text admits. After rescaling, A_g(0) matches lattice by construction, and the x-dependence of the gluon GPD is essentially the input PDF's. The paper acknowledges this, but the uncertainty bands omit the rescaling uncertainty, so they overstate what the lattice data constrain. I would read the gluon GPDs as a model-dependent projection, not a true extraction.\n\nFourth, no code or data are released. For an NN method that is a real reproducibility gap, though not a physics error.\n\nWho this is for: pion structure, lattice-to-phenomenology comparisons, EIC projections. It deserves a serious referee and probably major revision, not a desk reject. Fix the forward-limit and Mellin-moment statements, propagate the rescaling uncertainty into the gluon bands, release the code, and the quark-sector result stands on its own.","headline":"First NN-based pion GPD extraction; the quark sector is a solid, useful result, but the gluon rescaling and internal inconsistencies in the forward-limit and Mellin-moment claims make the paper's framing run ahead of its equations.","tokens_in":19346,"tokens_out":12257,"would_cite":true,"duration_ms":94208,"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":"This paper claims that a physics-informed neural network can extract the pion's quark and gluon GPDs from form-factor data while preserving PDF forward limits, with valence-quark results matching lattice QCD.","keywords":["pion","generalized parton distributions","deep neural networks","physics-informed neural networks","electromagnetic form factor","gravitational form factor","lattice QCD","hadron tomography"],"falsifier":"Directly computing the $x$-dependent pion gluon GPD on the lattice over the same $|t|$ range used here would settle the claim: if the rescaled $H^g(x,t)$ disagrees with the lattice result at small $x$, the constant-rescaling assumption fails. Alternatively, a future measurement of the pion gluon PDF that yields $\\int_0^1 dx\\,xg(x)\\approx0.55$ to $0.60$ at $\\mu^2=4\\,\\mathrm{GeV}^2$ would confirm the normalization correction, while a value near the input PDFs' 0.26 to 0.37 would indicate the rescaling is absorbing a lattice-systematic effect rather than a genuine gluon deficit.","tokens_in":18185,"feed_emoji":"🧠","tokens_out":11779,"duration_ms":93420,"temperature":0.7,"pith_summary":"This paper claims that a physics-informed neural network can extract the pion's unpolarized quark and gluon generalized parton distributions (GPDs) at zero skewness from the known pion PDFs, experimental pion electromagnetic form-factor data, and lattice-QCD determinations of the gluon gravitational form factor. The network is constructed so that each GPD reduces exactly to its input PDF in the forward limit, remains non-negative, and obeys the form-factor sum rules. The extracted valence-quark GPDs agree with independent lattice-QCD calculations over most of the explored kinematics, and the corresponding charge radius, about 0.67 fm, is close to the world average. The gluon sector matches the lattice gravitational form factor only after an overall rescaling of the gluon PDF input, which the paper traces to limited small-x gluon constraints. If the claim holds, the framework offers a flexible, largely model-independent route to pion tomography that extends to the nucleon.","feed_headline":"Neural nets reproduce pion structure seen in lattice QCD","feed_subtitle":"Quark distributions match lattice data across most kinematics; the gluon sector needs a rescaling factor.","key_machinery":"The load-bearing object is the multiplicative physics-informed parameterization $H^q(x,t)=q_v^{\\pi}(x)\\,e^{c|t|}\\,\\mathrm{NN}(x,t)$ and $H^g(x,t)=xg(x)\\,e^{c_g|t|}\\,\\mathrm{NN}(x,t)$, which confines the network to learning the residual correction beyond the forward PDF and the Regge-inspired exponential $t$ dependence. Positivity is built in through exponential output activations, with the gluon output written $\\exp[\\tanh(\\mathrm{NN}(x,t))|t|/(1+|t|)]$ so that $H^g(x,0)=xg(x)$ holds exactly. The loss function is a $\\chi^2$ sum over pion EMFF data and squared EMFF data, plus a charge-normalization penalty and a regularization term for the quark sector; the gluon sector is trained on the lattice $A_g(t)$ data with its own regularization. All results are quoted at $\\mu^2=4\\,\\mathrm{GeV}^2$, and the PDF replica ensemble is propagated through the network to produce $1\\sigma$ GPD uncertainty bands.","core_discovery":"The central claim is that a single leading-twist pion GPD in each sector can be determined nonparametrically through the multiplicative ansatz $H^q(x,t)=q_v^{\\pi}(x)\\,e^{c|t|}\\,\\mathrm{NN}(x,t)$ and $H^g(x,t)=xg(x)\\,e^{c_g|t|}\\,\\mathrm{NN}(x,t)$, where the input PDFs fix the forward limits $H^q(x,0)=q_v^{\\pi}(x)$ and $H^g(x,0)=xg(x)$, the exponential provides the momentum-transfer profile, and the network learns the residual $x$ and $t$ dependence. The network parameters are fixed by minimizing $\\chi^2$ losses built on the sum rules $F_\\pi(t)=\\sum_q e_q\\int_{-1}^{1}dx\\,H^q(x,t)$ and $A_g(t)=\\int_0^1 dx\\,x\\,H^g(x,t)$, with $\\chi^2/N\\approx1.4$ for 176 quark-sector data points and $\\chi^2/N\\approx0.84$ for 50 gluon-sector lattice points. The extracted quark GPDs are in good agreement with the renormalization-group-resummed lattice calculation except at large $x$ and large $|t|$, and the fit returns a pion charge radius of 0.668 and 0.667 fm for the two PDF inputs, close to the world-average 0.659 fm. Gluon results require rescaling the input gluon PDF by a factor of about 1.5 or 2.1 to reproduce the lattice $A_g(t)$; after that rescaling, the two PDF inputs yield nearly identical gluon GPDs.","pith_inferences":["If the paper's constant-rescaling picture is correct, the same network trained without rescaling should fail the lattice $A_g(t)$ constraint by a factor that is independent of $t$; a useful diagnostic is to plot the ratio $A_g^{\\mathrm{LQCD}}(t)/A_g^{\\mathrm{model}}(t)$ and check whether the residual is flat in $t$.","A stronger version of the paper's normalization argument predicts that direct lattice calculations of the $x$-dependent gluon GPD, once available, will reproduce the rescaled $H^g(x,t)$ at small $x$; if they instead follow the raw input gluon PDF scaled only at $A_g(0)$, the rescaling is absorbing a low-$x$ deficit the paper does not model.","The near identity of the quark-sector results from the two independent PDF inputs suggests the electromagnetic form-factor data, rather than the valence PDF choice, controls the extracted $H^q(x,t)$; replacing the valence input with a third phenomenological PDF set would provide a cheap cross-check of that claim."],"forward_implications":["A future measurement of the pion electromagnetic form factor at larger $|t|$ than the fitted range would directly test the trained network's extrapolation, because $F_\\pi(t)$ is an integral of the extracted quark GPD.","The same physics-informed ansatz can be applied to the nucleon, where two independent unpolarized GPDs enter and the constraints are richer; the paper identifies this as the natural next step.","The extraction quantifies pion tomography: Fourier transforming the zero-skewness GPDs yields transverse impact-parameter densities in which valence quarks localize at large $x$ and gluons dominate at small $x$.","The gluon-sector result implies that present pion gluon PDFs carry too little momentum at the reference scale unless rescaled, so improved small-$x$ pion gluon data would directly narrow the extracted gluon GPD uncertainty."],"supporting_citations":[{"why":"Supplies the pion valence-quark PDF input whose forward limit defines $H^q(x,0)$; its replica ensemble sets the quark-sector uncertainty band.","marker":"[60]"},{"why":"Supplies the second pion PDF set, including the gluon PDF input for $H^g(x,0)$; its replicas set the gluon-sector uncertainty band.","marker":"[61]"},{"why":"Provides the lattice-QCD $x$-dependent pion quark GPDs against which the extracted $H^q(x,t)$ is compared and found consistent except at large $x$ and $|t|$.","marker":"[48]"},{"why":"One of the two lattice-QCD pion EMFF calculations included in the quark-sector loss, labeled Lattice I.","marker":"[63]"},{"why":"The other lattice-QCD pion EMFF calculation included in the quark-sector loss, labeled Lattice II.","marker":"[64]"},{"why":"Provides the lattice-QCD gluon gravitational form-factor $A_g(t)$ data used to constrain the gluon GPD, labeled Lattice23.","marker":"[52]"},{"why":"Provides the earlier lattice-QCD $A_g(t)$ dataset used alongside Ref. [52] in the gluon-sector loss, labeled Lattice18.","marker":"[53]"},{"why":"NA7 elastic-scattering data anchor the small-$|t|$ EMFF behavior and thereby constrain the extracted charge radius.","marker":"[35]"},{"why":"Bebek electroproduction data extend the EMFF constraint to high $|t|$; this dataset carries the largest per-point $\\chi^2$ in the fit.","marker":"[31]"}],"fun_headline_variants":["Neural nets extract pion GPDs, match lattice for quarks","DNN pion GPDs: quark part matches lattice, gluon needs scaling","AI maps pion's quark and gluon distributions from data","Pion GPDs via deep learning: agreement with lattice for quarks","Gluon GPDs from neural nets need a rescaling factor of ~2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The gluon-sector results stand on a single, $x$- and $t$-independent rescaling factor (1.46 for one PDF analysis and 2.07 for the other) that forces the input gluon PDF's second moment to match lattice $A_g(0)$; if the factor-of-two mismatch is concentrated at small $x$, the extracted gluon GPD has the wrong $x$-dependence.","fun_headline_variants_meta":{"raw":{"variants":["Neural nets extract pion GPDs, match lattice for quarks","DNN pion GPDs: quark part matches lattice, gluon needs scaling","AI maps pion's quark and gluon distributions from data","Pion GPDs via deep learning: agreement with lattice for quarks","Gluon GPDs from neural nets need a rescaling factor of ~2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0012,"raw_usage":{"total_tokens":5077,"prompt_tokens":1206,"completion_tokens":3871,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":822,"completion_tokens_details":{"reasoning_tokens":3772}},"tokens_in":822,"tokens_out":3871,"duration_ms":21608,"temperature":1.0,"reasoning_tokens":3772,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:29:28.667085+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly computing the $x$-dependent pion gluon GPD on the lattice over the same $|t|$ range used here would settle the claim: if the rescaled $H^g(x,t)$ disagrees with the lattice result at small $x$, the constant-rescaling assumption fails. Alternatively, a future measurement of the pion gluon PDF that yields $\\int_0^1 dx\\,xg(x)\\approx0.55$ to $0.60$ at $\\mu^2=4\\,\\mathrm{GeV}^2$ would confirm the normalization correction, while a value near the input PDFs' 0.26 to 0.37 would indicate the rescaling is absorbing a lattice-systematic effect rather than a genuine gluon deficit.","supporting_citations":[{"cited_title":"|Fmodel π (ti)|2−|F exp π (ti)|2 σi #2 .(13) The FF sum rule condition Fπ(0)=1,(14) is imposed through the normalization penalty χ2 norm =","cited_arxiv_id":null,"evidence_quote":"Supplies the second pion PDF set, including the gluon PDF input for $H^g(x,0)$; its replicas set the gluon-sector uncertainty band."},{"cited_title":"Elfwing, E","cited_arxiv_id":null,"evidence_quote":"One of the two lattice-QCD pion EMFF calculations included in the quark-sector loss, labeled Lattice I."}],"review_version":2}