{"id":"d63d384f-9091-47fb-b304-5cbb2f84b524","arxiv_id":"2607.02647","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Exact Shapley values on PDF flavors treat χ² as the cooperative payoff, revealing data constraints and an unexpected intermediate-x gluon insensitivity.","lead":"Shapley values from cooperative game theory quantify how each parton flavor in a given x-region drives the χ² agreement between PDF predictions and collider data. The tool recovers textbook data-PDF mappings, exposes a gluon blind spot near x≈0.07, and can redesign NNPDF hyperparameter folds.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"The single-bump, same-sign-for-all-flavors probe may not faithfully isolate the local gluon sensitivity that underpins the central claim.","rationale":"The reader correctly isolates the single-bump, symmetric-coalition construction as the weakest modeling choice. That choice is load-bearing for the strongest claim (the universal gluon blind spot), because every quantitative statement about local sensitivity is obtained under precisely those restrictions. The theoretical argument that links the dip to the vanishing N=2 eigenvalue is elegant and independent of the probe, but the numerical evidence that the dip appears in x-space is not. The concrete test above isolates the probe dependence without requiring a full multi-bin Shapley calculation; if the dip survives both variants, the concern is largely dispelled and the CONDITIONAL verdict can stand. If it does not, the claim must be weakened to “a probe-dependent feature consistent with the N=2 flat direction.” No stronger objection (internal inconsistency, numerical pathology, or failure to recover known lore) is apparent, so the reader’s overall assessment remains appropriate.","tokens_in":31560,"tokens_out":641,"duration_ms":7096,"concrete_test":"Recompute the gluon Shapley scan of Fig. 4 (and the three-set comparison of Fig. 12) with two modified probes: (i) only the gluon is ever perturbed (all other flavors fixed at their baseline), and (ii) the width w is halved and doubled while keeping the amplitude fixed. If the dip at xµ ≈ 0.07 disappears or moves by more than a factor of two in either variant, the local-sensitivity interpretation of the central claim is not robust.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The strongest claim rests on the observation that a localized Gaussian perturbation of the gluon (amplitude = 68 % PDF uncertainty, fixed log-width w, sum rules restored) produces a pronounced dip in the exact Shapley value around xµ ≈ 0.07, and that this dip is universal across NNPDF4.0, CT18 and MSHT20. That observation is obtained under the restrictive probe of Sec. 3.2 (Eqs. 21–23): every flavor inside a coalition receives an identical-sign bump of the same width, the value function averages the + and − signs, and only one x-region is perturbed at a time. Because the momentum sum rule and the coupled singlet–gluon evolution are non-local, a simultaneous same-sign shift of several flavors can cancel or amplify the marginal contribution of the gluon in a way that a pure single-flavor, single-x probe would not. Consequently the reported dip could be an artifact of the particular coalition geometry rather than a genuine local loss of sensitivity of the PDF-to-observable map. The paper itself notes (end of Sec. 3.2) that a fuller feature space (multiple x-bins × signed perturbations) is combinatorially intractable, so the claim that the simplified probe is “faithful” remains an untested modeling assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces exact Shapley values as a model-agnostic diagnostic for parton distributions, treating the eight PDF flavors (in flavor or evolution basis) as players in a cooperative game whose value function is the experimental χ^{2}. A localized Gaussian perturbation of amplitude equal to the 68% PDF uncertainty, with sum rules restored after each coalition, is applied one x-region at a time; the resulting Shapley values φ_j(x_µ) quantify the marginal contribution of each flavor to the description of a chosen dataset. Applied to NNPDF4.0 (and cross-checked on CT18 and MSHT20), the method recovers the standard data-to-PDF map while revealing a previously unnoticed, universal dip in gluon sensitivity near x≈0.07. The dip is traced to the vanishing eigenvalue of the singlet-gluon anomalous-dimension matrix at N=2 (enforced by the momentum sum rule) and mapped to x-space via the saddle point of the Mellin inversion. Two applications are shown: a diagnostic of under-constrained regions relevant to BSM searches and Higgs production, and a proof-of-concept improvement of the K-fold design used in NNPDF hyperparameter optimization.","tokens_in":31855,"tokens_out":1128,"duration_ms":23489,"significance":"If the results hold, the work supplies a quantitative, methodology-independent tool that can be applied to any existing PDF set without re-fitting. Exact evaluation over the 128 coalitions (rather than the additive SHAP approximation) correctly retains flavor correlations and sum-rule constraints, a clear technical advance over prior XAI proposals in the field. The recovery of textbook expectations places those expectations on a numerical footing, while the gluon dip—confirmed on three independent global fits and given a clean QCD explanation—is a genuine new observation with direct phenomenological consequences for gluon-fusion Higgs and possible new-physics reabsorption. The fold-optimization example further demonstrates immediate practical utility for the NNPDF pipeline. These strengths make the paper a useful addition to the PDF-diagnostics literature.","major_comments":[{"comment":"Sec. 3.2, Eqs. (21–23): the central claim that the observed gluon dip is a genuine local loss of sensitivity rests on a highly restrictive probe (identical-sign Gaussian bump of fixed logarithmic width w applied simultaneously to every flavor inside a coalition, then averaged over + and - signs, one x-bin at a time). Because the momentum sum rule and the coupled singlet-gluon evolution are non-local, this coalition geometry can cancel or amplify the marginal contribution of the gluon relative to a pure single-flavor, single-x probe. The paper itself notes the combinatorial barrier to a fuller feature space, yet provides no robustness checks (variation of w, single-flavor-only games, or opposite-sign coalitions). Without such tests the dip could be an artifact of the chosen probe rather than a property of the PDF-to-observable map.","section":"Sec. 3.2, Eqs. (21-23)"},{"comment":"Appendix B and Table 9: the iterative removal of any dataset for which v_Di(S)>10^5 is an ad-hoc regularization whose threshold is a free parameter. The discarded points (chiefly fixed-target DY ratios) change with PDF set and with x_µ; it is not shown how much the gluon Shapley profile (especially the depth and location of the dip) moves when the threshold is varied or when those datasets are retained with a milder cut. Because the dip is the paper’s principal new result, its stability under this pruning must be quantified.","section":"Appendix B, Table 9"}],"minor_comments":[{"comment":"Fig. 4 and Fig. 5: the symlog color scale and the vertical offsets used for readability make quantitative comparison of the size of φ_j across x_µ difficult; a linear inset or a table of peak values would help.","section":"Figs. 4-5"},{"comment":"Sec. 4.3: the broader secondary dip near x~10^{-2} (visible for CT18 and, more weakly, for the other sets) is noted but not discussed; a short remark on its possible relation to the gluon-fusion Higgs region would strengthen the phenomenological claims.","section":"Sec. 4.3"},{"comment":"The Gaussian width w is fixed once and for all to the bin spacing of the 50-point scan; a one-sentence statement of the numerical value and a brief stability check would remove an unnecessary free parameter from the method.","section":"Sec. 3.2"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid, well-written contribution that sits comfortably within the scope of a hep-ph journal. The probe-limitation and regularization issues are real but addressable with a modest amount of additional numerical work; they do not undermine the overall soundness. I see no citation or novelty concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is an exact, correlation-preserving Shapley construction on the eight PDF flavors. They take a finished PDF set as a black box, apply a sum-rule-respecting Gaussian bump of height equal to the 68 % PDF uncertainty, and score coalitions by the experimental χ^{2}. Because there are only eight players they can sum all 128 coalitions without the independence approximation that breaks SHAP in HEP. That alone is useful: it works for NNPDF, CT18 or MSHT20 without ever looking inside the fit.\n\nWhat they get is solid. The method recovers the expected data-to-PDF map (HERA \to small-x singlet/gluon, fixed-target + DY \to valence and flavor separation, jets/top \to large-x gluon) and puts numbers on it. The real find is the pronounced dip in the gluon Shapley value around x ≈ 0.07. They give a clean Mellin-saddle explanation: the N=2 moment is protected by the momentum sum rule, so scaling violations cannot constrain it, and that moment maps to x ≈ 0.07. The same dip appears in CT18 and MSHT20, so it is not an NNPDF artifact. That is a genuine local loss of sensitivity that ordinary uncertainty bands hide by smoothness; it matters for gg\to H and for possible BSM bumps that could be reabsorbed. The fold-design check is a sensible secondary application.\n\nThe soft spot is exactly the one the stress-test flags: the probe is a single, same-sign, fixed-width bump applied to every flavor in a coalition and then averaged over ±. Non-local sum rules and singlet-gluon mixing mean this geometry could in principle cancel or amplify the gluon contribution. The paper itself notes that a multi-bin, signed feature space is combinatorially impossible, so the faithfulness of the simplified probe remains an assumption. Still, the fact that the dip is universal across three independent sets and sits exactly where the Mellin argument predicts makes an artifact less likely. Dataset pruning when ratios go pathological is transparent and minor. No code is shipped, which is annoying but not fatal; the description is enough to re-implement.\n\nThis is for people who care about PDF systematics, Higgs precision, or BSM reabsorption. It is not a revolution, but it is a practical tool that does something existing sensitivity methods cannot. I would send it to a serious referee and I would cite the gluon-dip result.","headline":"Clean, model-agnostic PDF diagnostic that recovers textbook lore and exposes a real intermediate-x gluon blind spot; the single-bump probe is a limitation but not a fatal one.","tokens_in":32463,"tokens_out":633,"would_cite":true,"duration_ms":6258,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Shapley values on PDF flavors reveal a hidden gluon blind spot at intermediate x and quantify which data constrain which partons.","keywords":["parton distribution functions","Shapley values","explainable AI","gluon PDF","PDF sensitivity","NNPDF","Mellin moments","hyperparameter optimization"],"falsifier":"Recompute the gluon Shapley profile after replacing the Gaussian bump with an alternative localized deformation (e.g., a Mellin-space moment shift or a different width) and check whether the dip at x ≈ 0.07 persists across NNPDF4.0, CT18 and MSHT20; its disappearance would falsify the claimed universality of the sensitivity loss.","tokens_in":32476,"feed_emoji":"⚛️","tokens_out":720,"duration_ms":5956,"temperature":0.7,"pith_summary":"Parton distribution functions (PDFs) sit at the heart of every hadron-collider prediction, yet the map from experimental data to PDFs is nonlocal, scale-dependent and often opaque. This paper treats the eight PDF flavors as players in a cooperative game whose payoff is the χ² agreement between theory and data. By applying a calibrated Gaussian bump to each flavor in a chosen x-region, restoring sum rules, and averaging the exact Shapley value over all coalitions, the authors obtain a quantitative score of how much each flavor, in each x-bin, contributes to the fit quality. The scores recover textbook expectations about which processes constrain which PDFs, but they also expose a previously unnoticed dip in gluon sensitivity around x ≈ 0.07. That dip is traced to the vanishing eigenvalue of the singlet-gluon anomalous-dimension matrix at the momentum sum-rule moment N = 2 and appears in three independent global PDF sets. The same diagnostic can be used to redesign the data folds employed in neural-network hyperparameter optimisation.","feed_headline":"Shapley values expose a hidden gluon blind spot at x≈0.07","feed_subtitle":"Exact game-theoretic scores recover which data constrain which PDFs and flag a universal loss of sensitivity.","key_machinery":"The exact Shapley value φ_j(x_µ) for flavor j at perturbation center x_µ, obtained by averaging the marginal change in χ² over all 128 coalitions of the eight flavors while a calibrated Gaussian bump is applied and sum rules are re-enforced.","core_discovery":"Exact Shapley values computed on the eight PDF flavors, after a localized Gaussian perturbation of amplitude equal to the 68 % PDF uncertainty and with sum rules restored, quantitatively recover the known data-to-PDF map and reveal a universal local loss of sensitivity of the gluon PDF around x ≈ 0.07 that is invisible to ordinary uncertainty bands.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Shapley values flag gluon PDF sensitivity loss at x≈0.07","Game-theoretic PDF scores uncover gluon blind spot near x=0.07","Exact Shapley recovery shows gluon soft spot at intermediate x","Cooperative PDF game exposes universal gluon gap around x≈0.07","Shapley on PDFs pinpoints data-map gaps and gluon loss at x~0.07"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"That a single, symmetric Gaussian bump of fixed logarithmic width, applied simultaneously to every flavor inside a coalition and then averaged over positive and negative signs, faithfully probes the local sensitivity of the full nonlinear PDF-to-observable map.","fun_headline_variants_meta":{"raw":{"variants":["Shapley values flag gluon PDF sensitivity loss at x≈0.07","Game-theoretic PDF scores uncover gluon blind spot near x=0.07","Exact Shapley recovery shows gluon soft spot at intermediate x","Cooperative PDF game exposes universal gluon gap around x≈0.07","Shapley on PDFs pinpoints data-map gaps and gluon loss at x~0.07"]},"model":"grok-4.5","effort":"low","cost_usd":0.005396,"raw_usage":{"total_tokens":1504,"prompt_tokens":811,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":53960000,"prompt_tokens_details":{"text_tokens":811,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":603,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":811,"tokens_out":90,"duration_ms":5205,"temperature":1.0,"reasoning_tokens":603,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T07:56:20.390305+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Recompute the gluon Shapley profile after replacing the Gaussian bump with an alternative localized deformation (e.g., a Mellin-space moment shift or a different width) and check whether the dip at x ≈ 0.07 persists across NNPDF4.0, CT18 and MSHT20; its disappearance would falsify the claimed universality of the sensitivity loss.","supporting_citations":[],"review_version":1}