{"id":"565e4a8a-00c1-4544-a893-7e0f3e414968","arxiv_id":"2504.12380","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Derives a factorization theorem and NLL resummation for SEMD-based event correlators, predicting a universal ledge in quark-jet non-Gaussianity.","lead":"Collider event data can be treated as a geometric space. This paper derives the first precision perturbative predictions on that space, using the Spectral Energy Mover's Distance as the metric, including a universal non-Gaussianity feature in quark jets.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Three-point non-Gaussianity ledge is computed with an uncontrolled Post inverse-Laplace truncation.","rationale":"The reader's weakest_assumption targets the ReLU soft-energy scaling in Sec. III.B/C. On close reading, that step is more solid than it first appears: the ReLU argument S_nl is min(S_A^+,S_B^+) - max(S_A^-,S_B^-), where S_A^+ and S_A^- differ only by boundary terms. For soft or collinear-subtraction terms, the leading hard-energy contributions cancel in the difference (if H_A != H_B the argument is negative and ReLU vanishes; if H_A = H_B it is zero), so an order-one ReLU term is not merely ruled out by non-negativity but by the min/max structure itself. The resulting linear-in-soft-energy scaling gives a soft-collinear mixing term suppressed by lambda^3, consistent with Eq. 23. Thus the factorization theorem's power-counting core is likely sound. The real weak point is the application to the three-point correlator: the non-Gaussianity eta_n-G(d) depends on f(q), which is evaluated with Post's formula at k=25. The paper explicitly shows that f(q)'s slope at the origin diverges as sqrt(k), and that the gluon non-Gaussianity is negative at high scales for smaller k. Because the advertised universal ledge is a central phenomenological result, this uncontrolled inversion is the most load-bearing concern. The reader's verdict of CONDITIONAL is appropriate; a concrete numerical test of the inversion would settle whether the ledge is physical or an artifact. I therefore keep the verdict unchanged while flagging a different primary concern than the reader's stated weakest assumption.","tokens_in":30957,"tokens_out":16880,"duration_ms":189103,"concrete_test":"Recompute f(q) in Eq. A28 using an independent numerical inverse Laplace transform (e.g., Talbot's method or a fixed-tolerance contour integral) rather than Post's k=25 formula, and regenerate Figs. 3 and 4 for the 500, 1000, and 2000 GeV quark and gluon jets. Track the position and height of the ledge (reported around d = 30, 50, 80 GeV) and the positivity of the gluon curves across the full plotted range. If the ledge moves outside the shaded soft-scale-variation bands, or if the gluon curves become non-negative in a qualitatively different region, then the universal-ledge claim is not robust to the inversion truncation and the non-Gaussianity results should be labeled preliminary. The two-point correlator results, which use an exact inverse Laplace transform, are unaffected by this test.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The factorization theorem Eq. 23 is defensible: the ReLU argument in Sec. III.B/C cancels leading hard-energy contributions through its min/max structure, so soft-collinear mixing is indeed suppressed by theta_cc * E_s ~ lambda^3, and the reader's ReLU-scaling concern is less fragile than it appears. The genuinely load-bearing weakness is the numerical inversion of the three-point correlator. The non-Gaussianity eta_n-G(d) is obtained via Post's inverse-Laplace formula truncated at k=25 derivatives (App. A.4, Eq. A27). The paper's own Figs. 7 and 8 show large variation with k, gluon distributions that go negative unless k is large, and Fig. 6 shows the derivative of f(q) at q=0 diverging like sqrt(k), indicating the truncated series is asymptotic rather than convergent in the region q near 0. The universal ledge central to the paper's phenomenological claim is extracted from a regime where the approximation is not demonstrably stable: the authors acknowledge significant variation with k and state that better approximations are needed. If a more accurate inversion shifts the ledge position or height beyond the quoted scale-variation bands, the claimed universality of eta_n-G(d) near the QCD scale becomes an artifact of the k=25 cutoff rather than a prediction of the factorization framework.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a factorization framework for precision calculations on the space of collider events using the spectral energy mover's distance (SEMD). In the small-distance limit d ≪ E, it argues that soft and collinear contributions to the SEMD between two jets factorize at leading power, leading to the factorization theorem in Eq. (23): p(d^2) = H_AB ∫ J_AB(d^2_J) S_AB(d^2_S) δ(d^2−d^2_J−d^2_S). The authors perform NLL resummation for the two- and three-point event correlators, define a non-Gaussianity measure η_n-G(d), and report a universal 'ledge' near the QCD scale that they interpret as validating previous simulation results. The appendices provide the resummation ingredients, a two-loop anomalous dimension for the jet function, and an argument that the SEMD equals the tangent EMD through NNLL accuracy.","tokens_in":31221,"tokens_out":10065,"duration_ms":105917,"significance":"If correct, this is the first systematically improvable calculational framework for collider dataspace observables and a substantial advance over fixed-order or Monte-Carlo-only studies. The two-point NLL predictions rest on well-established jet-mass anomalous dimensions, and the Sudakov-squared suppression argument is physically compelling. The explicit two-loop non-cusp anomalous dimension in Appendix B is a concrete new result, and the identification of the non-Gaussianity as a Sudakov-safe observable is conceptually interesting. However, the headline application—validation of a universal non-Gaussian structure—depends on a numerical inverse Laplace transform whose error is not controlled, so the strongest claims of the paper are not yet established.","major_comments":[{"comment":"The three-point correlator is evaluated with Post's inverse Laplace formula truncated at k=25 derivatives. Figure 6 shows that the derivative of f(q) at q=0 grows like the square root of k with the truncation order, so the series is not convergent in precisely the region q≈0 that controls the non-Gaussianity at the distances plotted in Figs. 3 and 4. Figures 7 and 8 show large variation with k and negative gluon distributions unless k is very large. The paper's own statements that the approximation is 'insufficient' and that 'better approximations' are needed (App. A.4; Sec. VI) confirm that the universal ledge is extracted from an uncontrolled approximation. The scale-variation bands in Figs. 3 and 4 do not quantify this inversion error. A quantitative inversion uncertainty, an alternative method, or a controlled k→∞ extrapolation is required before the ledge can be presented as a prediction.","section":"App. A.4, Eqs. (A26)–(A27), Figs. 6–8"},{"comment":"Even if the inversion were controlled, the claim that the ledge is a purely perturbative phenomenon is not supported by the parametric separation. The ledge positions for E=500, 1000, and 2000 GeV are approximately 30, 50, and 80 GeV, while the author's own estimate gives d_np ≈ 22, 32, and 45 GeV; the separation is only a factor of 1.3–1.8, not a parametric power. The text concedes that non-perturbative effects 'may be important' and could shift the ledge. To validate the universality observed in Ref. [62], the paper needs either a quantitative treatment of non-perturbative shifts or a demonstration that they are numerically negligible in the ledge region.","section":"Sec. V.B, Figs. 3 and 4, Eq. (46)"}],"minor_comments":[{"comment":"The heading contains a typo: 'Non-Guassianities' should be 'Non-Gaussianities'.","section":"Sec. V.B"},{"comment":"There is a placeholder citation '[ ? ? ]' for the non-perturbative shape-function shift of the jet-mass distribution; this should be replaced with the relevant literature.","section":"Sec. V.B, paragraph after Eq. (46)"},{"comment":"The color description 'sold green' should read 'solid green'.","section":"Fig. 5 caption"},{"comment":"The all-orders factorization statement would be easier to check if Secs. III.B–III.C included an explicit expansion of the min/max functions in Eq. (6) rather than only the scaling-based argument; the present argument is plausible but indirect.","section":"Sec. III.C, Eq. (20)"},{"comment":"The definition of η_n-G(d) in Eq. (37) drops the bin width and event-number factors used in Ref. [62]; the text should state explicitly whether the comparison in Figs. 3 and 4 is intended to match the absolute normalization of that simulation or only the shape.","section":"Eq. (37) and footnote 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript reuses the author's own SEMD construction and a previous simulation result, but the reuse is appropriately cited and there is no indication of fitting to data. The main concern is the mismatch between the abstract's claim of validated universality and the acknowledged limitations of the Post inversion used for the three-point correlator. I recommend requiring either a controlled inversion error estimate or a substantially softened claim about the non-Gaussian ledge, while recognizing that the factorization framework and two-point results are valuable and should not be rejected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is the first genuinely systematically-improvable perturbative calculation for collider dataspace metrics, and the two-point part is solid. The factorization theorem in Eq. (23), separating soft and collinear contributions to the SEMD at leading power, is the real contribution. I read Sec. III carefully. The reader's worry about the ReLU subtraction scaling is less serious than it looked: the min/max structure in S_nl cancels the leading hard-energy term, so the order-one ReLU possibility is excluded not just by metric non-negativity but by the actual form of the function. That argument is a scaling analysis, not a formal SCET proof, and the all-orders wording is a bit quick, but for the NLL results in the paper it is adequate.\n\nWhat is new: the factorization theorem, the NLL resummed two- and three-point SEMD correlators, the two-loop correlated anomalous dimension in App. B, and the demonstration that the SEMD matches EMD/TEMD through NNLL. The NLL anomalous dimensions reducing to known jet-mass results is a feature, not a defect: the two-point predictions inherit well-tested infrastructure. No fitting is done, no invented parameters enter, and the reuse of the author's earlier SEMD and simulation work is legitimate rather than circular.\n\nThe soft spot is not the factorization; it is the three-point inversion. The non-Gaussianity is obtained by Post's inverse Laplace formula truncated at k=25 derivatives. The paper's own Figs. 5 and 6 show the trouble: the slope of f(q) at q=0 grows like sqrt(k), which means no Taylor expansion exists there; Figs. 7 and 8 show the gluon distribution going negative at moderate k and significant variation from k=10 to k=25. That makes the \"universal ledge\" near the QCD scale a feature of a particular truncation, not yet a stable prediction of the framework. The author says as much in App. A.4 and the conclusions, but the abstract and Sec. V.B present the ledge as validated. Any referee should either demand a controlled inversion or a clearly downgraded claim. The two-point results and the NNLL setup are not affected.\n\nMinor issues: there is an unresolved '[ ? ? ]' citation placeholder in Sec. V.B, and that section would benefit from proper references on non-perturbative shape-function shifts. The citation pattern otherwise looks fair, and self-citations appear where the work genuinely builds on the author's own results.\n\nWho this is for: jet substructure and SCET practitioners, plus anyone using EMD-style event metrics. The two-point part deserves a serious referee; the three-point part needs revision before publication. My recommendation: send it to peer review, and make the inverse-Laplace problem the make-or-break issue.","headline":"A real first step toward precision perturbative calculations on collider dataspace, with a solid two-point factorization and a clearly-acknowledged weak spot in the inverse-Laplace approximation behind the headline non-Gaussianity.","tokens_in":31701,"tokens_out":3169,"would_cite":true,"duration_ms":35233,"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":"The SEMD jet-distance metric factorizes into soft and collinear parts, yielding the first resummed precision predictions on collider-event dataspace.","keywords":["collider event dataspace","Spectral Energy Mover's Distance","factorization theorem","resummation","jet substructure","dataspace non-Gaussianity","Sudakov safety","Energy Mover's Distance"],"falsifier":"Evaluate the exact $p=2$ SEMD in the limit where jet A has a collinear splitting and jet B has one soft wide-angle particle, and check whether the cross term $\\theta_{\\mathrm{cc},A}\\,\\omega_{s,B}\\,\\mathrm{ReLU}(S_{cs,AB})$ is suppressed as $\\lambda^3$ rather than $\\lambda^2$; a direct fixed-order soft-collinear computation that finds a $\\lambda^2$ remainder would falsify the factorization theorem.","tokens_in":30729,"feed_emoji":"⚛️","tokens_out":7294,"duration_ms":68187,"temperature":0.7,"pith_summary":"This paper establishes the first systematically improvable calculational framework on the space of collider physics events, using the Spectral Energy Mover's Distance (SEMD) as the metric. Its central claim is that in the limit of small metric distance $d$ compared with the jet energy $E$, soft and collinear contributions to the SEMD factorize from one another to all orders in the QCD coupling. That factorization reduces the two-point event correlator to a hard function times a convolution of a jet function and a soft function, so logarithms of $d$ can be resummed by renormalization-group evolution. The paper presents next-to-leading-logarithmic predictions for the two-point jet correlator and for the dataspace non-Gaussianity, and shows that the original Energy Mover's Distance agrees with the SEMD through next-to-next-to-leading-logarithmic accuracy.","feed_headline":"Jet-dataspace metric factorizes: NLL predictions arrive","feed_subtitle":"A factorization theorem turns event-space geometry into resummed QCD, including a universal ledge near the QCD scale.","key_machinery":"The central object is the $p=2$ Spectral Energy Mover's Distance, the only IRC-safe collider event metric with an exact closed form, whose expression is sums of squared pair energies times angular weights minus ReLU subtraction terms built from cumulative spectral functions. The argument is carried by a power-counting analysis in the limit $d=\\lambda E$: collinear angles are rescaled by $\\lambda$ and soft energies by $\\lambda^2$, and each of the metric's three sectors (collinear-collinear, soft-soft, collinear-soft) is shown homogeneous of degree $\\lambda^2$. The load-bearing result is that the soft-collinear ReLU term is suppressed by $\\theta_{\\mathrm{cc}} E_s \\sim \\lambda^3$, so to leading power the metric separates into a jet-mass-like collinear part plus a soft part. Renormalization-group evolution of the hard, jet, and soft functions in Laplace space then resums logarithms, with one-loop anomalous dimensions inherited from the jet mass and a computed two-loop correlation correction $\\Delta\\gamma^{(2)}_{AB} = -2(\\alpha_s/2\\pi)^2 C_J^2(6\\pi^2\\log 2 - 7\\zeta_3)$.","core_discovery":"The discovery is that the $p=2$ SEMD, although a global optimal-transport-style metric, collapses at leading power into a sum of per-jet squared masses plus soft squared-mass contributions, with no soft-collinear mixing term. The paper proves this by power counting the metric's ReLU subtraction functions: collinear angles scale as $\\lambda$, soft energies as $\\lambda^2$, and the mixed ReLU term scales as $\\theta_{\\mathrm{cc}} E_s \\sim \\lambda^3$, hence is subleading; the argument uses metric non-negativity to rule out an order-one soft-energy-independent ReLU term. Consequently the two-point correlator obeys $p(d^2)=H_{AB}\\int J_{AB}(d^2_J)S_{AB}(d^2_S)\\delta(d^2-d^2_J-d^2_S)$, which is the basis for NLL resummation. The paper argues that the non-Gaussianity's universal ledge near the QCD scale is a perturbative phenomenon, and that measuring two events makes the two-point correlator Sudakov-suppressed twice over, so it is perturbatively calculable over nearly all of its support.","pith_inferences":["Beyond the paper: if the factorization holds, geometric observables such as local dimensionality and scalar curvature of the dataspace manifold can be assigned perturbative error bars rather than simulation-based estimates.","Beyond the paper: the near-total perturbative calculability of the two-point correlator opens a route to a high-precision $\\alpha_s$ extraction from jet-pair distance distributions, provided the non-perturbative shape-function shift is controlled.","Beyond the paper: the metric distance between real and virtual phase-space points could serve as a process-independent regulator for fixed-order subtraction; the paper gestures at this idea but does not develop it.","Beyond the paper: computing $\\eta_{n\\text{-}G}(d)$ at NNLL for quark jets would test whether the ledge's near-universal value around 0.2 persists or is an artifact of the NLL truncation."],"forward_implications":["The two-point jet-event correlator is Sudakov-suppressed twice over, and its NLL distribution is perturbatively reliable from its peak down to distances near $\\sqrt{\\Lambda_{\\mathrm{QCD}} E R}$.","The non-Gaussianity $\\eta_{n\\text{-}G}(d)$ is a Sudakov-safe observable: fixed-order perturbation theory is singular when the two distances are set equal, but the resummed expression regulates and exponentially suppresses that region.","The universal ledge in $\\eta_{n\\text{-}G}(d)$ near the QCD scale, previously seen in simulated data, is reproduced by pure perturbative resummation and is not a sign of parton-to-hadron physics.","The SEMD and the original (tangent) EMD produce identical event-correlator distributions through NNLL accuracy.","All ingredients except fixed-order matching are in place for NNLL resummation of the two-point correlator, via the two-loop anomalous dimension computed in the appendix."],"supporting_citations":[{"why":"Defines the Energy Mover's Distance, the original collider-event metric that the SEMD is compared against.","marker":"[3]"},{"why":"Introduced the SEMD, supplied its closed form, and identified its double-logarithmic relation to jet mass.","marker":"[12]"},{"why":"Provided the fast evaluation method for the spectral EMD that makes semidatasets practical to analyze.","marker":"[17]"},{"why":"Reported the simulated universal ledge in the dataspace non-Gaussianity that this paper validates perturbatively.","marker":"[62]"},{"why":"Supplies the well-known jet-mass resummation and one-loop anomalous dimension that the NLL factorization relies on.","marker":"[84]"},{"why":"Established the concept of Sudakov-safe observables used here to define the non-Gaussianity.","marker":"[91]"},{"why":"Provides Post's inversion formula used to approximate the inverse Laplace transform of the three-point correlator.","marker":"[97]"},{"why":"Gives the matching Laplace inversion theory supporting the Post approximation.","marker":"[98]"}],"fun_headline_variants":[],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the metric's subtraction term cannot be independent of soft-particle energies because that would make the metric negative, so it must vanish linearly with those energies; if it instead stayed order one in some kinematic corner, soft and collinear contributions would mix at leading power and the factorization would collapse.","fun_headline_variants_meta":{"error":"Client error '402 Payment Required' for url 'https://api.deepseek.com/chat/completions'\nFor more information check: https://developer.mozilla.org/en-US/docs/Web/HTTP/Status/402"},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:32:49.505489+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the exact $p=2$ SEMD in the limit where jet A has a collinear splitting and jet B has one soft wide-angle particle, and check whether the cross term $\\theta_{\\mathrm{cc},A}\\,\\omega_{s,B}\\,\\mathrm{ReLU}(S_{cs,AB})$ is suppressed as $\\lambda^3$ rather than $\\lambda^2$; a direct fixed-order soft-collinear computation that finds a $\\lambda^2$ remainder would falsify the factorization theorem.","supporting_citations":[{"cited_title":"Resumming the color-octet contribution to radiative Upsilon decay","cited_arxiv_id":"hep-ph/0106316","evidence_quote":"Established the concept of Sudakov-safe observables used here to define the non-Gaussianity."}],"review_version":1}