{"id":"e75180fd-21b5-4615-a247-9263a2d12fd1","arxiv_id":"2608.12923","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"In unitary multi-pomeron exchange models, the entropy computed from probabilities derived from inclusive cross-sections peaks near rapidity 12 to 17 and then slowly decreases, and the probabilistic construction matches unitarity only in the infinite-energy limit.","lead":"Simple pomeron-exchange models with unitarity produce particle-number probabilities from scattering cross-sections; the resulting entropy rises with collision energy, peaks around rapidity 12 to 17, and then slowly falls, instead of growing without bound. The result challenges a standard expectation of logarithmic entropy growth in high-energy scattering and exposes when the probabilistic reading of inclusive cross-sections is consistent.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The RG entropy fall for Y≳12 may be a truncation artifact: Eq. (15) is summed only to nmax=40 while at Y≈30 the mean emitted-particle number is ≈75, so a large high-n tail is omitted and no convergence study is given.","rationale":"The paper contains a clean analytic result: Eq. (16) and the exp(-cY) consistency deviation are checkable by hand and I do not dispute them. The advertised entropy turnover, however, is purely numerical. The most load-bearing condition for that turnover is not the Poisson-emission ansatz or the dropped BFKL phase factor, although those matter; it is whether the numerical P(n) distribution is actually converged. The text explicitly states that Eq. (15) was summed with nmax=40 and gives no convergence study. A simple estimate shows that at Y=30, the mean particle number is about 75, so the n=40 cutoff removes a substantial fraction of the probability mass and a correspondingly non-negligible piece of the entropy. The fact that the observed maximum sits near the Y where the mean passes the cutoff is a warning that Fig. 5 may describe the truncation rather than the physics. This concern can be settled by a direct rerun, so I do not reject the paper; I keep the verdict conditional, matching the reader's verdict. I partially agree with the reader: the reader identified the Poisson/drop-phi assumptions as weakest and mentioned the missing convergence study only in the rationale, whereas I would put the numerical tail treatment at the center of the entropy claim.","tokens_in":15638,"tokens_out":17971,"duration_ms":203181,"concrete_test":"Recompute S(Y) for Y=10, 12, 15, 20, 25, 30 from Eq. (15) at the same RG parameters, but with the sum over n and m extended well beyond 40 (e.g., m up to 100 and n up to 300, or using the equivalent Poisson-mixture form summed to all n), and compare the resulting curve with Fig. 5. If S(30)≥S(20), or if the slope after Y=12 changes substantially, then the claimed maximum and fall are not established until a full convergence study is supplied.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Sect. 2.2 the entropy is computed from Eq. (15) after summing n and m up to nmax=40, with no convergence check. The claimed fall of S(Y) for Y≳12 in Fig. 5 sits exactly where the cutoff starts to bite. For the RG parameters, r(30)≈6, so F(2r)≈3.1 and the cut-pomeron mean is <m>=2r/F(2r)≈3.9; with c=1 and j=30, the mean particle number is <n>=[F(2r)/(2F(r))] j <m> ≈ 0.64×30×3.9≈75. The n=40 cutoff is therefore below the mean of the emitted distribution. The first few cut-pomeron components have probabilities p_m≈Γ(m,2r)/(m!F(2r))≈0.32, 0.16, 0.11 for m=1,2,3; at Y=30 the mass beyond n=40 is at least O(0.15), and the missing tail contributes a non-negligible amount to the Shannon entropy. The numerical maximum near Y≈12 corresponds to <n> crossing the cutoff, which is the signature expected from a truncation artifact. The paper's own asymptotic (17)-(19) predicts linear growth at large Y; discarding it as 'too crude' without first demonstrating numerical convergence leaves the headline non-monotonicity unsupported. For the BFKL case the cutoff is much less severe because c≈0.057, but there the turnover lies in the region Y<20 that the paper itself flags as inconsistent with unitarity.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the construction of probabilities P(n) for n produced particles from inclusive cross-sections in a unitary multi-pomeron exchange model without pomeron interactions. For both a Regge-Gribov pomeron and a BFKL pomeron, the author derives P(n) from k-fold inclusive moments via binomial inversion, expresses the inelastic cross-section ratio as F(2r(1-e^{-j}))/(2F(r)), and quantifies its deviation from the unitary value, which vanishes only as j=cY approaches infinity. The paper then computes P(n) numerically and reports that the Shannon entropy first rises, peaks at rapidity Y approximately 12-17, and then slowly falls, contrary to the analytic large-Y estimate E(Y) proportional to (Delta/4)Y. The BFKL case is additionally shown to fail probability construction when the exponential rapidity factor phi(y_i) is retained, because the moments grow too fast for the inversion to converge.","tokens_in":1587,"tokens_out":2073,"duration_ms":69140,"significance":"If the reported non-monotonic entropy were established, it would be a notable result: it contradicts the linear-in-rapidity entropy growth found in earlier one-dimensional pomeron studies and would imply that the naive probabilistic interpretation of inclusive cross-sections is only asymptotically consistent with unitarity. The derivation has genuine strengths: the relation between cut-pomeron probabilities and particle-emission probabilities is derived explicitly from the inclusive moments, and the unitarity inconsistency is reduced to a clean analytic expression with exponentially small large-Y discrepancy. However, the central quantitative claim rests on numerical sums that are truncated at a value below the mean multiplicity at the energies of interest, and no convergence analysis is provided. The BFKL corroboration is also confined to a rapidity range that the paper itself identifies as inconsistent with unitarity. Thus the headline claim is currently unsupported, although it may be salvageable with additional numerical evidence.","major_comments":[{"comment":"The entropy turnover is not supported without a convergence study. In Eq. (15) the sums over n and m are truncated at nmax=40, but for the RG parameters at Y=30 one has r approximately 6, F(2r)/(2F(r)) approximately 0.64 and <m> approximately 3.9, giving a mean multiplicity <n> approximately 0.64 times 30 times 3.9, which is approximately 75 and lies far above the cutoff. The omitted high-n tail therefore has non-negligible probability mass and contributes substantially to the Shannon entropy. The claimed maximum near Y=12 coincides with the rapidity at which the mean multiplicity crosses the truncation threshold, which is precisely the signature expected from a cutoff artifact. The paper's own asymptotic estimate (17)-(19) predicts linear growth E(Y) approximately (Delta/4)Y; dismissing this as 'too crude' is not sufficient without demonstrating numerical convergence. Please report the tail probability, the entropy as a function of nmax at fixed Y=20 and Y=30, and an extrapolated or analytically corrected result.","section":"Section 2.2, Eq. (15), Fig. 5"},{"comment":"The BFKL entropy result cannot serve as independent corroboration of a peak near Y=12. Equation (43) and Fig. 9 show that the model is inconsistent with unitarity for Y<20, with delta being large in precisely the region where the claimed maximum occurs. Since retaining phi(y_i) makes the moments grow as I_n proportional to e^(gamma n^2/4)/n! and the inversion (2) divergent, the calculable BFKL case is that with phi=0, and it is only physically admissible for Y>20. The entropy maximum at Y approximately 12 therefore lies outside the reliable domain. Please state explicitly which Y intervals are physically trustworthy for the BFKL model and separate the RG and BFKL claims in the abstract and conclusions.","section":"Sections 3.4-3.5, Eq. (43), Fig. 9, Fig. 11"},{"comment":"The predicted position of the entropy maximum is not parameter-free: it depends on the multiplicity parameter c through j=cY. For the RG model the paper adopts c=1 without showing how this value is extracted from data, and no sensitivity study in c is given. Since the abstract states a quantitative peak range 12-17, the dependence of the peak position and of the turnover on c should be established, or the abstract should be softened to a parametric statement. The same issue affects the comparison of RG and BFKL cases, where c differs by nearly a factor of 20.","section":"Section 2.1, Eq. (11), Fig. 5"}],"minor_comments":[{"comment":"There are several typos in the abstract: 'achievung', 'cros s-sections', 'inelasic', and 'achievun g' should be corrected.","section":"Abstract"},{"comment":"The sentence 'The elastic cross-section is evidently sigma_el = sigma_tot = sigma_in' is not correct as written; the intended relation is presumably sigma_tot = sigma_el + sigma_in, with sigma_in = 8*pi*lambda*F(2r)/2 and sigma_el = sigma_tot - sigma_in. Please rewrite this passage.","section":"Section 2.1, after Eq. (8)"},{"comment":"In the text before Eq. (15) the reference to 'P(n) given by (40)' should be to Eq. (15), since (40) is in the BFKL section.","section":"Section 2.1, Eq. (15)"},{"comment":"The entropy is denoted E(Y) in Eq. (18) but S(Y) in the figures and text; please unify the notation.","section":"Eq. (18) and Figs. 5, 11"},{"comment":"The asymptotic expression for P(n) contains an internal inconsistency: theta(r-n) is first written and then theta(2r-n) is used. This should be clarified, since the normalization and the derivation of Eq. (19) depend on the correct upper limit.","section":"Section 2.1, Eq. (17)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central numerical claim is likely affected by the n=40 truncation, and the requested convergence tests are essential before publication. The editor may also wish to verify the citations to [5], [16], and [18], which appear to be preprint numbers dated after the arXiv submission identifier; if these are not typographical errors, they could indicate incomplete verification of references. The author's reliance on [34] for the BFKL emission formula is heavy, but the formula is quoted explicitly and the calculation is self-contained enough for the main argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe one result worth remembering here is Eq. (16): in the eikonal multi-pomeron model, the inelastic cross-section reconstructed from inclusive moments is σ_in/σ_tot = F(2r(1−e^{−cY}))/(2F(r)), which approaches the unitarity value only as cY→∞. That is a genuine analytic observation, and the derivation of P(n) from binomial inversion of factorial moments is clean. The paper is honest about its most fragile input—independent Poisson emission from each cut pomeron—and about the BFKL case where keeping the φ(y_i) factor makes the moment series diverge. I would not question the analytic consistency condition.\n\nThe entropy turnover, however, is not supported. The numerics sum Eq. (15) only to n_max=40, and the stress-test arithmetic checks out: with c=1 and r(30)≈6, the mean multiplicity is j·r/F(r)≈76, so the cutoff lies below the mean of the emitted-particle distribution. The missing high-n tail is of order 0.15 or more at Y=30, and Shannon entropy is sensitive to that tail. The maximum near Y≈12–17 is exactly where the mean starts to cross the cutoff. The paper's own asymptotic says E(Y)∼(Δ/4)Y, and dismissing that as “too crude” without a convergence study is not enough. For the BFKL case the cutoff is less severe because c is small, but there the turnover sits in the Y<20 region the paper itself flags as inconsistent with unitarity, and the calculation drops the exponential φ that the text says is “apparently observed in experiment.” So the headline non-monotonicity looks like a truncation artifact, not a robust feature.\n\nThe citation pattern is fine; the relevant pomeron literature is cited, and I do not see self-citation propping up the new result. The paper does have editorial problems: equation numbers are wrong in places, and typos make checking harder than it should be. None of that affects the logic of Eq. (16).\n\nWho is this for? Someone working on pomeron models and particle-production entropy will want the analytic consistency relation, but should not take the entropy turnover at face value. It deserves referee attention, because Eq. (16) is significant enough to publish once the numerics are redone with a convergence check. I would not cite the entropy claim in its present form.","headline":"Eq. (16) is a clean new consistency condition, but the headline entropy turnover is very likely a truncation artifact from n_max=40.","tokens_in":16603,"tokens_out":3702,"would_cite":true,"duration_ms":38787,"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":"Pomeron-model entropy peaks near rapidity 12–17 and then falls","keywords":["entropy","inclusive cross-sections","multiple pomeron exchange","BFKL pomeron","AGK cutting rules","Poisson emission","unitarity","multiplicity distribution"],"falsifier":"Measure (or compute without imposing the paper's Poisson ansatz) the entropy of the particle-number distribution from inclusive cross-sections at rapidities from $Y=5$ to $Y=30$: if the entropy keeps rising beyond $Y\\approx 17$, or if the ratio $F(2r(1-e^{-j}))/(2F(r))$ fails to approach the unitary value with an $\\exp(-cY)$ deficit, the central claim is wrong. In the BFKL case, retaining $\\varphi(y_i)$ and computing the moments $I_n$ numerically should show growth like $e^{\\gamma n^2/4}$; finding instead that the moment sum converges would falsify the claimed obstruction to defining probabilities.","tokens_in":15237,"feed_emoji":"📊","tokens_out":7252,"duration_ms":71333,"temperature":0.7,"pith_summary":"The paper asks whether the probabilities that enter the entropy of produced particles can be reconstructed from inclusive cross-sections in a unitary model of multiple pomeron exchanges with no interaction between pomerons. It shows that within this model the n-particle probability is the convolution of a cut-pomeron distribution and a Poisson emission law, but the inelastic cross-section this construction implies, $F(2r(1-e^{-j}))/(2F(r))$, agrees with the unitarity value $F(2r)/(2F(r))$ only in the limit $j=cY\\to\\infty$. The discrepancy falls as $\\exp(-cY)$, so consistency is restored only at asymptotically high energies. Numerically, the entropy built from these probabilities rises with rapidity, reaches a maximum near $Y=12$–$17$, and then slowly falls, in both the local and BFKL pomeron versions. A sympathetic reader would care because this gives a concrete, testable prediction for the energy dependence of the multiplicity entropy, and it exposes a finite-energy inconsistency inherent in the probabilistic interpretation of inclusive data.","feed_headline":"Entropy peaks near rapidity 12–17, then falls","feed_subtitle":"Probabilities from inclusive cross-sections match unitarity only at infinite energy; the entropy turnover is the signature.","key_machinery":"The central object is the set of cut-pomeron diagrams (AGK cuts) in an eikonal sum of non-interacting pomeron exchanges, together with the assumption of independent Poisson emission from each cut pomeron, with mean $j=cY$. The argument is carried by the incomplete-Gamma identities $\\Gamma(m,2r)$ for the cut-pomeron distribution and by the sum identity leading to $\\sigma_{in}/\\sigma_{tot}=F(2r(1-e^{-j}))/(2F(r))$, where $F(r)=\\sum_{n\\ge1}(-r)^n/(n\\,n!)$ and $r(Y)$ is the energy-dependent rescattering parameter. This identity simultaneously yields the probabilities $P(n)$ by inversion of the inclusive moments and exposes the unitarity mismatch, since unitarity requires $F(2r)$ in the numerator. The BFKL version replaces $r$ by the BFKL growth factor and the emission constant $c$ by a number fixed by the gluon-emission vertex; dropping the non-leading factor $\\varphi(y_i)$ is what allows the whole probabilistic construction to go through.","core_discovery":"The central claim is that in unitary multi-pomeron exchange models the probability $P(n)$ of $n$ produced particles is $P(n)=\\frac{a}{n!}\\sum_{m\\ge 1}\\frac{1}{m!}(jm)^n e^{-jm}\\Gamma(m,2r)$, the convolution of the cut-pomeron distribution with Poisson emission of mean $j=cY$ per pomeron. Summing $P(n)$ for $n\\ge 1$ yields $\\sigma_{in}/\\sigma_{tot}=F(2r(1-e^{-j}))/(2F(r))$, which coincides with the eikonal unitary value $F(2r)/(2F(r))$ only when $j\\to\\infty$. Thus the probabilistic emission picture disagrees with unitarity at finite energies, with a relative error vanishing as $\\exp(-cY)$; for a local pomeron with $c$ of order unity the discrepancy is already at the $10^{-4}$ level near $Y=10$, while for the BFKL pomeron, with its smaller $c$, it is still about 2% at $Y=20$. The numerical entropy from this probability distribution rises and then falls, with its maximum in the interval $Y\\approx 12$–$17$, contradicting the logarithmic growth one would expect from crude large-energy asymptotics. For the BFKL pomeron, retaining the full rapidity-dependent factor $\\varphi(y_i)$ in the n-gluon cross-section makes the moments $I_n$ grow like $e^{\\gamma n^2/4}$, so the inversion used to build probabilities diverges and no probabilities can be constructed at all.","pith_inferences":["Because the turnover comes from the balance between the growth of $r(Y)$ (which widens the cut-pomeron distribution) and the linear growth of the Poisson mean $j=cY$, it is a generic feature of independent-source emission models, not a detail specific to pomeron trajectories.","If measured multiplicity distributions in high-energy hadron collisions show an entropy that keeps growing beyond the predicted interval, that would point either to pomeron–pomeron interactions or to a non-Poisson emission law, both outside this paper's model.","The divergence of the moment inversion when $\\varphi$ is retained suggests that entropy defined from inclusive cross-sections may require a regulator in any theory with long-range correlations, an issue the paper leaves open.","One testable extension is to use the identity for $\\sigma_{in}/\\sigma_{tot}$ to estimate the unitarity deficit from measured inclusive multiplicities and compare its rapidity dependence with $\\exp(-cY)$, separating the emission intensity $c$ from the trajectory parameters."],"forward_implications":["If the probability formula is correct, the entropy of the produced-particle distribution is not monotonic: it peaks near $Y\\approx 12$–$17$ and then falls, so fits that assume logarithmic growth will fail at high rapidity.","The inelastic cross-section reconstructed from inclusive cross-sections underestimates the unitary inelastic cross-section at finite rapidity, with a deficit that scales as $\\exp(-cY)$; a model that claims exact consistency at finite energy must account for this deficit.","For BFKL pomerons the parameter $c$ is nearly twenty times smaller than in the local model, so the model becomes approximately unitary only above $Y\\approx 20$; below that rapidity the deficit is large.","In the BFKL case, the non-leading rapidity factor $\\varphi(y_i)$, though beyond leading-logarithmic accuracy, cannot be dropped without losing the ability to define probabilities at all; keeping it makes the moment inversion divergent."],"supporting_citations":[{"why":"Supplies the AGK cutting rules used to sum the k-fold inclusive cross-sections and derive the cut-pomeron distribution.","marker":"[33]"},{"why":"Gives the explicit single-pomeron n-gluon inclusive cross-section and the constant c used for BFKL emission.","marker":"[34]"},{"why":"Frames the two-stage emission scenario (cut pomerons then particles) that the convolution formula realizes.","marker":"[35]"},{"why":"First reported the finite-energy inconsistency between emission cross-sections and the unitary inelastic cross-section that this paper derives analytically.","marker":"[23]"},{"why":"Provides the seminal one-dimensional model whose logarithmic entropy growth is the baseline this paper's turnover is measured against.","marker":"[6]"},{"why":"Shows extra emission from the pomeron vertex in QCD, motivating the choice of a model without pomeron interactions.","marker":"[21]"}],"fun_headline_variants":["Entropy peaks at rapidity 12–17, then drops","Inclusive scattering entropy rises then falls","High-energy entropy turnover near rapidity 12–17","Entropy from inclusive cross-sections peaks and declines","Pomeron model entropy peaks then falls with energy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that each cut pomeron emits particles independently according to a Poisson law with mean $cY$, and, in the BFKL case, that the rapidity-dependent factor $\\varphi(y_i)$ in the n-gluon cross-section can be set to zero; if emission is not Poisson or that factor is not negligible, the constructed probabilities and the entropy turnover do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Entropy peaks at rapidity 12–17, then drops","Inclusive scattering entropy rises then falls","High-energy entropy turnover near rapidity 12–17","Entropy from inclusive cross-sections peaks and declines","Pomeron model entropy peaks then falls with energy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000152,"raw_usage":{"total_tokens":1213,"prompt_tokens":963,"completion_tokens":250,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":579,"completion_tokens_details":{"reasoning_tokens":173}},"tokens_in":579,"tokens_out":250,"duration_ms":2768,"temperature":1.0,"reasoning_tokens":173,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:33:48.640963+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure (or compute without imposing the paper's Poisson ansatz) the entropy of the particle-number distribution from inclusive cross-sections at rapidities from $Y=5$ to $Y=30$: if the entropy keeps rising beyond $Y\\approx 17$, or if the ratio $F(2r(1-e^{-j}))/(2F(r))$ fails to approach the unitary value with an $\\exp(-cY)$ deficit, the central claim is wrong. In the BFKL case, retaining $\\varphi(y_i)$ and computing the moments $I_n$ numerically should show growth like $e^{\\gamma n^2/4}$; finding instead that the moment sum converges would falsify the claimed obstruction to defining probabilities.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the AGK cutting rules used to sum the k-fold inclusive cross-sections and derive the cut-pomeron distribution."},{"cited_title":"J C 4 (1998) 685; [arXiv: 9710263/hep-ph]","cited_arxiv_id":null,"evidence_quote":"Gives the explicit single-pomeron n-gluon inclusive cross-section and the constant c used for BFKL emission."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Frames the two-stage emission scenario (cut pomerons then particles) that the convolution formula realizes."},{"cited_title":"Hladik, H.J","cited_arxiv_id":null,"evidence_quote":"First reported the finite-energy inconsistency between emission cross-sections and the unitary inelastic cross-section that this paper derives analytically."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the seminal one-dimensional model whose logarithmic entropy growth is the baseline this paper's turnover is measured against."}],"review_version":1}