{"id":"47216240-8430-4cb6-b1d0-17b432bdd88c","arxiv_id":"2412.17267","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"The U-matrix unitarization scheme forces a geometric, correlated pomeron distribution, while the eikonal scheme gives a Poisson distribution, changing predicted particle multiplicities.","lead":"This paper calculates how many pomerons, the effective exchange objects of the strong force, are produced in high-energy proton collisions under two competing unitarity schemes. It finds that the U-matrix scheme produces stronger fluctuations and correlations between pomerons, and argues this scheme may describe ultra-high-energy collisions better.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The U-matrix σ_n sum (Eq. 47) fails the unitarity sum rule against the paper's own amplitude (Eq. 46), so Eq. (32) is not a valid pomeron topological cross-section for the U-matrix scheme.","rationale":"I read the paper in good faith and find a genuine, internally checkable defect in the central argument. The U-matrix pomeron topological cross-section is presented as Eq. (32), and all subsequent statistical results — geometric distribution, variance exceeding the mean, positive f2, factorial moments, and multiparton-enhancement claims — follow from that formula together with the exponential spectral density. The reader's verdict correctly identifies Eq. (26) as a weakly sourced premise, but the stronger problem is that the paper itself exhibits a unitarity sum-rule violation between Eq. (46) and Eq. (47). Since a topological cross-section is defined by cutting the same amplitude whose imaginary part gives the inelastic cross-section, the two must agree; their disagreement means the proposed σ_n is not the cut-pomeron cross-section of the U-matrix amplitude used here. The remedy is not merely to cite [16] more carefully: either the U-matrix amplitude or the topological cross-section formula must be modified, and the AGK inconsistency must be resolved. Until then, the central comparison between eikonal Poissonian pomerons and U-matrix geometric/correlated pomerons is unsupported. I do not see a way to accept the current preprint as a reliable derivation of the advertised conclusions; a corrected derivation, with the factor-of-2 error fixed and the AGK sum rule checked at finite Im χ, could change that assessment.","tokens_in":19127,"tokens_out":10766,"duration_ms":106667,"concrete_test":"Set χ = iγ with γ = 1 in the U-matrix formulas and evaluate three quantities directly. (a) From Eq. (31)/(42), compute S = 1 + iA and hence G_inel^unit = 1 − |S|^2 = 8/9. (b) From Eq. (32) or (45), compute σ_n = γ^n/(1 + γ)^(n+1) and sum over n ≥ 1, giving Σσ_n = 1/2. (c) Compute A from the spectral representation S = ∫_0^∞ ρ(τ)e^{iτχ}dτ with ρ(τ) = 2e^(−2τ), obtaining A = i(1 − S), and compare with Eq. (31). If (a) ≠ (b) for any γ > 0, then Eq. (32) is not the topological cross-section of the same unitary amplitude, and the multiplicity and correlation conclusions built on it are not established.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central result is that Eq. (32), σ_n(s,b) = (Im χ)^n/(1 + Im χ)^(1+n), is the U-matrix pomeron topological cross-section and implies geometric, correlated pomeron exchanges. This result is internally inconsistent with the U-matrix amplitude used in the same paper. For a purely imaginary Born term, χ = iγ, the amplitude in Eq. (31)/(42) gives, via Eq. (46), G_inel = 2γ/(1 + γ/2)^2. Summing the proposed topological cross-section over n ≥ 1 gives, as the paper itself writes in Eq. (47), Σσ_n = γ/(1 + γ). These two expressions are not equal for generic γ: at γ = 1 they give 8/9 and 1/2, respectively, and as γ → ∞ they give 0 and 1. A topological cross-section must satisfy the AGK/cut-unitarity identity Σ_{n≥1} σ_n = G_inel; the paper acknowledges the disagreement but does not resolve it. Consequently, W_n = σ_n/Σσ_n is not the probability distribution of cut pomerons for the U-matrix amplitude used in the paper. There is also a factor-of-2 inconsistency: from S = ∫ρ e^{iτχ} = 1/(1 − iχ/2), one obtains A = i(1 − S) = χ/(2 − iχ), not χ/(1 − iχ/2) as in Eq. (31)/(42). Correcting this mismatch, or re-deriving Eq. (26) from [16] in an AGK-consistent way, is a precondition for the geometric-distribution and correlation claims.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper applies Kancheli's generalized representation of the unitarized elastic amplitude, S[χ] = ∫ dτ ρ(τ) e^{iτχ}, to derive what it calls the pomeron topological cross-section σ_n(s,b) for two unitarization schemes: the eikonal, with ρ(τ)=δ(τ−1), and the U-matrix, with ρ(τ)=e^{−τ/c}/c and c=1/2. The resulting expressions, σ_n = (2 Im χ)^n e^{−2Im χ}/n! and σ_n = (Im χ)^n/(1+Im χ)^{n+1}, are interpreted as a Poisson and a geometric distribution of the number of exchanged pomerons, respectively. On this basis the paper computes pomeron multiplicity distributions, mean and variance of the number of pomerons, the two-particle correlation f2, normalized factorial moments, and a hadron multiplicity model for pp collisions, concluding that the U-matrix scheme inherently produces larger pomeron fluctuations and positive higher-order correlations, and that correlated pomeron exchanges enhance multi-parton collisions. The eikonal results reproduce known Poisson behavior, while the U-matrix results are advertised as a qualitative distinction between unitarization schemes.","tokens_in":19643,"tokens_out":18836,"duration_ms":153184,"significance":"If the framework were internally consistent, the paper would offer a clean, falsifiable separation between unitarization schemes: in the eikonal case f2 is predicted to be negative below roughly 10^4 GeV and positive above it, whereas in the U-matrix case f2 is predicted to be positive at all energies, with normalized factorial moments growing with rank. The manuscript is transparent in several respects: the Poisson-exponential mixture algebra leading to the geometric distribution is shown explicitly (Eqs. 33–34), the fit parameters are tabulated, and the conflict with the AGK-based results of Luna and Ryskin [22] is acknowledged rather than hidden. The generalized-representation formalism follows Kancheli [16] in a readable way, and the eikonal limit is textbook-correct. However, the central results are analytic consequences of the assumed exponential spectral density and of the unproven input formula (26), and the paper's own equations demonstrate that the U-matrix σ_n violates the cut-unitarity sum rule that any topological cross-section must satisfy.","major_comments":[{"comment":"The U-matrix amplitude in Eqs. (31) and (42) is inconsistent with the spectral representation (17)–(30) used in the same paper. With ρ(τ)=e^{−τ/c}/c and c=1/2, Eq. (17) gives S(s,b)=1/(1−iχ/2). Combined with the standard definition A=i(1−S) used in Eqs. (21)–(28) and (38), this yields A=χ/(2−iχ), not A=χ/(1−iχ/2) as stated in (31) and (42). The discrepancy is a factor of two in the numerator of the U-matrix amplitude; it propagates into the inelastic cross-section (46) and affects the fitted parameters and all impact-parameter and energy-dependent results of Section III. The revision must correct the amplitude or redefine the Born term consistently and recompute the derived quantities.","section":"§III.A, Eqs. (31) and (42)"},{"comment":"The proposed pomeron topological cross-section (32) fails the cut-unitarity sum rule against the paper's own U-matrix amplitude. For a purely imaginary Born term χ=iγ, Eq. (46) gives G_inel = 2γ/(1+γ/2)^2, whereas the sum in Eq. (47) gives Σ_{k≥1} σ_k = γ/(1+γ); at γ=1 these are 8/9 and 1/2, respectively, and for γ→∞ they tend to 0 and 1. The manuscript acknowledges the disagreement but does not resolve it: the claim that the discrepancy arises from the omission of diffractive-state contributions in the AGK application of [22] is an assertion, not a calculation, and the sentence stating that G_inel→1 is 'closer to the expected asymptotic behavior of G_inel→0' is self-contradictory. Because W_n in Eq. (52) is defined by normalizing σ_n with Σσ_n, the identification of W_n as the probability of n cut pomerons requires Σ_{n≥1}σ_n = G_inel; without this identity, Eq. (32) is not a valid topological cross-section for the amplitude used in this paper, and the geometric-distribution and correlation claims of Sections III.B and III.C lack a valid basis. The surrounding text is also internally inconsistent, stating first that [22]'s result differs by 'an additional multiplicative factor of 2' and then that Eq. (45) is 'exactly the same as our result (32)'.","section":"§III.A, Eqs. (46)–(47)"},{"comment":"Equation (26), the pomeron topological cross-section expressed as a superposition of Poisson distributions with mean 2τ Im χ, is quoted from [16] without derivation, and it is the only input that separates the statistics of the two schemes. Combined with the assumed exponential density (30), Eq. (26) yields the geometric distribution (32) through the elementary Poisson-exponential mixture identity demonstrated in Eqs. (33)–(34). The central results of the paper—geometric pomeron statistics, variance exceeding the mean at all energies, positive f2, and growing factorial moments—are therefore analytic consequences of the chosen spectral density rather than derived consequences of U-matrix unitarization. Moreover, the manuscript itself cites [22] as having shown that U-matrix unitarization is inconsistent with the AGK rules for a pomeron with intercept greater than 1. The revision must derive Eq. (26) from the AGK cutting rules, including the diffractive-state contributions invoked in the text, and demonstrate that the resulting σ_n satisfies the unitarity sum rule; otherwise the comparison between the eikonal and U-matrix pomeron weights reduces to a comparison of two assumed ρ(τ) functions.","section":"§II, Eq. (26)"},{"comment":"The hadron multiplicity model assumes a Poisson distribution for the number of particles produced in n showers, with mean n⟨N1⟩, even though the manuscript acknowledges in the same paragraph that this ignores the known violation of KNO scaling. This is acceptable for an illustrative toy model, but the abstract and conclusion advertise as a finding that 'correlated pomeron exchanges within the U-matrix summation play a key role in enhancing multi-parton collisions.' That claim is drawn from the Poisson-shower model combined with the disputed σ_n; the revision should either strengthen the model (for example, by using a negative-binomial or KNO-violating shower distribution) or explicitly qualify the multi-parton-enhancement claim as a model-dependent illustration.","section":"§III.D, Eqs. (59)–(62)"}],"minor_comments":[{"comment":"Several cross-references to equation numbers are wrong: the text near Eq. (15) refers to 'the series' structure in (38)', and §III.A refers to 'the generalized representation of the S matrix 38', but Eq. (38) is defined only later and is the same object as Eq. (15). Renumber or fix the citations.","section":"§II and §III.A"},{"comment":"The text says the factorial moments are computed 'specifically at 13 GeV and 57 GeV', while Fig. 10 and Figs. 13–14 use 13 TeV and 57 TeV. The GeV/TeV discrepancy should be corrected.","section":"§III.C, Fig. 10 and surrounding text"},{"comment":"Equation (37) equates σ_n(s,b) with the probability P(X=n) before the normalization W_n = σ_n/Σσ_n is introduced in Eq. (52). This is only consistent because σ_n in (32) sums to 1 when n=0 is included; the distinction between the n=0 diffractive term and the n≥1 inelastic terms should be stated explicitly.","section":"§III.A, Eq. (37)"},{"comment":"Both footnotes say only 'There appears to be a typo in the formula in [16]' without specifying the typo or the corrected formula. Since Eq. (30) is load-bearing, the corrected spectral density should be stated explicitly.","section":"§III.A, footnotes [20] and [23]"},{"comment":"The table reports best-fit parameters and χ²/d.o.f. for the eikonal and U-matrix fits, but neither the data sets nor the energy range used in the fits are specified. This information is needed to assess whether the two schemes are fitted on the same footing.","section":"§III.B, Table I"},{"comment":"There are numerous typos and grammatical slips, including 'Poisonian' for 'Poissonian' (twice in §I), 'with the coefficients ... are :' in Eq. (42) context, and 'the odds' style phrasing in §III.B. These should be cleaned up.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is the manuscript's relation to [22] (Luna and Ryskin). The paper concedes that [22] found U-matrix unitarization inconsistent with AGK rules for a supercritical pomeron, yet uses an AGK-based formula to derive its central result and dismisses the inconsistency with a paragraph whose asymptotic statement is self-contradictory. Combined with the factor-of-two error in the U-matrix amplitude (Eqs. 31/42 vs. Eqs. 17/30), the central claim is currently unsupported. I do not think this is a reject-and-forget case: the generalized-representation framework is legitimate, the eikonal side is correct, and the Poisson-exponential mixture identity is clean, so a genuine re-derivation of Eq. (26) with a consistent treatment of the unitarity sum rule could salvage the geometric-distribution claim. But the revision must be substantive—a corrected amplitude and a paragraph of hand-waving will not suffice. Note also the heavy self-citation pattern ([8], [9], [13], [24] are all the author's own works); the reliance on [24] for the fitted parameters is appropriate, but the fit's data coverage should be stated explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Frankly, this paper has a load-bearing problem that the authors half-see. The central result, that the U-matrix scheme gives a geometric pomeron distribution with positive correlations, does not follow from the U-matrix amplitude used in the same paper. With the stated spectral density rho(tau)=e^{-tau/c}/c, c=1/2, the S-matrix is 1/(1-i chi/2), so the unitarized amplitude is chi/(2-i chi), not chi/(1-i chi/2) as in Eq. (31). That factor of 2 is not cosmetic: for chi=i gamma, the inelastic cross-section from Eq. (46) is 8 gamma/(2+gamma)^2, while the sum of the proposed topological cross-sections in Eq. (47) is gamma/(1+gamma). The two disagree for every gamma, and the paper acknowledges this but hand-waves about diffractive states instead of doing a calculation. A topological cross-section has to satisfy sum_n sigma_n = G_inel; Eq. (32) does not, so the geometric distribution is not the cut-pomeron distribution of the U-matrix amplitude actually used.\n\nWhat is genuinely useful here is the presentation of the generalized S-matrix formalism and the numerical exploration. The figures showing mean, variance, f2, and factorial moments are clearly presented, and the fits to cross-section data give the schemes a concrete parameterization. The qualitative claim that the U-matrix gives larger fluctuations and stronger correlations than the eikonal is plausible and worth taking seriously. The paper also deserves credit for openly flagging the AGK inconsistency instead of burying it.\n\nBut the newness is thin. Eq. (32) matches Eq. (45) of Luna-Ryskin up to the factor of 2, and the geometric form is just the standard compound-Poisson result for an exponential rate. The results are baked in by choosing an exponential spectral density and quoting Eq. (26) from Kancheli without derivation. The multiplicity calculation assumes Poissonian shower distributions despite known KNO violation, which limits its contact with data.\n\nIf the authors fix the factor of 2, derive Eq. (26) in an AGK-consistent way, and reconcile the unitarity sum, the qualitative U-matrix picture might survive. Right now, the central claim is not established. I'd send it to a referee because the topic is relevant and the paper is readable, but I'd expect major revision or rejection as it stands.","headline":"The paper's central geometric-distribution claim for the U-matrix does not survive the factor-of-2 inconsistency and failed AGK sum rule, though the formalism review and numerical exploration have some value.","tokens_in":20057,"tokens_out":4859,"would_cite":false,"duration_ms":43021,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that in the U-matrix unitarization scheme the number of exchanged pomerons is geometrically distributed rather than Poisson, making multi-pomeron exchanges correlated and their fluctuations larger at all energies.","keywords":["pomeron weights","U-matrix unitarization","eikonal unitarization","multi-pomeron exchange","geometric distribution","pomeron correlations","multi-parton interactions","high-energy QCD"],"falsifier":"At collision energies below about $10^4$ GeV, the geometric distribution predicts the number of exchanged pomerons is overdispersed ($\\mathrm{Var}>\\langle n\\rangle$ and $f_2>0$), while the eikonal scheme predicts a Poisson distribution with $\\mathrm{Var}=\\langle n\\rangle$ and $f_2=0$. Measuring the pomeron multiplicity distribution in that energy range and checking whether the variance exceeds the mean, or recomputing $\\sigma_n$ from the standard cutting rules with a different answer, would settle the claim.","tokens_in":18918,"feed_emoji":"📊","tokens_out":14431,"duration_ms":116973,"temperature":0.7,"pith_summary":"This paper argues that the choice of unitarization—how one ensures that scattering probabilities respect the unitarity bound—changes the statistics of the soft pomerons exchanged in high-energy proton collisions. In the standard eikonal scheme, the number of exchanged pomerons is Poisson-distributed, so each exchange is statistically independent. In the U-matrix scheme, the paper derives a geometric distribution, meaning one exchanged pomeron makes additional pomeron exchanges more likely. Because pomerons are exchanged between composite, strongly interacting protons, the paper contends that such correlated exchanges are more physical than independent ones. If this is right, the U-matrix scheme is the better unitarization for QCD processes with cross-sections that keep growing with energy.","feed_headline":"Pomeron count is geometric, not Poisson, in U-matrix scheme","feed_subtitle":"Correlated multi-pomeron exchanges broaden proton multiplicity and favor the U-matrix scheme at high energy.","key_machinery":"The central object is the generalized S-matrix $S[\\chi(s,b)]=\\int_0^\\infty d\\tau\\,\\rho(\\tau)e^{i\\tau\\chi(s,b)}$, where $\\chi(s,b)$ is the single-pomeron Born amplitude in impact-parameter space and $\\rho(\\tau)$ is a spectral density encoding the pomeron weights. The argument works by fixing $\\rho(\\tau)$: the delta function $\\delta(\\tau-1)$ gives the eikonal scheme, while the exponential $e^{-\\tau/c}/c$ with $c=1/2$ gives the U-matrix scheme. This object carries the derivation because Eq. (26), $\\sigma_n=\\int d\\tau\\,\\rho(\\tau)(2\\tau\\,\\mathrm{Im}\\,\\chi)^n e^{-2\\tau\\,\\mathrm{Im}\\,\\chi}/n!$, turns each scheme choice into a concrete pomeron multiplicity distribution, from which the mean, variance, $f_2$, and higher factorial moments are computed.","core_discovery":"Using the generalized representation of the unitarized elastic amplitude of reference [16], $S[\\chi(s,b)]=\\int d\\tau\\,\\rho(\\tau)e^{i\\tau\\chi(s,b)}$, the paper derives the pomeron topological cross-section $\\sigma_n(s,b)=\\int d\\tau\\,\\rho(\\tau)\\,\\frac{(2\\tau\\,\\mathrm{Im}\\,\\chi)^n}{n!}e^{-2\\tau\\,\\mathrm{Im}\\,\\chi}$. A delta-function spectral density reproduces the eikonal amplitude and gives the Poisson law $\\sigma_n=(2\\,\\mathrm{Im}\\,\\chi)^n e^{-2\\,\\mathrm{Im}\\,\\chi}/n!$. An exponential spectral density $\\rho(\\tau)=e^{-\\tau/c}/c$ with $c=1/2$ reproduces the U-matrix amplitude $A=\\chi/(1-i\\chi/2)$ and gives the geometric distribution $\\sigma_n=(\\mathrm{Im}\\,\\chi)^n/(1+\\mathrm{Im}\\,\\chi)^{1+n}$. From this distribution the paper concludes that in the U-matrix scheme the variance of the number of exchanged pomerons always exceeds the mean, the two-pomeron correlation $f_2$ stays positive at every energy, and normalized factorial moments grow with rank, so pomeron exchanges are correlated rather than independent. It further uses this geometric weighting in a string-model calculation of proton-proton multiplicities and finds that correlated pomeron exchanges enhance multi-parton collisions, especially double-parton collisions.","pith_inferences":["A testable extension the paper does not pursue is to fit $\\rho(\\tau)$ as a free function to total, elastic, and inelastic cross-section data; if the preferred density is far from exponential, the geometric distribution would be an artifact of the $c=1/2$ choice.","Because compounding a Poisson with an exponential rate is what produces the geometric distribution, the same formalism could accommodate other rate distributions, for example a gamma distribution, yielding negative-binomial-like pomeron multiplicities that still satisfy unitarity; the paper does not explore these alternatives.","The correlated exchanges imply a non-factorizable two-parton distribution $F(x_1,x_2)$ inside the proton, which could be probed in same-event double parton scattering; the paper flags this question but leaves the calculation for future work.","The paper compares schemes using separately fitted parameters for each; a stricter test would fix one scheme's parameters and evaluate the other scheme's predictions with the same input, isolating the effect of unitarization from parameter fitting."],"forward_implications":["If the U-matrix scheme is right, pomeron multiplicity in proton-proton collisions is overdispersed at every collision energy, so hadron multiplicity distributions should be broader than a Poisson distribution even below the $10^4$ GeV threshold where the eikonal predicts Poissonian behavior.","The geometric distribution implies that the exchange of one pomeron increases the chance of further exchanges, which changes the interpretation of double and multi-parton collisions and enhances double-parton scattering in the U-matrix scheme.","The unitarity-limit decrease of each multi-pomeron topological cross-section occurs at a slightly higher energy in the U-matrix than in the eikonal, giving an energy-dependent signature that data on total, elastic, and inelastic cross-sections can distinguish.","Monte Carlo event generators built on eikonal or quasi-eikonal unitarization should swap the Poissonian weight for the geometric distribution to obtain more realistic high-energy and cosmic-ray predictions.","The factorial moments $F_q$ grow with rank $q$ and depend on impact parameter, so correlation measurements in central versus peripheral collisions can test which unitarization scheme is realized."],"supporting_citations":[{"why":"supplies the generalized S-matrix expansion and the Poisson-superposition formula (26) from which both pomeron topological cross-sections are derived.","marker":"[16]"},{"why":"defines the eikonal and U-matrix unitarization forms that the two spectral densities reproduce.","marker":"[19]"},{"why":"establishes the baseline result that the eikonal scheme makes the number of exchanged pomerons Poisson-distributed.","marker":"[5]"},{"why":"gives the mixed-Poisson framework showing that exponential compounding of a Poisson rate produces a geometric distribution.","marker":"[21]"},{"why":"provides the alternative derivation of the pomeron topological cross-section from cutting rules that the paper compares with its Eq. (32).","marker":"[22]"},{"why":"supplies the fitted pomeron-trajectory and form-factor parameters used in the numerical plots for both schemes.","marker":"[24]"},{"why":"provides the string-model multiplicity parametrization used to compute proton-proton multiplicity distributions.","marker":"[27]"}],"fun_headline_variants":["Geometric pomeron weights reshape proton multiplicity","Pomeron correlations favor U-matrix unitarization","U-matrix pomerons boost multi-parton collisions","Geometric pomeron distribution emerges from U-matrix","Pomeron variance exceeds mean in U-matrix scheme"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole comparison rests on an unproved counting rule for the number of cut pomerons, taken without derivation from reference [16], plus the specific exponential weighting that defines the U-matrix scheme; if either differs from the correct physics, the geometric-versus-Poisson conclusion does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Geometric pomeron weights reshape proton multiplicity","Pomeron correlations favor U-matrix unitarization","U-matrix pomerons boost multi-parton collisions","Geometric pomeron distribution emerges from U-matrix","Pomeron variance exceeds mean in U-matrix scheme"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000337,"raw_usage":{"total_tokens":1929,"prompt_tokens":1076,"completion_tokens":853,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":692,"completion_tokens_details":{"reasoning_tokens":774}},"tokens_in":692,"tokens_out":853,"duration_ms":7297,"temperature":1.0,"reasoning_tokens":774,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:39:02.376197+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At collision energies below about $10^4$ GeV, the geometric distribution predicts the number of exchanged pomerons is overdispersed ($\\mathrm{Var}>\\langle n\\rangle$ and $f_2>0$), while the eikonal scheme predicts a Poisson distribution with $\\mathrm{Var}=\\langle n\\rangle$ and $f_2=0$. Measuring the pomeron multiplicity distribution in that energy range and checking whether the variance exceeds the mean, or recomputing $\\sigma_n$ from the standard cutting rules with a different answer, would settle the claim.","supporting_citations":[{"cited_title":"On the eikonal unitarisation at high energies","cited_arxiv_id":"1309.5860","evidence_quote":"supplies the generalized S-matrix expansion and the Poisson-superposition formula (26) from which both pomeron topological cross-sections are derived."},{"cited_title":"New analytic unitarization schemes","cited_arxiv_id":"0812.0735","evidence_quote":"defines the eikonal and U-matrix unitarization forms that the two spectral densities reproduce."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes the baseline result that the eikonal scheme makes the number of exchanged pomerons Poisson-distributed."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the alternative derivation of the pomeron topological cross-section from cutting rules that the paper compares with its Eq. (32)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the fitted pomeron-trajectory and form-factor parameters used in the numerical plots for both schemes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the string-model multiplicity parametrization used to compute proton-proton multiplicity distributions."}],"review_version":1}