arxiv: 2605.08736 · v1 · submitted 2026-05-09 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Higher-order local constraints from reciprocal symmetry and entanglement entropy of charged-particle multiplicity distributions in pp collisions

Alex Prygarin, Claudelle Capasia Madjuogang Sandeu, Mustapha Ouchen

Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:40 UTC · model grok-4.3

classification ✦ hep-ph

keywords reciprocal symmetrymultiplicity distributionsKNO scalingentanglement entropypp collisionsATLAS experimentcharged particleslocal constraints

0 comments

The pith

Reciprocal symmetry of the scaling function implies local algebraic constraints on multiplicity distributions at the mean.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Reciprocal symmetry f_s(z) equals f_s(1/z) in the KNO-violating term makes h(u) = f_s(e^u) an even function of u = ln z. Consequently, the odd derivatives of h at u=0 vanish and translate into relations among the derivatives of the probability P at n = . The paper derives the k=1 constraint relating the third and second derivatives to the distribution value itself and verifies that it is satisfied by ATLAS data at 13 TeV. The same symmetry and normalization conditions also yield a model-independent formula for the entanglement entropy of the distribution.

Core claim

The reciprocal symmetry f_s(z)=f_s(1/z) of the KNO-violating term implies that h(u)≡f_s(e^u) is even in u=ln z. Each odd derivative of h at u=0 therefore supplies a local algebraic constraint on the multiplicity distribution at n=<n>. The k=1 constraint is <n>^3 P'''(<n>)+6<n>^2 P''(<n>)=5 P(<n>), which is equivalent to the unconditional residual δ_3=1-2ρ_0+ρ_1=0. In ATLAS 13 TeV data with the largest fit window, δ_3=-0.02±0.11, consistent with the symmetric value. A χ² test rejects global symmetry at 13 TeV but the local picture near z=1 holds at the first two non-trivial orders. The entanglement entropy is given by the model-independent expression S=ln<n>+1−(1/2)∫ e^{-z} f_s²(z) dz + O(f_s

What carries the argument

The even function h(u)=f_s(e^u) obtained from the reciprocal symmetry f_s(z)=f_s(1/z), whose vanishing odd derivatives at u=0 produce algebraic constraints on the derivatives of P at the mean multiplicity.

Load-bearing premise

That the reciprocal symmetry is sufficiently accurate near z=1 for the Taylor expansion of h(u) to produce local constraints that are borne out by the data, despite the global symmetry being rejected at 13 TeV.

What would settle it

A measurement with smaller uncertainties showing that δ_3 differs from zero by several standard deviations, or that the combination <n>^3 P'''(<n>)+6<n>^2 P''(<n>)-5 P(<n>) is nonzero near the mean, would falsify the local constraint derived from the symmetry.

Figures

Figures reproduced from arXiv: 2605.08736 by Alex Prygarin, Claudelle Capasia Madjuogang Sandeu, Mustapha Ouchen.

**Figure 2.** Figure 2: FIG. 2. Local-fit determination of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Comparison of the multiplicative-noise prediction [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: shows the integrand e −zf 2 s (z) across the measured z range for the three energies. The largest contributions come from z ∼ 1 and from the high-z tail where f 2 s → 1, with the central window 1/3 < z < 3 contributing only ∼ 25% of the total ∆S. 0 1 2 3 4 5 6 7 8 z = n/⟨n⟩ 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 e − z f2 s (z) integrand of ΔS= − 1 2 ∫ ∞ 0 e −z f2 s (z)dz −2ΔS contribution 7 TeV, ΔS= −0.008… view at source ↗

read the original abstract

The reciprocal symmetry $f_s(z)=f_s(1/z)$ of the KNO-violating term in proton--proton charged-multiplicity distributions, observed at $\sqrt{s}=7$, $8$ and $13$ TeV, implies that the function $h(u)\equiv f_s(e^u)$ is even in $u=\ln z$. Each odd derivative of $h$ at $u=0$ then provides a local algebraic constraint on the multiplicity distribution at $n=\langle n\rangle$. The $k=0$ constraint $P'(\langle n\rangle)=-P(\langle n\rangle)/\langle n\rangle$ has been verified previously. We derive the $k=1$ constraint, $\langle n\rangle^3 P'''(\langle n\rangle)+6\langle n\rangle^2 P''(\langle n\rangle)=5\,P(\langle n\rangle)$, equivalent to the unconditional residual $\delta_3\equiv 1-2\rho_0+\rho_1=0$, and test it in the ATLAS data: at $13$~TeV with the largest fit window we find $\delta_3=-0.02\pm 0.11$, consistent with the leading-order symmetric value, while the $7$ and $8$~TeV results are inconclusive at the available precision. A $\chi^2$ test of the global symmetry across the window $1/3<z<3$ is consistent with the symmetry at $7$ and $8$~TeV but rejects it decisively at $13$~TeV, where the smaller experimental uncertainties expose a residual departure from $z\to 1/z$ invariance away from $z=1$. The overall picture is therefore that the symmetry holds at the leading two non-trivial orders near $z=1$ but breaks down globally at the precision afforded by the $13$~TeV data, suggesting that $f_s(z)=f_s(1/z)$ is at best an approximate symmetry. We derive a model-independent expression for the entanglement entropy $S=\ln\langle n\rangle+1-\tfrac{1}{2}\int_0^\infty e^{-z}f_s^2(z)\,dz+\mathcal{O}(f_s^3)$, in which the linear term in $f_s$ cancels by virtue of normalisation and the constraint $\langle z\rangle=1$, and evaluate it numerically on the ATLAS data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a clean k=1 local constraint on P(n) from the evenness of h(u) and gives a model-independent entropy formula with a nice cancellation, but the locality assumption for the Taylor step is not quantified even though global symmetry fails at 13 TeV.

read the letter

The core result is a new algebraic relation ³ P'''() + 6 ² P''() = 5 P() that follows directly once you accept f_s(z) = f_s(1/z) near z=1 and expand h(u) = f_s(e^u) to third order. They also write down an entanglement entropy expression S = ln + 1 - ½ ∫ e^{-z} f_s²(z) dz + O(f_s³) where the linear piece drops out from normalization and =1 alone. Both pieces are new relative to the KNO literature they cite, and the entropy formula is genuinely model-independent on its own terms. The data test at 13 TeV gives δ₃ = -0.02 ± 0.11 inside the largest window, which sits inside the symmetric expectation, while the χ² test rejects the symmetry over 1/3 < z < 3 at the same energy. That contrast is the honest picture the paper presents: approximate local reciprocity but clear global departure once errors shrink. The derivation itself is straightforward and does not hide extra assumptions beyond the evenness of h(u). The entropy cancellation is particularly clean and does not lean on the symmetry at all. The soft spot is the unquantified radius of the Taylor expansion. The constraint uses derivatives at exactly u=0, yet the paper shows the symmetry already fails when integrated over a wider interval at 13 TeV. Without a plot or table that tracks |f_s(z) - f_s(1/z)| versus |ln z| near zero, it is hard to judge whether the reported δ₃ is protected from the asymmetry that appears farther out. The ±0.11 error bar is large enough that the test is not decisive either way. This work is for people who already care about multiplicity distributions, KNO scaling violations, and information measures in soft hadron physics. A reader fitting models to ATLAS or CMS data could plug the k=1 relation into their fits as an extra check, and the entropy formula offers a quick way to compute S without choosing a specific parametrization. It is narrow but internally consistent on the points it actually derives. I would send it to peer review. The derivations are transparent, the data comparison is explicit, and the locality question is something referees can press the authors to address with a short auxiliary plot. The paper does not overclaim once you read the global χ² result alongside the local δ₃ number.

Referee Report

1 major / 1 minor

Summary. The paper reports that the reciprocal symmetry f_s(z)=f_s(1/z) observed in the KNO-violating component of charged-particle multiplicity distributions in pp collisions at LHC energies implies that h(u)≡f_s(e^u) is even, yielding local algebraic constraints on P(n) at n=⟨n⟩ via Taylor expansion. It derives the k=1 constraint ⟨n⟩³P'''(⟨n⟩)+6⟨n⟩²P''(⟨n⟩)=5P(⟨n⟩) (or δ₃=0), tests it on ATLAS data (finding δ₃=-0.02±0.11 at 13 TeV consistent with zero), notes that global χ² rejects the symmetry at 13 TeV while local consistency holds near z=1, and derives the model-independent entanglement entropy S=ln⟨n⟩+1−½∫e^{-z}f_s²(z)dz+O(f_s³).

Significance. If the local symmetry near z=1 is robust, the work supplies parameter-free higher-order constraints on multiplicity distributions and a compact, model-independent expression for entanglement entropy that cancels the linear term in f_s by normalization and ⟨z⟩=1. This could constrain phenomenological models of particle production and link multiplicity statistics to information-theoretic quantities in high-energy collisions. The explicit data tests and derivation of the entropy formula are strengths.

major comments (1)

[Abstract and data-analysis section] Abstract and data-analysis section: The central claim that 'the symmetry holds at the leading two non-trivial orders near z=1' rests on δ₃ consistency at 13 TeV, yet the same data set yields a decisive χ² rejection of global reciprocity over 1/3<z<3. No auxiliary figure or table quantifies the scale |ln z| at which |f_s(z)−f_s(1/z)| first exceeds the local statistical uncertainty, leaving open whether asymmetry already contaminates the Taylor truncation used to extract the k=1 constraint at the reported ±0.11 precision.

minor comments (1)

[Entropy derivation] The numerical evaluation of the integral in the entropy expression is stated to be performed on ATLAS data, but the binning, interpolation, or cutoff procedure used for f_s(z) is not specified, which affects reproducibility of the quoted S values.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to better quantify the range of validity of the local reciprocal symmetry. We agree that an auxiliary figure or table would strengthen the distinction between local consistency near z=1 and global deviations, and we will incorporate this in the revision.

read point-by-point responses

Referee: [Abstract and data-analysis section] Abstract and data-analysis section: The central claim that 'the symmetry holds at the leading two non-trivial orders near z=1' rests on δ₃ consistency at 13 TeV, yet the same data set yields a decisive χ² rejection of global reciprocity over 1/3<z<3. No auxiliary figure or table quantifies the scale |ln z| at which |f_s(z)−f_s(1/z)| first exceeds the local statistical uncertainty, leaving open whether asymmetry already contaminates the Taylor truncation used to extract the k=1 constraint at the reported ±0.11 precision.

Authors: We agree that an explicit quantification of the |ln z| scale is needed to confirm that the local Taylor truncation remains uncontaminated. The δ₃ constraint is obtained from a local fit to P(n) in a window around n=⟨n⟩ (i.e., z near 1), while the global χ² integrates over 1/3<z<3. In the revised manuscript we will add a new figure displaying |f_s(z)−f_s(1/z)|/σ(z) versus |ln z|, where σ(z) is the point-wise uncertainty propagated from the ATLAS data. This will show that the normalized deviation remains consistent with zero (within 1σ) for |ln z| ≲ 0.1, which covers the fit window used for δ₃, while statistically significant departures appear only at larger |ln z| that drive the global χ² rejection. We will also update the abstract and discussion to state more precisely that the leading two orders hold locally near z=1, with the global symmetry breaking exposed only by the higher-precision 13 TeV data at larger |ln z|. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from observed symmetry and normalization

full rationale

The paper observes reciprocal symmetry f_s(z)=f_s(1/z) in data at multiple energies, defines h(u)≡f_s(e^u) as even, and derives the k=1 algebraic constraint on P(n) at n=<n> via Taylor expansion of the even function (odd derivatives vanish at u=0). This is a direct mathematical consequence, not a fit or self-definition. The entropy expression follows from an expansion where the linear f_s term cancels explicitly by normalization and <z>=1, independent of the symmetry. The χ² global rejection at 13 TeV is reported separately and does not enter the local derivation or entropy formula. No self-citations, ansatze smuggled via citation, or renaming of known results occur; the central claims reduce to the input symmetry assumption plus standard normalization, with data tests serving as external checks rather than tautological outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the assumption that the observed f_s(z) is even in ln z near z=1, on standard Taylor expansion, and on normalization plus <z>=1; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The function h(u) = f_s(e^u) is even, so all odd derivatives vanish at u=0.
Invoked to obtain the local algebraic constraints from the reciprocal symmetry observed in data.
standard math Normalization ∫ P(n) dn = 1 and <z> = 1 hold exactly.
Used to cancel the linear term in the entanglement entropy expansion.

pith-pipeline@v0.9.0 · 5772 in / 1761 out tokens · 50560 ms · 2026-05-12T01:40:49.908342+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
The reciprocal symmetry f_s(z)=f_s(1/z) ... implies that the function h(u)≡f_s(e^u) is even in u=ln z. Each odd derivative of h at u=0 then provides a local algebraic constraint

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 3 internal anchors

[1]

Reciprocal symmetry and KNO scaling violation in proton-proton collisions

M. Ouchen and A. Prygarin, “Reciprocal symmetry and KNO scaling violation in proton-proton collisions,” arXiv:2605.00128 [hep-ph] (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

KNO scaling, memoryless- ness and maximal entanglement at universal fixed point,

M. Ouchen and A. Prygarin, “KNO scaling, memoryless- ness and maximal entanglement at universal fixed point,” arXiv:2511.18446 [hep-ph] (2025)

work page arXiv 2025
[3]

Soft gluons in the infinite-momentum wave function and the BFKL Pomeron,

A. H. Mueller, “Soft gluons in the infinite-momentum wave function and the BFKL Pomeron,” Nucl. Phys. B 415, 373 (1994)

work page 1994
[4]

Deep inelastic scattering as a probe of entanglement

D. E. Kharzeev and E. M. Levin, “Deep inelastic scatter- ing as a probe of entanglement,” Phys. Rev. D95, no. 11, 114008 (2017) [arXiv:1702.03489 [hep-ph]]. 9

work page Pith review arXiv 2017
[5]

Deep inelastic scat- tering as a probe of entanglement: Confronting experi- mental data,

D. E. Kharzeev and E. M. Levin, “Deep inelastic scat- tering as a probe of entanglement: Confronting experi- mental data,” Phys. Rev. D104, no. 3, L031503 (2021) [arXiv:2102.09773 [hep-ph]]

work page arXiv 2021
[6]

Quantum information approach to high energy interactions,

D. E. Kharzeev, “Quantum information approach to high energy interactions,” Phil. Trans. Roy. Soc. A380, no. 2216, 20210063 (2022) [arXiv:2108.08792 [hep-ph]]

work page arXiv 2022
[7]

Einstein- Podolsky-Rosen paradox and quantum entanglement at subnucleonic scales,

Z. Tu, D. E. Kharzeev and T. Ullrich, “Einstein- Podolsky-Rosen paradox and quantum entanglement at subnucleonic scales,” Phys. Rev. Lett.124, no. 6, 062001 (2020) [arXiv:1904.11974 [hep-ph]]

work page arXiv 2020
[8]

Andreevet al.(H1), Measurement of charged particle multiplicity distributions in DIS at HERA and its implication to entanglement entropy of partons, Eur

V. Andreevet al.[H1 Collaboration], “Measurement of charged particle multiplicity distributions in DIS at HERA and its implication to entanglement en- tropy of partons,” Eur. Phys. J. C81, 212 (2021) [arXiv:2011.01812 [hep-ex]]

work page arXiv 2021
[9]

Hentschinski and K

M. Hentschinski and K. Kutak, “Evidence for the max- imally entangled low x proton in deep inelastic scat- tering from H1 data,” Eur. Phys. J. C82, 111 (2022) [arXiv:2110.06156 [hep-ph]]

work page arXiv 2022
[10]

Maximally entangled proton and charged hadron multiplicity in deep inelastic scattering,

M. Hentschinski, K. Kutak and R. Straka, “Maximally entangled proton and charged hadron multiplicity in deep inelastic scattering,” Eur. Phys. J. C82, 1147 (2022) [arXiv:2207.09430 [hep-ph]]

work page arXiv 2022
[11]

Probing the onset of maximal entanglement inside the proton in diffractive deep inelastic scattering,

M. Hentschinski, D. E. Kharzeev, K. Kutak and Z. Tu, “Probing the onset of maximal entanglement inside the proton in diffractive deep inelastic scattering,” Phys. Rev. Lett.131, 241901 (2023) [arXiv:2305.03069 [hep- ph]]

work page arXiv 2023
[12]

QCD evolution of entanglement entropy,

M. Hentschinski, D. E. Kharzeev, K. Kutak and Z. Tu, “QCD evolution of entanglement entropy,” Rep. Prog. Phys.87, no. 12, 120501 (2024) [arXiv:2408.01259 [hep- ph]]

work page arXiv 2024
[13]

Entanglement entropy, Monte Carlo event generators, and soft gluons DIScovery

M. Hentschinski, H. Jung and K. Kutak, “Entanglement entropy, Monte Carlo event generators, and soft gluons DIScovery,” Phys. Rev. D113, no. 5, 054024 (2026) [arXiv:2509.03400 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[14]

Kutak and M

K. Kutak and M. Praszalowicz, “Entropy, purity and gluon cascades at high energies with recombinations and transitions to vacuum,” Eur. Phys. J. C85, 1215 (2025) [arXiv:2508.13781 [hep-ph]]

work page arXiv 2025
[15]

Kutak and S

K. Kutak and S. L¨ ok¨ os, “Entropy and multiplicity of hadrons in the high energy limit within dipole cas- cade models,” Phys. Rev. D112, no. 9, 096017 (2025) [arXiv:2509.07898 [hep-ph]]

work page arXiv 2025
[16]

Universality of scaling entropy in charged hadron mul- tiplicity distributions at the LHC,

L. S. Moriggi, F. S. Navarra and M. V. T. Machado, “Universality of scaling entropy in charged hadron mul- tiplicity distributions at the LHC,” Phys. Rev. D112, 074019 (2025) [arXiv:2506.09899 [hep-ph]]

work page arXiv 2025
[17]

Charged-particle multiplicities in pp interactions measured with the ATLAS detector at the LHC

G. Aadet al.[ATLAS Collaboration], “Charged-particle multiplicities inppinteractions measured with the AT- LAS detector at the LHC,” New J. Phys.13, 053033 (2011) [arXiv:1012.5104 [hep-ex]]

work page Pith review arXiv 2011
[18]

Charged-particle distributions in $pp$ interactions at $\sqrt{s}=8$ TeV measured with the ATLAS detector

G. Aadet al.[ATLAS Collaboration], “Charged-particle distributions in √s= 8 TeVppinteractions at the LHC,” Eur. Phys. J. C76, no. 7, 403 (2016) [arXiv:1603.02439 [hep-ex]]

work page Pith review arXiv 2016
[19]

Charged-particle distributions in $\sqrt{s}=13$ TeV $pp$ interactions measured with the ATLAS detector at the LHC

G. Aadet al.[ATLAS Collaboration], “Charged-particle distributions in √s= 13 TeVppinteractions measured with the ATLAS detector at the LHC,” Phys. Lett. B 758, 67 (2016) [arXiv:1602.01633 [hep-ex]]

work page Pith review arXiv 2016
[20]

Study of the KNO scaling inppcollisions at √sfrom 0.9 to 13 TeV using results of the ATLAS at the LHC,

Y. Kulchitsky and P. Tsiareshka, “Study of the KNO scaling inppcollisions at √sfrom 0.9 to 13 TeV using results of the ATLAS at the LHC,” Eur. Phys. J. C82, no. 5, 462 (2022) [arXiv:2202.06697 [hep-ex]]

work page arXiv 2022
[21]

Charged particle multiplicities in pp interactions at sqrt(s) = 0.9, 2.36, and 7 TeV

V. Khachatryanet al.[CMS Collaboration], “Charged particle multiplicities inppinteractions at √s= 0.9, 2.36, and 7 TeV,” JHEP01, 079 (2011) [arXiv:1011.5531 [hep- ex]]

work page Pith review arXiv 2011
[22]

Pomeron evolution, entan- glement entropy and Abramovskii-Gribov-Kancheli cut- ting rules,

M. Ouchen and A. Prygarin, “Pomeron evolution, entan- glement entropy and Abramovskii-Gribov-Kancheli cut- ting rules,” Phys. Rev. D112, no. 9, 094027 (2025) [arXiv:2508.12102 [hep-ph]]

work page arXiv 2025
[23]

The bare pomeron in quantum chromo- dynamics,

L. N. Lipatov, “The bare pomeron in quantum chromo- dynamics,” Sov. Phys. JETP63, 904 (1986)

work page 1986
[24]

High-energy asymptotics of multicolor QCD and two-dimensional conformal field theories,

L. N. Lipatov, “High-energy asymptotics of multicolor QCD and two-dimensional conformal field theories,” Phys. Lett. B309, 394 (1993)

work page 1993
[25]

Spin-spin entanglement in diffractive heavy-quark production,

M. Fucilla and Y. Hatta, “Spin-spin entanglement in diffractive heavy-quark production,” Phys. Rev. D113, no. 3, L031504 (2026) [arXiv:2509.05267 [hep-ph]]

work page arXiv 2026
[26]

Probing quantum en- tanglement with Generalized Parton Distributions at the Electron-Ion Collider,

Y. Hatta and J. Schoenleber, “Probing quantum en- tanglement with Generalized Parton Distributions at the Electron-Ion Collider,” arXiv:2511.04537 [hep-ph] (2025)

work page arXiv 2025
[27]

Quantum entanglement in electron-nucleus collisions: Role of the linearly polarized gluon distribution

M. Fucilla, Y. Hatta, and B.-W. Xiao, “Quantum entan- glement in electron-nucleus collisions: Role of the linearly polarized gluon distribution,” arXiv:2604.11697 [hep-ph] (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026