Information geometry of DGLAP evolution

Alex Prygarin; Sergey Bondarenko

REVIEW

Information geometry of DGLAP evolution

Sergey Bondarenko, Alex Prygarin

High Energy Physics - Phenomenology · Submitted 20 Oct 2026

Pith's verdict: A genuine first: the Fisher-Rao speed of the DGLAP momentum density with a closed-form, three-moment lower bound — sound as a conditional theorem, with the unproved derivative-exchange assumption Hex as the one real gap in scope.

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arXiv ID2610.28115 (version 3)

Pith IDB5T5ILGW, Pith's permanent record of this paper and its review

CitationAlex Prygarin, Sergey Bondarenko. "Information geometry of DGLAP evolution." arXiv:2610.28115 (2026). https://pith.science/paper/B5T5ILGW

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Editor's note

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Two things to know. The paper has a real new result: the first evaluation of the Fisher-Rao speed of the momentum-weighted density P = xΣ + xg under LO DGLAP, plus a closed-form lower bound B(ζ) that needs only three input moments and the coupling, with no need to solve for the evolved density. Second, the theorem is conditional on a derivative-exchange assumption, Hex, which is checked on the evaluated inputs but not proved for all admissible inputs. The authors say so in Table I and in the conclusions; the stress-test note is right that this is the load-bearing gap.

Credit where it is due. The derivation is clean: N=2 Mellin moment equations, the finite-scale variance of Eq. (25), the drift interval of Eq. (21), then Cauchy-Schwarz. The checks are meaningful — independent generating-function differentiation, Talbot inversion, and an NLO drift identity that matches CT18 grid drift to 0.05 percent. They ship code and stored outputs. Novelty is scoped honestly: the Cramér-Rao-type inequality is credited to Ito-Dechant and Nicholson et al., the cumulant method to jet physics, and the genuinely new content is the first speed evaluation for this density and the closed-form width with quark-gluon species memory. They also resist overclaiming: the H1 single-power equality is flagged as a property of the form rather than of the data, and a √ξ trial slope gives nearly the same correlation, so near-equality does not single out a power reweighting.

Soft spots, in proportion. Hex is the one real gap. The Appendix C counterexamples — moving endpoints, insufficient decay — are explicitly not DGLAP paths, so they show the exchange is not automatic rather than refuting the theorem. A referee should push for a checkable sufficient condition for Hex, or a theorem statement restricted to the verified class. Tightness is input-dependent: two of five model inputs give (B/κ)² from 0.721 to 0.922, and the 200-input scan at Q0 spans 0.695 to 0.998. The paper is honest about this, but the abstract leans on the favorable three. The NLO extension covers only the drift identity, and the speed is not an observable — both acknowledged. The manuscript also carries many unresolved 'Sec.??' placeholders and at least one corrupted sentence; that is editing, not substance.

Who it is for: people building information-geometric descriptions of QCD evolution and people fitting PDFs who want a shape-change diagnostic. The conditional theorem holds up; the gap is scope, not correctness. Send it to a serious referee, with the request that they press on the Hex condition.

Referee report

Significance

If the result holds, this is a genuinely new and clean application of information geometry to collinear QCD: the first Fisher-speed evaluation for the momentum-weighted DGLAP density (the authors hedge this claim appropriately), a moment-only bound that avoids solving the evolution, and a set of independent numerical checks — generating-function differentiation, Talbot inversion of the Mellin solution, an NLO drift identity with residuals at or below 0.061 percent, and a 200-input scan — that make the claims reproducible from the supplied ancillary material. The paper is unusually honest in its reporting: Table I states that Hex is checked but not proved for every admissible input; Sec. IV A reports the full scan range rather than only the favorable cases and notes that no lower bound on (B/κ)² holds for all inputs; the √ξ trial-slope comparison shows that near equality does not by itself single out a power reweighting; and the H1 single-power equality is explicitly identified as a property of the form, not a result about the data. These features, together with the consistency checks on the moment algebra (for example the limits of Eqs.

Major objections (1)

[Table I (Hex); Eqs. (26)–(28); Appendix C] The central inequality κ ≥ B of Eqs. (27)–(28) is reached through Eq. (26), d⟨ξ⟩/dτ = Cov_P(sτ, ξ), and this identity requires both parts of the derivative-exchange assumption Hex of Table I: differentiation under the normalization integral (giving ⟨sτ⟩_P = 0) and differentiation under the first logarithmic moment integral. Table I states Hex is 'checked on the evaluated inputs but not proved for every admissible LO input,' and Appendix C shows the exchange is not automatic for smooth nonnegative densities: the moving-endpoint example of Eq. (C2) has ⟨sτ⟩_P = −1/(1+τ) ≠ 0 and d⟨ξ⟩/dτ ≠ Cov_P(sτ, ξ), and Eq. (C3) shows that vanishing at a moving boundary does not by itself validate the exchange. These examples are not LO DGLAP paths, so they do not refute the theorem, but they demonstrate that 'sufficiently regular' (abstract) is not sufficient by itself, and the paper gives no argument ruling out an analogue of these failures inside the admissible class. Since §IV presents Eq. (28) as bounding κ 'at every scale of the LO path,' the authors should either prove Hex for a well-characterized class of admissible inputs — for example, finite sums of beta-distribution terms with stated parameter ranges, using the large-x endpoint behavior of Eq. (C8) and the small-x input-power analysis of Appendix E — or else restate the main theorem, the abstract, and the conclusions as conditional on a per-input verification of Hex via the majorant condition of §II A. The limitation is acknowledged honestly by the authors, but it is load-bearing for the advertised universal scope and should be closed or explicitly scoped before publication. The authors acknowledge this: “Checked on the evaluated inputs but not proved for every admissible LO input.”

Minor objections (5)

[Throughout] Several cross-references remain as unresolved placeholders — 'Sec.??' appears in §II B, §III A, §IV A, §IV C, §VI, and Appendix G3 (as 'Fig.??'), and 'Eq. (??)' appears in Appendix B — and these must be resolved before the manuscript can be typeset.
[§IV, paragraph after Eq. (29)] The sentence beginning 'In this displays keeps its definition as the t slope' is garbled; it should read, for example, 'In this display, s keeps its definition as the t-slope.'
[§V and Fig. 2 / Table VI] For the hard-quark, steep-gluon and soft-quark, hard-gluon inputs, §V states that the joint Fisher information diverges at Q0 and that the Markov-contraction assumptions fail there; because the marginal, species-summed information can remain finite, the (B/κ)² values quoted for these inputs in Fig. 2 and Table VI are not necessarily outside the scope of Eq. (27), but the manuscript should state explicitly that the mean-bound hypotheses (finite marginal information, Hex, positive variance) hold at the scales where those ratios are reported.
[Eq. (27)] The display 'κ|corr_P(s,ξ)| = B' is introduced one sentence before the qualification that the correlation form requires positive information; moving that qualification ahead of the display would prevent a transient misreading of the equality.
[Table II and §VI] The footnote reporting that lowering xmin from 10⁻⁶ to 10⁻⁹ changes the CT18 speed at 10⁴ GeV by 1.72 percent, outside the quoted uncertainty, is an important systematic statement; a matching remark in the main text of §VI would help readers distinguish normalization-interval systematics from the PDF and coupling uncertainties quoted in the table.

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The paper

The paper claims

Under LO DGLAP evolution, the speed of the proton's momentum-distribution change has a closed-form lower bound from three input numbers, within a few percent of the true speed for realistic inputs.

The reading

The paper recasts the proton's momentum distribution $P=x\Sigma+xg$ as a probability law, which is legitimate because momentum conservation normalizes it, and then measures how fast its shape changes with resolution through the Fisher information, the statistical speed at which neighboring scales can be told apart. Its main result is a closed-form lower bound $B$ on this speed at every evolution time, built from the drift and width of $\xi=\ln(1/x)$ and fixed by just three numbers of the input density plus the coupling, with no need to compute the evolved density. For proton-like, steep, and valence-like inputs, $(B/\kappa)^2$ ranges from 0.948 to 0.995 at 2, 10, 100, and $10^4$ GeV, and a realistic LO trajectory started from a next-to-next-to-leading-order global fit gives 0.977 to 0.988. Near-saturation means the evolved shape change is almost exactly a reweighting by a power of $1/x$, with the logarithmic rate of change nearly affine in $\ln(1/x)$. If correct, this turns "how fast is the proton rearranging" into a three-number calculation.

What carries the argument

The load-bearing object is the momentum-weighted density $P=x\Sigma+xg$ on $0<x<1$, together with the coordinate $\xi=\ln(1/x)$ and the Fisher-Rao speed $\kappa$ per unit evolution time $\tau=\int (\alpha_s/2\pi)\,dt$. The argument runs through Mellin differentiation of the LO singlet equation at $N=2$: the column sums of the splitting matrix vanish there by momentum conservation, so the drift $d\langle\xi\rangle/d\tau$ becomes a convex combination of the two species' logarithmic loss rates, and the variance $\mathrm{Var}_P(\xi)$ obeys a closed moment system whose solution separates persistent drift-and-broadening terms proportional to $\zeta$ from decaying species-mixing terms proportional to $E=e^{-\omega\zeta}$. The bound itself is the Cauchy-Schwarz inequality $\kappa\ge |d\langle\xi\rangle/d\tau|/\mathrm{sd}_P(\xi)=B$, turned into a closed form by substituting those moment solutions; the equality condition, a logarithmic slope affine in $\xi$, identifies saturation with infinitesimal reweighting by a power of $1/x$. A secondary mechanism maps normalized densities through $u=2\sqrt{P}$ onto a sphere in square-integrable functions, giving the shortest-path distance $D_{FR}=2\arccos\int_0^1 dx\,\sqrt{P(t_1)P(t_2)}$ used for the length comparisons.

Extended reading notes

Core claim

The paper's central claim is that the speed $\kappa$ at which the momentum density $P=x\Sigma+xg$ changes under leading-order DGLAP evolution is bounded from below at every scale by $B=|d\langle\xi\rangle/d\tau|/\mathrm{sd}_P(\xi)$, and that $B$ has a closed form at $n_f=5$ with one-loop running: once the coupling is specified, only the input quark momentum fraction, the quark part $C_{q0}$ of the centered first moment, and the input variance of $\xi=\ln(1/x)$ enter, through the drift and variance solutions of Eqs. (24) and (25). The bound is the Cauchy-Schwarz inequality applied to $d\langle\xi\rangle/d\tau=\mathrm{Cov}_P(s_\tau,\xi)$, where $s_\tau$ is the logarithmic scale slope; equality holds exactly when that slope is affine in $\xi$, the infinitesimal version of reweighting the density by a power of $1/x$. For the proton-like, steep, and valence-like inputs, $(B/\kappa)^2$ ranges from 0.948 to 0.995 at 2, 10, 100, and $10^4$ GeV, and for an LO trajectory started from a global next-to-next-to-leading-order fit it ranges from 0.977 to 0.988, so the residual slope $r=s_\tau-\beta(\xi-\langle\xi\rangle)$ carries only a few percent of the momentum-weighted mean-square rate. The paper further establishes the drift interval $3.3865\le d\langle\xi\rangle/d\tau\le 7.9252$ per unit $\tau$, the large-time coefficients $A_\kappa=3.024$ and $A_L=1.578$, contraction of the joint quark-gluon Fisher information within a flavor segment, and evolution-path lengths that exceed the shortest Fisher-Rao distance by only 1.7 to 7.7 percent.

Load-bearing premise

The load-bearing premise is the derivative-exchange assumption Hex of Table I: differentiating under the integrals that define the normalization and the first logarithmic moment must be legitimate for the input and along the entire LO path, which the authors check on the inputs they evaluate but do not prove for every admissible input, and the moving-endpoint counterexamples of Appendix C show the identities can fail when support or decay is not controlled.

Editorial extensions

If this is right

At every scale of a forward LO path with nonnegative input quark and gluon densities, the true speed $\kappa$ is at least the closed-form $B$, so three input numbers plus the coupling certify how fast the momentum distribution is changing without evolving the density.
For proton-like, steep, and valence-like inputs, $(B/\kappa)^2\ge 0.948$ at the four tested scales, meaning the momentum-weighted logarithmic slope of the evolved density is dominated by its part affine in $\ln(1/x)$; the deficit $1-(B/\kappa)^2$ equals the momentum-weighted mean-square share of the residual slope.
The drift $d\langle\xi\rangle/d\tau$ stays between 3.3865 and 7.9252 per unit $\tau$ along the whole path exactly when the input quark momentum fraction lies in $[0,1]$, and it relaxes monotonically at rate $31/9$ toward the limit 5.7290.
The large-time forms $\liminf_{\zeta\to\infty}\sqrt{\zeta}\,\kappa\ge A_\kappa=3.024$ and $\liminf_{\zeta\to\infty}L/\sqrt{\zeta}\ge A_L=1.578$ give asymptotic speed and length coefficients fixed only by the one-loop matrix.
The five model evolution paths from the input scale to $10^4$ GeV exceed the shortest Fisher-Rao distance between their end densities by only 1.7 to 7.7 percent, and a future electron-ion collider experiment could measure the analogous excess for the normalized structure function $F_2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the near-saturation holds beyond the tested inputs, the essential shape change of the proton's momentum distribution under LO evolution is a one-parameter tilt family, powers of $1/x$; the residual slope $r$ would then be the natural diagnostic for whatever part of scaling violations a single effective power misses.
The moment-only bound doubles as a cheap consistency test for any new parton-density set: compute the three input moments, evolve numerically, and compare $B$ with the computed $\kappa$; large deficits would flag scales or $x$ ranges where the slope is far from affine, as the paper's own scan already shows for hard-quark or hard-gluon inputs.
The closed-form variance carries decaying species-mixing terms with timescales $9/31$ in $\tau$ for the quark-fraction relaxation and $69/62$ per unit $\zeta$ for the variance eigenmode, suggesting the input's quark-gluon division stops shaping the geometry after a modest evolution time; one could test whether the deficit $1-(B/\kappa)^2$ at fixed scale tracks the surviving species separation $C_q
The same drift-width Cauchy-Schwarz construction applies to any momentum-weighted probability law whose logarithmic slope satisfies a sum rule, such as fragmentation functions, where relative entropy has already been studied; an analogous three-moment bound could be written down without new machinery.

Original abstract

Quark and gluon splittings shift proton momentum toward smaller fractions with increasing resolution. Momentum conservation makes the summed momentum-weighted quark and gluon densities, when nonnegative, a probability distribution. We apply information geometry, the geometry of probability distributions, to this probabilistic form of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution. We evaluate, to our knowledge for the first time, the speed at which the shape of the proton's momentum distribution changes, measured by how well distributions at neighboring resolutions can be told apart statistically. We show that this speed has a closed-form lower bound at leading order, with one-loop splitting functions and running coupling, five fixed quark flavors, nonnegative quark and gluon densities and sufficiently regular inputs. Three numbers of the input (the distribution at the starting resolution) fix the bound at every scale without computing the evolved distribution, once the coupling is specified. The squared bound is at least 94.8 percent of the squared speed for three of the five model inputs at four scales from 2 GeV to 10 TeV. The bound is reached when the distribution's relative rate of change is linear in the logarithm of the inverse momentum fraction. The relative rate of change is close to this linear form in mean square weighted by momentum for these three model inputs at all four scales, and closer still to linear in the square root of the logarithm at the three higher scales. These descriptions are compatible because the logarithm and its square root are strongly correlated when weighted by these momentum distributions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-21: 1 request, 6,949 input tokens.

What the abstract itself states, as probabilities: claims first 0.91 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.13 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Quark and gluon splittings shift proton momentum toward smaller fractions with increasing resolution. Momentum conservation makes the summed momentum-weighted quark and gluon densities, when nonnegative, a probability distribution. We apply information geometry, the geometry of probability distributions, to this probabilistic form of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution. We evaluate, to our knowledge for the first time, the speed at which the shape of the proton's momentum distribution changes, measured by how well distributions at neighboring resolutions can be told apart statistically. At leading order, with one-loop splitting functions and running coupling, five fixed quark flavors, nonnegative quark and gluon densities and sufficiently regular inputs, we obtain a closed-form lower bound on this speed. Once the coupling is specified, three numbers of the input (the distribution at the starting resolution) fix the bound at every scale without computing the evolved distribution. For three of the five model inputs at four scales from 2 GeV to 10 TeV, the squared bound is at least 94.8 percent of the squared speed. The bound is reached when the distribution's relative rate of change is linear in the logarithm of the inverse momentum fraction. For these three model inputs at the four scales, the relative rate of change is close to this linear form in mean square weighted by momentum.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Quark and gluon splittings shift proton momentum toward smaller fractions with increasing resolution.	background 1.00	0.08	0.04	asserted 1.00
s1	Momentum conservation makes the summed momentum-weighted quark and gluon densities, when nonnegative, a probability distribution.	background 0.95	0.09	0.03	asserted 1.00
s2	We apply information geometry, the geometry of probability distributions, to this probabilistic form of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution.	method 0.65	0.86	0.05	asserted 0.97
s3	We evaluate, to our knowledge for the first time, the speed at which the shape of the proton's momentum distribution changes, measured by how well distributions at neighboring resolutions can be told apart statistically.	contribution 0.96	0.96	0.23	asserted 1.00
s4	At leading order, with one-loop splitting functions and running coupling, five fixed quark flavors, nonnegative quark and gluon densities and sufficiently regular inputs, we obtain a closed-form lower bound on this speed.	result 0.79	0.94	0.21	asserted 1.00
s5	Once the coupling is specified, three numbers of the input (the distribution at the starting resolution) fix the bound at every scale without computing the evolved distribution.	result 0.85	0.89	0.16	asserted 1.00
s6	For three of the five model inputs at four scales from 2 GeV to 10 TeV, the squared bound is at least 94.8 percent of the squared speed.	result 1.00	0.94	0.29	asserted 1.00
s7	The bound is reached when the distribution's relative rate of change is linear in the logarithm of the inverse momentum fraction.	result 0.91	0.86	0.18	asserted 1.00
s8	For these three model inputs at the four scales, the relative rate of change is close to this linear form in mean square weighted by momentum.	result 0.94	0.92	0.14	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Mean logarithmic momentum drift per unit

**Figure 2.** FIG. 2. Fisher speeds and squared mean-bound ratios. Upper-panel black solid, blue dashed, and red dash-dotted curves give

Figures reproduced from arXiv:2610.28115 by the authors.

References

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Fisher-Rao Geometry of Gluon Saturation Fronts
hep-ph 2026-11 conditional novelty 7.0 of 10

The transverse Fisher-Rao metric between gluon saturation fronts decays as t^-4 and tau^-4, driven by a spatial translation plus time delay of a universal relaxation flow.
Parton wave functions, density matrices and entropy in DGLAP evolution
hep-ph 2026-11 conditional novelty 6.0 of 10

A one-parton density matrix constructed from Lipatov's wave functions separates sampled-parton entropy from a maximum mode entropy, and the evolution of the history-averaged state obeys a Lindblad equation that reduce...

Discussion

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