{"id":"e254d78e-6462-4abc-a4e1-febb3986cb59","arxiv_id":"2610.28115","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For leading-order DGLAP evolution, the Fisher information speed of the proton momentum density is bounded below by a closed-form expression fixed by three input moments.","lead":"This paper measures how fast the shape of the proton's momentum distribution changes as we look at it with higher resolution. It derives a simple closed-form lower bound on that rate using only three numbers from the starting distribution, without running the full DGLAP evolution.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The main claim's universal scope rests on unproved derivative-exchange Hex: without d/dτ∫ξP = ∫ξ∂τP dx, Eq. (26) and hence κ≥B fail to be established for all admissible inputs. The conditional theorem is sound; the gap is unverified scope.","rationale":"The reader's weakest-assumption identification is correct: Hex is the hinge on which Eq. (27) depends. My independent reading found no internal inconsistency in the derivation of the bound from Eq. (26), nor in the LO moment equations leading to the closed form Eq. (28). The near-equality ratios are finite computations with supplied code; without a specific numerical error, they are not the load-bearing point. The unresolved 'Sec.??' cross-references and OCR-corrupted formulas, such as the η expression in Eq. (24) and coefficients in Eq. (B6), are real editorial defects and should be fixed, but they do not by themselves undermine the central argument. The theorem is explicitly conditional, and the paper is unusually candid about this limitation. The proposed commutator test would settle whether Hex is merely unproved or actually false for a natural admissible class; if it finds a violation, the central claim must be restricted, and if it finds none, the remaining need is an analytic majorant proof. Either way, the reader's CONDITIONAL verdict remains appropriate, so no change is recommended.","tokens_in":32634,"tokens_out":21265,"duration_ms":363101,"concrete_test":"Compute the exact exchange defect Δ(τ) = d/dτ∫₀¹ ξP dx − ∫₀¹ ξ∂τP dx, and the analogous normalization defect, for an adversarial but LO-admissible family: Pq = c_q B_{a_q,b_q}, Pg = c_g B_{a_g,b_g} with a_j ∈ (−1,1), b_j > 0, covering the 200-input scan and endpoint limits a→−1⁺, b→0⁺. Evaluate both sides by high-order Mellin inversion and by direct quadrature of the LO convolution integrals at τ = 0 and small τ. If any nonzero Δ appears, Eq. (26) and hence κ≥B fail for that admissible input; if Δ ≡ 0 for the whole grid, Hex is verified numerically for this class and the remaining task is a majorant proof, not a counterexample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central inequality is Eq. (27), κ ≥ B. It follows from Cauchy-Schwarz applied to Eq. (26), d⟨ξ⟩/dτ = Cov_P(sτ, ξ). That covariance identity requires the derivative exchanges in Hex: d/dτ∫P dx = ∫∂τP dx and d/dτ∫ξP dx = ∫ξ∂τP dx. Table I marks Hex as checked on evaluated inputs but not proved for every admissible LO input, and Appendix C supplies genuine derivative-exchange failure mechanisms, moving endpoints and insufficient decay. Those explicit counterexamples are not DGLAP paths, so they do not disprove the theorem, but they show the exchange is not automatic. If Hex fails for some input that otherwise satisfies the LO moment, positivity, and Fisher assumptions, then ⟨s⟩P need not vanish and d⟨ξ⟩/dτ is not the covariance of sτ with ξ; Eq. (28) can then exceed the true speed, contradicting the advertised 'at every evolution time' bound. The conditional statement under Hex is derived consistently and the numerical tightness claims are supported by the supplied evaluation material; the unresolved issue is the scope of the admissible class. This is the load-bearing gap: an unproved hypothesis at the hinge of the main theorem, acknowledged by the authors rather than hidden.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper applies information geometry to the momentum-weighted, normalized parton density P = xΣ + xg, whose derivative in t = ln Q² is governed at leading order by the DGLAP equations. The central object is the Fisher speed κ per unit evolution time τ = ∫(αs/2π)dt, and the advertised main result is the closed-form lower bound B(ζ) of Eq. (28) with κ ≥ B at every scale, where B is determined solely by three input numbers (the quark momentum fraction, the centered quark moment Cq0, and the input variance of ξ = ln(1/x)) together with QCD constants, so that no evolved density is needed. The derivation combines N = 2 Mellin moment equations giving the drift (Eqs. (19)–(24)) and the variance at all finite scales (Eq. (25)), the covariance identity d⟨ξ⟩/dτ = Cov_P(sτ, ξ) (Eq. (26)), and the Cauchy–Schwarz inequality (Eq. (27)). The theorem is conditional on the assumptions of Table I, including the derivative-exchange hypothesis Hex, which the paper checks numerically on the evaluated inputs but does not prove for the whole admissible class. Numerically, (B/κ)² ranges from 0.948 to 0.995 at Q = 2, 10, 100, 10⁴ GeV for three selected model inputs and from 0.977 to 0.988 for an LO trajectory started from CT18 NNLO densities, while a 200-input scan gives a wider range (0.695–0.998 at Q0 and 0.941–0.985 at 10⁴ GeV). Additional results include the drift interval of Eq.","tokens_in":33079,"tokens_out":26096,"duration_ms":159788,"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_comments":[{"comment":"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.","section":"Table I (Hex); Eqs. (26)–(28); Appendix C","authors_acknowledge":"Checked on the evaluated inputs but not proved for every admissible LO input.","authors_acknowledge_p":0.82}],"minor_comments":[{"comment":"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.","section":"Throughout","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.'","section":"§IV, paragraph after Eq. (29)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"§V and Fig. 2 / Table VI","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Eq. (27)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Table II and §VI","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The manuscript is one of a series from these authors with extensive disclosed use of LLM tools (Appendix G3); the disclosure is unusually complete, the numerical claims are backed by supplied code and stored outputs, and I found no sign of fabricated results, but the editor may wish to be aware of the degree of AI involvement when considering this venue's policies. On scope: the paper is a formal theory contribution with phenomenological illustrations, and the authors correctly stress that the speed is not an observable; the proposed LHeC/EIC test is framed cautiously as a qualitative excess measurement. The reference list is appropriate, including the recent preprints [17] and [18]; no citation-pattern concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"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.\n\nCredit 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.\n\nSoft 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.\n\nWho 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.","headline":"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.","tokens_in":33577,"tokens_out":4695,"would_cite":true,"duration_ms":31824,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["12.38.Bx","13.60.Hb"],"model":"deepseek/deepseek-v4-flash","headline":"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.","keywords":["DGLAP evolution","information geometry","Fisher information","parton distribution functions","momentum density","speed bound","Mellin moments","deep inelastic scattering"],"falsifier":"Take an admissible LO input with nonnegative quark and gluon densities, finite logarithmic moments and positive variance, but with small-$x$ behavior vanishing faster than any power or with an effective endpoint that shifts with scale; evolve it at $n_f=5$ and compute directly the momentum-weighted mean slope $\\langle s\\rangle_P$ and the ratio $(B/\\kappa)^2$ at several scales. If $\\langle s\\rangle_P$ deviates from zero, or if $(B/\\kappa)^2$ exceeds 1 at any scale, the derivative-exchange assumption fails and the bound as stated does not hold for that input.","tokens_in":34124,"feed_emoji":"⚛️","tokens_out":13474,"duration_ms":80743,"temperature":0.7,"pith_summary":"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.","feed_headline":"Three input numbers bound the proton's DGLAP evolution speed","feed_subtitle":"The bound sits within about 5 percent of the true speed for realistic inputs, without ever evolving the density.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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.",""],"forward_implications":["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 $F_2$.",""],"supporting_citations":[{"why":"Supplies the one-loop singlet splitting matrix whose N=2 column derivatives give the drift and width entering the closed-form bound.","marker":"[3, 4]"},{"why":"Provides the Cauchy-Schwarz/Cramer-Rao inequality on which $\\kappa\\ge B$ rests, together with the Fisher-information speed formalism used throughout.","marker":"[36]"},{"why":"States the same time-information inequality in stochastic thermodynamics and is cited alongside [36] as the bound's basis.","marker":"[39]"},{"why":"Establishes the momentum-weighted density as a probability law and the sum rule $\\langle s\\rangle_P=0$ that turns the Fisher information into the variance of the logarithmic slope.","marker":"[18]"},{"why":"Supplies the global next-to-next-to-leading-order parton-density fit from which the realistic LO trajectory with $(B/\\kappa)^2=0.977$-$0.988$ is evolved, and the fitted-density speed comparisons.","marker":"[46]"},{"why":"Gives the N=2 relaxation of the quark momentum fraction used in the closed-form mean and variance solutions.","marker":"[61]"},{"why":"Identifies the quadratic coefficient of small-separation relative entropy with the Fisher-Rao metric, supporting the speed interpretation in the QCD context.","marker":"[26]"}],"fun_headline_variants":["Three input numbers fix a lower bound on proton DGLAP speed","Proton DGLAP speed bound from three numbers, within five percent","Information geometry sets DGLAP speed bound from three inputs","DGLAP momentum-shape speed: lower bound with just three inputs","Three numbers bound proton DGLAP evolution speed, no evolution needed"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Three input numbers fix a lower bound on proton DGLAP speed","Proton DGLAP speed bound from three numbers, within five percent","Information geometry sets DGLAP speed bound from three inputs","DGLAP momentum-shape speed: lower bound with just three inputs","Three numbers bound proton DGLAP evolution speed, no evolution needed"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000142,"raw_usage":{"cost":0.0001418326,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001418326,"upstream_inference_prompt_cost":3.24506e-05,"upstream_inference_completions_cost":0.000109382},"total_tokens":1872,"prompt_tokens":1271,"completion_tokens":601,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1143,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":523}},"tokens_in":1271,"tokens_out":601,"cost_source":"openrouter_usage","duration_ms":7519,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":523,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-21T09:07:32.922458+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Take an admissible LO input with nonnegative quark and gluon densities, finite logarithmic moments and positive variance, but with small-$x$ behavior vanishing faster than any power or with an effective endpoint that shifts with scale; evolve it at $n_f=5$ and compute directly the momentum-weighted mean slope $\\langle s\\rangle_P$ and the ratio $(B/\\kappa)^2$ at several scales. If $\\langle s\\rangle_P$ deviates from zero, or if $(B/\\kappa)^2$ exceeds 1 at any scale, the derivative-exchange assumption fails and the bound as stated does not hold for that input.","supporting_citations":[{"cited_title":"Ito \\ and\\ author A","cited_arxiv_id":null,"evidence_quote":"Provides the Cauchy-Schwarz/Cramer-Rao inequality on which $\\kappa\\ge B$ rests, together with the Fisher-information speed formalism used throughout."},{"cited_title":"Nicholson, Luis Pedro García-Pintos, Adolfo del Campo, Jason R","cited_arxiv_id":null,"evidence_quote":"States the same time-information inequality in stochastic thermodynamics and is cited alongside [36] as the bound's basis."},{"cited_title":"Golec-Biernat,Partonic entropy of the proton from DGLAP evolution, arXiv:2512.23102 [hep-ph]","cited_arxiv_id":null,"evidence_quote":"Establishes the momentum-weighted density as a probability law and the sum rule $\\langle s\\rangle_P=0$ that turns the Fisher information into the variance of the logarithmic slope."},{"cited_title":"New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC","cited_arxiv_id":null,"evidence_quote":"Supplies the global next-to-next-to-leading-order parton-density fit from which the realistic LO trajectory with $(B/\\kappa)^2=0.977$-$0.988$ is evolved, and the fitted-density speed comparisons."},{"cited_title":"title QCD and collider physics , volume 8 , publisher Cambridge University Press","cited_arxiv_id":null,"evidence_quote":"Gives the N=2 relaxation of the quark momentum fraction used in the closed-form mean and variance solutions."},{"cited_title":"Investigating QCD dynamical entropy in high-energy nuclear collisions, Phys.Lett.B868, 139737","cited_arxiv_id":null,"evidence_quote":"Identifies the quadratic coefficient of small-separation relative entropy with the Fisher-Rao metric, supporting the speed interpretation in the QCD context."}],"review_version":1}