{"id":"b8b939c3-f38a-4e24-93bc-30cd5193e853","arxiv_id":"2611.02459","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"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.","lead":"This paper applies Fisher-Rao information geometry to the traveling waves that describe gluon saturation in high-energy quantum chromodynamics. It finds that differences between starting conditions decay with a universal t^-4 power law, driven by a mechanism where changing the initial state is equivalent to nudging the wave sideways and delaying its universal relaxation by a fixed amount.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The BK τ^-4 universality law is the least secured load-bearing claim: it rests on τ ≤ 40 numerics, with the local exponent shifting from 4.02 to 4.06 on time-step halving and 30–40% leading-edge grid sensitivity, and it has no asymptotic support.","rationale":"The reader's CONDITIONAL verdict is the right one, and my stress test does not move it. The delay mechanism is well tested: Table I shows percent-level agreement between the independently predicted a'^2 and the measured g⊥θθ/g⊥shift for FKPP and for BK at τ = 15, and the FKPP t^-4 law has both a conditional analytic derivation and sub-percent numerical extrapolation toward C*/4. My concern is narrower than the reader's weakest-assumption list: I would not put the primary weight on the three Appendix A hypotheses, because the FKPP result has independent numerical support and the hypotheses are stated explicitly. The QCD-relevant claim, however, is the BK τ^-4 law, and that claim currently rests on a single numerical campaign ending at τ = 40. The time-step dependence of the local exponent, the 2% convergence floor on a metric that has already decayed by eight orders of magnitude, and the 30–40% sensitivity to leading-edge truncation all point in the same direction: the τ^-4 exponent is not yet resolved to the accuracy the abstract implies. This is why I would keep the verdict CONDITIONAL rather than ACCEPT, but also not REJECT: the mechanism is supported at intermediate rapidities, the FKPP analogue is convincing, and the BK observation is consistent with τ^-4 even if not yet decisive. The proposed longer run with refinement checks would settle whether the BK power law is a genuine asymptotic statement or a finite-window effect.","tokens_in":16206,"tokens_out":10069,"duration_ms":63544,"concrete_test":"Run the BK evolution to τ = 80 with the leading-edge boundary scaled accordingly (e.g., ρmin ≈ -800 at Δρ = 0.15, with a check at ρmin ≈ -1200), and repeat with at least two grid spacings (Δρ = 0.15, 0.10) and two time steps (dτ = 0.02, 0.01). Compute β_loc = -d ln g⊥θθ / d ln τ over [30,40] and [60,80] for both GBW and MV, and track τ^4 g⊥shift for both initial conditions. If β_loc moves outside 4 ± 0.1 under refinement or between the two rapidity intervals, or if the two τ^4 g⊥shift curves fail to approach a common plateau, the announced universal BK τ^-4 law is not established. As a secondary reproducibility check, deposit the run outputs and the scripts behind Table I and Fig. 4 so the claimed 0.55% residual at τ = 40 can be independently recomputed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The delay mechanism g⊥θθ ≃ a'^2 g⊥shift is genuinely supported by the independent neighboring-run test in Table I, and I do not contest that part of the argument. The load-bearing step that is least secure is the transfer of that mechanism into a universal τ^-4 power law for full BK. Section V-D reports the local exponent as 4.02 (GBW) and 4.01 (MV) at τ = 40, shifting to 4.06 when the time step is halved; Section III states only ~2% convergence of the transverse metric at τ = 40; and Section V-A reports that truncating the leading edge at ρmin = -40 changes g⊥θθ by 30–40%. The authors explicitly acknowledge at b26, b108 and b127 that the BK τ^-4 behavior is a numerical observation without an asymptotic derivation, so this is not a hidden flaw, but it remains the point on which the abstract's QCD universality claim depends. The Levin-Tuchin interior comparison explains the factor 3.5 in Iρ, but it does not by itself imply τ^-4. If the continuum exponent is actually 4.1, or if the τ^4-rescaled shift controls for GBW and MV drift apart beyond τ = 40, the central claim is weakened to a statement about the computed rapidity window.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The manuscript develops a Fisher-Rao information-geometric description of gluon saturation fronts. With the coordinate-space front density p = ∂N/∂ln r^2, a normalized probability density in the logarithmic dipole-size variable ρ, the authors define the Fisher metric on the family of fronts parameterized by (τ, θ), factor out the longitudinal propagation component via the Schur complement (Eq. 9), and study the transverse component g⊥θθ as a measure of shape distinguishability between trajectories. The main claim is a universal delay mechanism: to leading order, deforming the initial condition is equivalent to a spatial translation plus a temporal delay of the universal relaxation flow, giving g⊥θθ ≃ a'^2 g⊥shift (Eq. 18). This relation is tested out-of-sample in Table I, where a' and the delay t'0 are extracted from neighboring runs only; the predicted ratio matches the directly computed metric to within 0.7% for FKPP and to 0.74% for BK at τ = 15, and to about 3% at τ = 40 (0.55% on a finer grid). For FKPP the mechanism yields g⊥θθ ∝ t^-4 with a universal, parameter-free amplitude C* ≃ 1.112, derived conditionally on three explicitly stated analytic hypotheses (Appendix A) and verified numerically at the sub-percent level after extrapolation (Table II). For BK the paper offers numerical evidence, without an asymptotic derivation, for the corresponding τ^-4 law up to τ = 40, with the local exponent reaching 4.01-4.02.","tokens_in":16582,"tokens_out":27718,"duration_ms":152815,"significance":"The paper is a serious, carefully executed contribution with a genuinely new object at its center: the transverse Fisher-Rao metric as a quantitative memory-loss measure for saturation fronts. The outstanding strength is the independence of the delay-mechanism test: in Table I the translation and delay are fit from neighboring simulations only, and the prediction a'^2 g⊥shift is then compared with the tangent-equation metric without reusing the prediction to set any constant, so the 0.05-0.7% agreement for FKPP and the 0.74% figure for BK at τ = 15 constitute genuine evidence rather than construction. The FKPP amplitude C* is parameter-free, the derivation is honestly labeled as conditional, and its numerical verification (extrapolated t^4 g⊥shift within sub-percent of C*/4; residual Fisher cosine 0.99995 at t = 1280) is convincing. The paper also draws the right distinction between the intrinsic p-metric and the experimental likelihood Fisher matrix (Eq. 20), and it discloses its numerical sensitivities rather than concealing them.","major_comments":[{"comment":"The claim that full BK evolution reveals a universal τ^-4 relaxation law is the least quantitatively secured load-bearing statement in the paper, and the error budget the authors report does not yet support the precision implied by the quoted exponents. The local exponent at τ = 40 is 4.02 (GBW) and 4.01 (MV) but shifts to 4.06 when the time step is halved (Sec. V-D); the transverse metric itself is converged only to about 2% at τ = 40 (Sec. III); and truncating the leading edge at ρmin = −40 changes g⊥θθ by 30-40% (Sec. V-A). The absence of an asymptotic derivation for BK is acknowledged explicitly ([b26], [b108]), so this comment is not about disclosure; it is about the support level of an abstract-level universality claim. I request either (i) a systematic convergence study of the local exponent, with independent variation of ρmin, Δτ, Δρ, and of the fitting window in τ, reporting an extrapolated continuum exponent with an uncertainty estimate, or (ii) a commensurate softening of the claim in the abstract and in Sec. VI-C to 'compatible with τ^-4 within the studied rapidity window τ ≤ 40'. The skeptic's concern about this point lands on reading the paper; the paper's own numbers make the fragility explicit.","section":"Sec. V-D, Sec. III, Sec. V-A","authors_acknowledge":"For BK, theτ −4 behav- ior is a numerical observation, without a corresponding asymptotic derivation.","authors_acknowledge_p":0.67},{"comment":"The sentence 'Truncating the grid at ρmin = −40 changes ṁ by only 1-2% at τ = 7 but g⊥θθ by 30-40%' is ambiguous: the rapidity at which the g⊥θθ change is evaluated is not stated, and no truncation sensitivity at τ = 40 is given. Since the τ^-4 claim is made near τ = 40 and the leading edge dominates the transverse metric, this diagnostic should be reported at τ = 40, and its contribution to the quoted ~2% convergence (Sec. III) should be stated explicitly.","section":"Sec. V-A","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The first occurrence of the t^-4 claim ('Transverse deformations exhibit a universal t^-4 relaxation in the Fisher-KPP asymptotic analysis') appears before the conditional qualifier is stated; I recommend attaching the qualifier at first mention, because Appendix A's hypotheses (i)-(iii) are unproven and hypothesis (iii) concerns exactly the exponentially decaying initial data u0 = min(1, e^{-γ0 x}) used in all numerical tests, so the numerics remain the primary evidence in the tested regime.","section":"Abstract and Sec. I","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The statement that the rescaled shift control τ^4 g⊥shift 'converges towards a common value' for GBW and MV is central to the universality claim but is not quantified; please report the τ^4 g⊥shift values for GBW and MV at τ = 40 together with their difference, so the reader can judge the precision of this convergence against the ~2% metric error.","section":"Sec. V-D and Fig. 4","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The quoted ~2% convergence of the BK transverse metric at τ = 40 is attributed jointly to the time step and the upper limit of u; please give the individual contributions, especially since halving the time step alone changes the local exponent from 4.02 to 4.06.","section":"Sec. III and Sec. V-D","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"For the independence claim of the Table I test, state explicitly in the text that the relaxation rate S = ∂t p|ξ is evaluated from the evolution operator (the right-hand side of the PDE) of the unperturbed run and not from the tangent equation; the caption's phrase 'extracted only from the profiles of runs at θ ± 0.01' is accurate for a' and t'0 only if S's provenance is made unambiguous.","section":"Sec. IV-C and Table I","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The HERA extrapolation (nine units of rapidity at ᾱ ≈ 0.2 giving τ ≈ 1.8, hence 'τ ≲ 2') should be written out explicitly, since the phenomenological consistency argument in this section depends on that numerical mapping.","section":"Sec. VI-B","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest and well executed, and the convergence analysis requested in Major Comment 1 appears feasible within the existing framework (the code and stored run outputs are available from the author). If the authors either supply a systematic error budget for the BK exponent or soften the universality wording accordingly, I would support publication. I would not require an asymptotic derivation for BK, which may be very hard; an explicit numerical extrapolation with error bars is a reasonable standard for the claim level stated. One scope note: the 'universality' referred to in the abstract is universality across the studied families of initial conditions (GBW, MV, and the γ-family for FKPP), not a proof over all admissible initial data; the text documents this, but the abstract's phrasing invites over-reading."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Read Spezzano's Fisher-Rao geometry paper. The short version: the central mechanism claim is solid and honestly tested, the headline universality law is more fragile than the abstract suggests, and the paper knows it.\n\nWhat's actually new: this is the first Fisher-Rao treatment of nonlinear saturation fronts. Earlier information-geometry work, Peschanski's dynamical entropy and the DGLAP preprints, follows single trajectories. Spezzano factors out the longitudinal motion and studies the transverse metric g_theta_theta, which measures how fast distinct initial conditions become indistinguishable. The delay mechanism (varying the initial condition is, to leading order, a translation plus a time shift of the universal relaxation) is the load-bearing result, and it is genuinely validated: the translation a' and delay t'_0 are extracted from neighboring runs only, and the predicted ratio g_theta_theta/g_shift matches the tangent-equation measurement to about 1% for FKPP and BK at tau = 15 (Table I). That is a real out-of-sample check, not fitting to the claim. The FKPP t^-4 law with universal amplitude C* is a conditional derivation that agrees with numerics to sub-percent, and the conditions are stated plainly in Appendix A. The factor-3.5 discrepancy in translation information, attributed to the Levin-Tuchin interior, is a falsifiable observation backed by an explicit curvature check.\n\nThe soft spot, in proportion: the BK tau^-4 universality law rides on numerics to tau = 40 with about 2% convergence, 30-40% sensitivity to leading-edge grid truncation, and a local exponent that shifts from 4.02 to 4.06 when the time step is halved. The paper says all of this explicitly (b26, b108, b127) and flags that there is no asymptotic derivation for BK. This is not a hidden flaw, but it is the point on which the abstract's QCD-universality claim depends, and the worry that the continuum exponent could be 4.1, or that GBW and MV drift apart past tau = 40, is legitimate. The FKPP t^-4 law is conditional on three unproven hypotheses (b159): the formal t^-2 term of Graham's expansion, differentiability in theta and t, and extension to exponentially decaying data. If any fail, the analytic claim reduces to numerics. Data and code are not public, which makes the BK power law harder to trust.\n\nWho this is for: the small-x community, people working on geometric scaling and initial-condition sensitivity, and front-propagation theorists. It deserves a serious referee: the methodology is clean, the independent test is well designed, and the delay mechanism is a contribution regardless of how the power-law question settles. Send it to peer review, and push for released code and data and for BK runs past tau = 40 before the tau^-4 claim is quoted as established.","headline":"Fresh information-geometric angle on saturation fronts with a genuinely independent test of the central delay mechanism; the BK tau^-4 law is honestly labeled numerical and is the fragile part.","tokens_in":17151,"tokens_out":4472,"would_cite":true,"duration_ms":26166,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["12.38.-t"],"model":"deepseek/deepseek-v4-flash","headline":"Deforming a saturation front's initial shape is, to leading order, a spatial shift plus a time delay — a universal relaxation that drives the transverse shape-distinguishability metric down as $t^{-4}$ in Fisher-KPP and $\\tau^{-4}$ in BK.","keywords":["Fisher-Rao metric","information geometry","gluon saturation","Balitsky-Kovchegov equation","Fisher-KPP fronts","geometric scaling","transverse metric","small-x QCD"],"falsifier":"Compute the weighted Fisher integrals of the formal $t^{-2}$ term of the asymptotic expansion: if they fail to converge, or if that term does not describe the comoving relaxation at large $t$, the predicted limit $C_*/4 = 0.2780$ will not match the extrapolated value of $t^4 g^\\perp_{\\mathrm{shift}}$, and the $t^{-4}$ law reduces to an empirical fit. For BK, evolve beyond $\\tau = 40$ with deeper leading-edge grids and finer time steps: if the local decay exponent $-\\partial\\ln g^\\perp_{\\theta\\theta}/\\partial\\ln\\tau$ keeps approaching 4 and the rescaled shift control $\\tau^4 g^\\perp_{\\mathrm{shift}}$ converges to a common value for GBW and MV initial conditions, the claim stands; if the exponent drifts with grid size, the $\\tau^{-4}$ law is falsified.","tokens_in":17567,"feed_emoji":"📉","tokens_out":17927,"duration_ms":93874,"temperature":0.7,"pith_summary":"As a saturation front propagates, nonlinear gluon saturation erases the memory of its initial condition, and this paper asks exactly how fast. It answers with an information-geometric quantity, the transverse Fisher-Rao metric $g^\\perp_{\\theta\\theta}$, which measures how distinguishable two saturation fronts remain once their shared forward motion is factored out, and claims this distinguishability dies as a steep universal power law: $t^{-4}$ in the Fisher-KPP diffusive limit and $\\tau^{-4}$ in the full BK equation. The mechanism is a single relaxation mode — to leading order, deforming the initial profile merely translates the front in space and delays it in time — which yields the identity $g^\\perp_{\\theta\\theta} \\simeq a'^2 g^\\perp_{\\mathrm{shift}}$, verified to about 1% using only neighboring simulations. If correct, this gives a parameter-free measure of the approach to geometric scaling and rationalizes why pre-asymptotic fits to HERA data remain sensitive to initial shape parameters.","feed_headline":"Fronts forget: saturation erases shape memory at a universal rate","feed_subtitle":"A new information-geometric measure shows saturation fronts lose initial-shape memory at a steep universal rate.","key_machinery":"The carrying object is the normalized front density $p(\\rho|\\tau,\\theta) = \\partial N/\\partial \\ln r^2$, a probability density on the logarithmic dipole-size axis, together with the Fisher-Rao metric it induces, $g_{ij} = \\int p^{-1}(\\partial_i p)(\\partial_j p)\\,d\\rho$ — an intrinsic Riemannian measure of distinguishability between front profiles. The transverse component $g^\\perp_{\\theta\\theta}$ is the Schur complement $g_{\\theta\\theta} - g_{\\tau\\theta}^2/g_{\\tau\\tau}$: the Fisher norm of the tangent residual after the best time-shift projection is removed, so it isolates genuine shape distinguishability. The load-bearing identity is the delay form $\\partial_\\theta p \\simeq t'_0\\,\\partial_\\tau p - a'\\,\\partial_q p$, which converts a shape variation into a time delay plus a translation and yields $g^\\perp_{\\theta\\theta} \\simeq a'^2 g^\\perp_{\\mathrm{shift}}$; the asymptotic Fisher-KPP expansion $u = \\phi(\\xi) + \\psi(\\xi)/t + \\cdots$ supplies the coefficients and the universal constant $C_* = \\|(1-\\Pi_{f'})Q\\|^2/f$, the Fisher-projected norm of the relaxation profile $Q = -\\psi'$ that fixes the coefficient of the $t^{-4}$ law.","core_discovery":"In the comoving frame, a pulled front forgets its initial condition through a single universal relaxation mode. The paper's central claim is that, to leading order, varying the initial condition is equivalent to a spatial translation of the front combined with a temporal delay of the universal relaxation flow: $\\partial_\\theta p \\simeq t'_0\\,\\partial_\\tau p - a'\\,\\partial_q p$, with delay $t'_0 = -\\tfrac{2}{3}\\partial_\\theta\\alpha_1$ and displacement $a' = \\partial_\\theta\\sigma - \\dot{\\sigma}t'_0$ determined by the two non-universal constants $\\alpha_0$ and $\\alpha_1$ of the asymptotic front expansion. Because the translation part is the rigid-shift tangent, the transverse metric obeys $g^\\perp_{\\theta\\theta} \\simeq a'^2 g^\\perp_{\\mathrm{shift}}$, and the paper verifies this identity independently — from the profiles of neighboring runs alone, without tangent integration — to within 1% for Fisher-KPP and for BK at $\\tau = 15$. The transverse component then decays as $g^\\perp_{\\theta\\theta} \\propto t^{-4}$ in Fisher-KPP, with universal amplitude $C_* \\simeq 1.112$ reproduced to sub-percent accuracy by extrapolation, and numerically as $\\tau^{-4}$ in BK. The same framework shows nonlinear saturation freezes the longitudinal Fisher speed, and that the BK front interior is non-diffusive: its translation information exceeds the diffusive estimate by a factor of 3.5, with the mean-drift ratio $R_B \\simeq 1.175$ pointing to a universal non-Gaussian front.","pith_inferences":["Because the shift-plus-delay equivalence holds for two very different kernels and two initial-condition families, the natural test it suggests is whether it survives running coupling and modified leading-edge tails; if it does, the collapse to a two-dimensional parameter space would reshape global fits at the EIC.","The paper's distinction between the density metric and the experimental likelihood Fisher matrix $F_{ab}$ is worth pushing: observables sample $p$ through a kinematic window around $r \\sim 1/Q$, so the same memory-loss rate need not appear in cross-section fits, and computing $F_{ab}$ on HERA kinematics is the direct way to see whether shape sensitivity actually dies as fast as the metric says.","The universal constant $C_*$ is independent of the initial condition, so it could in principle be pinned down by any setup that resolves the front's shape-relaxation mode; conversely, once a full BK asymptotics exists to supply the analogous constant, comparing it with $C_*$ would quantify how much of the memory-loss rate is diffusive in origin."],"forward_implications":["The transverse metric $g^\\perp_{\\theta\\theta}$ provides a parameter-free quantitative measure of memory loss toward geometric scaling: it decays as $t^{-4}$/$\\tau^{-4}$, far steeper than the $t^{-1}$ Fisher-speed decay of a freely diffusing linear front, because saturation arrests broadening.","Pre-asymptotic data such as HERA's, which reach only $\\tau \\lesssim 2$, sit where $g^\\perp_{\\theta\\theta}$ is still orders of magnitude larger than at $\\tau \\gtrsim 10$, quantitatively rationalizing the persistent sensitivity of small-x fits to initial shape parameters like $\\gamma$.","At late times the delay mechanism makes different initial shapes nearly interchangeable up to a front position and a time offset: the Fisher cosine between the shape-parameter residual and the shift residual reaches 0.9999, and the ratio $g^\\perp_{\\theta\\theta}/g^\\perp_{\\mathrm{shift}}$ is fixed by $a'^2$ alone, so the effective parameter space of fronts collapses toward two directions.","For BK, the translation information $I_\\rho^{\\mathrm{BK}} \\simeq 0.120$ exceeds the diffusive Fisher-KPP estimate by a factor of 3.5, and the mean-drift ratio $R_B \\simeq 1.175$ sits above its FKPP counterpart 1.1502 — direct evidence that the saturation interior carries information a diffusive approximation misses."],"supporting_citations":[{"why":"Supplies the all-orders formal expansion of the Fisher-KPP front whose $t^{-2}$ term the $t^{-4}$ derivation invokes beyond any proven remainder bound.","marker":"[12]"},{"why":"Supplies the proven refined long-time asymptotics with remainder of order $t^{-3/2+\\gamma}$ that the derivation must extend.","marker":"[19]"},{"why":"Establishes convergence of Fisher-KPP solutions to traveling waves, the basis for the universal relaxation flow.","marker":"[9]"},{"why":"Derives the universal algebraic convergence of pulled-front velocities used to benchmark the BK front velocity.","marker":"[10]"},{"why":"Maps BK saturation onto pulled Fisher-KPP fronts and defines the geometric-scaling framework with the critical parameters $v_c$, $\\gamma_c$.","marker":"[2]"},{"why":"Supplies the mean-drift, Cramer-Rao-type bound on the Fisher speed that the paper's $R_B$ diagnostics test against.","marker":"[17]"},{"why":"Provides the Cramer-Rao-type inequality underlying the mean-drift bound.","marker":"[18]"},{"why":"Defines the Fisher-Rao metric and the spherical embedding used throughout the calculation.","marker":"[26]"},{"why":"Supplies the saturation-region law for the scattering matrix whose Gaussian interior explains why BK translation information exceeds its diffusive estimate by a factor of 3.5.","marker":"[28]"}],"fun_headline_variants":["Saturation fronts lose shape memory through one universal mode","Gluon saturation erases initial-shape memory at a universal rate","Fisher-Rao geometry reveals fronts forget at a t^-4 rate","Translating a front plus time delay explains memory loss","Saturation freezes Fisher speed; transverse shape fades as t^-4"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing premise is that the Fisher-KPP front expansion can be pushed to its formal $t^{-2}$ term, differentiated in the initial-condition parameter, and extended from step-like data to the exponentially decaying inputs used here — three hypotheses the paper states in Appendix A but does not prove — while the BK $\\tau^{-4}$ law rests on numerical evidence alone, with the transverse metric shifting by 30–40% when the grid's leading edge is truncated.","fun_headline_variants_meta":{"raw":{"variants":["Saturation fronts lose shape memory through one universal mode","Gluon saturation erases initial-shape memory at a universal rate","Fisher-Rao geometry reveals fronts forget at a t^-4 rate","Translating a front plus time delay explains memory loss","Saturation freezes Fisher speed; transverse shape fades as t^-4"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.00014,"raw_usage":{"cost":0.0001399398,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001399398,"upstream_inference_prompt_cost":3.63818e-05,"upstream_inference_completions_cost":0.000103558},"total_tokens":1884,"prompt_tokens":1315,"completion_tokens":569,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1144,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":494}},"tokens_in":1315,"tokens_out":569,"cost_source":"openrouter_usage","duration_ms":7540,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":494,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-03T21:54:56.214035+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Compute the weighted Fisher integrals of the formal $t^{-2}$ term of the asymptotic expansion: if they fail to converge, or if that term does not describe the comoving relaxation at large $t$, the predicted limit $C_*/4 = 0.2780$ will not match the extrapolated value of $t^4 g^\\perp_{\\mathrm{shift}}$, and the $t^{-4}$ law reduces to an empirical fit. For BK, evolve beyond $\\tau = 40$ with deeper leading-edge grids and finer time steps: if the local decay exponent $-\\partial\\ln g^\\perp_{\\theta\\theta}/\\partial\\ln\\tau$ keeps approaching 4 and the rescaled shift control $\\tau^4 g^\\perp_{\\mathrm{shift}}$ converges to a common value for GBW and MV initial conditions, the claim stands; if the exponent drifts with grid size, the $\\tau^{-4}$ law is falsified.","supporting_citations":[{"cited_title":"Precise asymptotics for Fisher-KPP fronts","cited_arxiv_id":"1712.02472","evidence_quote":"Supplies the all-orders formal expansion of the Fisher-KPP front whose $t^{-2}$ term the $t^{-4}$ derivation invokes beyond any proven remainder bound."},{"cited_title":"Refined long time asymptotics for Fisher-KPP fronts","cited_arxiv_id":"1607.08802","evidence_quote":"Supplies the proven refined long-time asymptotics with remainder of order $t^{-3/2+\\gamma}$ that the derivation must extend."},{"cited_title":"Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc.44, 1","cited_arxiv_id":null,"evidence_quote":"Establishes convergence of Fisher-KPP solutions to traveling waves, the basis for the universal relaxation flow."},{"cited_title":"Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Physica D146, 1","cited_arxiv_id":null,"evidence_quote":"Derives the universal algebraic convergence of pulled-front velocities used to benchmark the BK front velocity."},{"cited_title":"Traveling wave fronts and the transition to saturation","cited_arxiv_id":"hep-ph/0310357","evidence_quote":"Maps BK saturation onto pulled Fisher-KPP fronts and defines the geometric-scaling framework with the critical parameters $v_c$, $\\gamma_c$."},{"cited_title":"Information geometry of DGLAP evolution","cited_arxiv_id":"2610.28115","evidence_quote":"Supplies the mean-drift, Cramer-Rao-type bound on the Fisher speed that the paper's $R_B$ diagnostics test against."},{"cited_title":"Stochastic time-evolution, information geometry and the Cramer-Rao Bound","cited_arxiv_id":"1810.06832","evidence_quote":"Provides the Cramer-Rao-type inequality underlying the mean-drift bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Fisher-Rao metric and the spherical embedding used throughout the calculation."},{"cited_title":"Solution to the evolution equation for high parton density QCD","cited_arxiv_id":"hep-ph/9908317","evidence_quote":"Supplies the saturation-region law for the scattering matrix whose Gaussian interior explains why BK translation information exceeds its diffusive estimate by a factor of 3.5."}],"review_version":1}