{"id":"95e84897-866c-47c2-a364-cb447840102b","arxiv_id":"2506.05311","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Micro-heterogeneities in a viscoelastic material produce non-Gaussian probe displacement distributions and power-law tails in the time-dependent diffusion coefficient, confirmed by generalized Langevin simulations with a Prony-series memory kernel.","lead":"The paper uses computer simulations of probe particles in a model semisolid material with patchy stiffness and viscosity, described by a non-Markovian Langevin equation. It finds that the patchiness creates non-Gaussian motion and clear signatures in diffusion and rheology that could help characterize soft materials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Prony-series approximation, valid only for τ≥0.1 s, changes the late-time D(τ) scaling and low-frequency G″ behavior, so the simulations do not actually test the central NM-KVMH predictions they are claimed to validate.","rationale":"The reader's weakest_assumption identifies the inherited gamma-distribution ansatz (Eq. 8) as the main unvalidated input, which is an external-validity concern. My stress-test instead focuses on an internal consistency issue: the numerical simulations, which are the paper's new evidence, do not reproduce the predicted late-time D(τ) power law or the low-frequency G″ power law because the Prony-series approximation is explicitly non-equivalent to Eqs. 5–6 and is restricted to τ≥0.1 s. The paper acknowledges these limitations but still claims that the simulations validate non-exponential D(τ) behavior and the hallmark of heterogeneity; the data in Fig. 2(b) show a crossover between regimes, not the clean power-law tail of Eq. 6. This does not invalidate the model or the experimental fits in Fig. 1, but it weakens the paper's claim to provide numerical validation of the central mechanism. Since the reader already recommended CONDITIONAL acceptance pending clarification, my concern does not move the verdict; it adds a specific technical condition: the simulation must be repeated with a more accurate Prony representation (or another method) to confirm that the predicted D(τ) and G″ scalings emerge. I therefore keep the verdict unchanged. The agreement is partial because my primary concern is different from the reader's weakest_assumption, though the reader did mention the Prony approximation's limited validity in the rationale.","tokens_in":14441,"tokens_out":6199,"duration_ms":68398,"concrete_test":"Repeat the GLE simulations with a Prony representation covering a much wider time window (e.g., N=12–15 modes fitted to Eq. 3 over τ∈[10^{-6},10^3] s instead of N=7 fitted over τ≥0.1 s), using the same p=0.7 parameters. If the ensemble-averaged D(τ) then follows Eq. 6 (τ^{−(1+αn)}) for at least two decades after the MSD plateau, the discrepancy is due to the limited Prony approximation and the central claim survives. Additionally, run a case with smaller p (e.g., p=0.3, giving fitted α with αn<1) and check whether the low-frequency G″ approaches ω^{αn}; if it does not, the numerical method fails to reproduce the predicted G″ hallmark.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that micro-heterogeneities produce a smooth MSD crossover and a power-law tail in D(τ), and that fitting Eq. 5 yields an α that predicts low-frequency loss modulus behavior. The paper's simulation, however, does not reproduce the predicted D(τ) tail. The authors approximate the local NM-KV MSD by a finite Prony series (Eq. 9, N=7, Table I), explicitly valid only for τ≥0.1 s, and then derive the averaged forms Eqs. 11–12. They state these are ‘not strictly equivalent’ to Eqs. 5–6 and that lim_{τ→0} D(τ) is finite, not τ^{n−1}. Figure 2(b) shows the discrepancy concretely: for p=0.7, Eq. 6 predicts D(τ)∝τ^{−2.5}, while Eq. 12 gives D(τ)∝τ^{−0.7}, and the numerical results ‘seem to change from one regime to the other at later times’ rather than following Eq. 6. Consequently, the simulation validates only the Prony-averaged model (Eqs. 11–12), not the NM-KVMH hallmarks in Eqs. 5–6 that the paper uses to interpret experimental data. The paper's own acknowledgment that Eq. 5 fits the simulated MSD with α≈3 but that D(τ) follows a different tail than Eq. 6 means the fitted α does not predict the D(τ) behavior. Similarly, the low-frequency G″ hallmark G″∝ω^{αn} is only valid for αn<1, yet the simulated case has αn≈1.5, yielding linear G″; thus the regime in which the hallmark is claimed is not tested. This is not a mathematical error in the Prony derivation, but it is a load-bearing gap: the numerical evidence does not actually support the central claim as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents GLE-based Brownian simulations intended to validate the non-Markovian Kelvin-Voigt model with micro-heterogeneities (NM-KVMH). The authors approximate the local NM-KV mean squared displacement by a Prony series (Eq. 9, Table I), average over a generalized gamma distribution of local relaxation times (Eqs. 7-10), and derive approximate closed forms for the ensemble-averaged MSD and time-dependent diffusion coefficient (Eqs. 11-12). They then simulate the corresponding overdamped GLE, report agreement between simulation and Eqs. 11-12, and examine non-Gaussian displacement distributions and shear moduli. The paper claims that the hallmark of micro-heterogeneity is a smooth crossover in the MSD and a power-law tail in D(τ), and that fitting Eq. 5 yields a single heterogeneity parameter α that predicts low-frequency loss-modulus behavior.","tokens_in":14932,"tokens_out":2756,"duration_ms":36590,"significance":"If the central claim is correct, the paper offers a practical framework: a single parameter α extracted from a microrheology MSD would characterize micro-heterogeneity and predict both D(τ) tails and low-frequency G''(ω) behavior. The analytical Prony averaging is carried out explicitly, and the numerical implementation appears to reproduce Eqs. 11-12 faithfully, which is a useful technical validation of the simulation scheme. The non-Gaussian displacement distributions in Fig. 3 provide a concrete, potentially falsifiable prediction. However, the significance is reduced by the fact that the simulations do not directly test the original NM-KVMH equations, Eqs. 5-6, and the validation is partly by construction because the averaging is performed over the same assumed gamma distribution used to derive the model.","major_comments":[{"comment":"The simulations do not test the original NM-KVMH hallmarks because the Prony approximation is used and is valid only for τ ≥ 0.1 s, with a finite limit for D(τ→0) instead of the true diverging τ^{n-1} behavior. The paper itself states that Eqs. 11-12 are 'not strictly equivalent' to Eqs. 5-6, and Fig. 2(b) shows that Eq. 6 predicts D(τ) ∝ τ^{-2.5} while Eq. 12 gives D(τ) ∝ τ^{-0.7}, with the numerical data changing from one regime to the other at later times. Consequently, the agreement between simulation and Eqs. 11-12 validates only the Prony-averaged model, not the model used for experimental interpretation in Fig. 1 and for the central claim that D(τ) ∝ τ^{-(1+αn)} at later times. This is a load-bearing gap that must be addressed, either by rephrasing the simulation claims in terms of the approximate model or by providing numerical evidence for the original equations.","section":"Sec. IV, Fig. 2, Eqs. 11-12 vs Eqs. 5-6"},{"comment":"The fitted value α ≈ 3 from the MSD does not predict the simulated D(τ) tail: Eq. 6 with α ≈ 3 gives D(τ) ∝ τ^{-2.5}, while the simulation follows the Prony-averaged prediction τ^{-0.7}. Because the paper explicitly proposes that fitting Eq. 5 yields an α that characterizes heterogeneity and predicts the later-time D(τ) behavior and low-frequency G''(ω), this discrepancy undermines the predictive link. A direct test of Eq. 6 on simulated trajectories that actually follow the NM-KVMH local dynamics, or a clear demonstration of how the Prony approximation preserves the α-dependence at relevant timescales, is needed.","section":"Sec. IV, Fig. 2(b), Eq. 6"},{"comment":"The claimed low-frequency loss-modulus hallmark G''(ω) ∝ ω^{αn} is not tested by the presented simulations. The paper notes that for the simulated case α ≈ 3 and n = 0.5, so αn ≈ 1.5, which yields a linear G''(ω) rather than the power-law regime that is the stated hallmark for αn < 1. Thus Fig. 4 cannot support the conclusion that micro-heterogeneities produce the power-law loss modulus observed in Fig. 1. The authors should either simulate a parameter set with αn < 1 or obtain G''(ω) directly from Eq. 5 to demonstrate the predicted scaling.","section":"Sec. IV, Fig. 4"},{"comment":"The generalized gamma distribution and the scaling relation γ_{j,ξ} = (p/ξ)γ*_j are assumed rather than independently validated. Since the simulation averages over this assumed distribution, the agreement with Eqs. 11-12 is expected by construction and cannot serve as independent evidence for the gamma-distribution ansatz. The paper should discuss this limitation explicitly and, ideally, test sensitivity to alternative heterogeneity distributions or compare against independent measurements of local viscoelastic properties or van Hove distributions beyond the fits to the model's own expressions.","section":"Sec. II, Eqs. 8 and 10"}],"minor_comments":[{"comment":"The text contains typographical issues such as 'V oigt' instead of 'Voigt' in the introduction; these should be corrected in a final revision.","section":"Throughout"},{"comment":"The notation for the stochastic terms is slightly confusing: ε_j(t) is defined as N_j(0,1)√Δt, but later in the same equation the noise is written with ε_0(t) and ε_j(t); clarifying that these are independent increments would improve readability.","section":"Sec. III, Eq. 17"},{"comment":"The kurtosis figure labels (a)-(h) and the corresponding text are not fully aligned with the panel descriptions; adding explicit time values to the panel labels in the text would help the reader connect the figure to the narrative.","section":"Sec. IV, Fig. 3(i)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is largely built on the corresponding author's own prior models (Refs. 4 and 6), which is not itself a flaw, but it raises the bar for independent validation. The current simulations validate the Prony-averaged variant rather than the original NM-KVMH expressions, and the low-frequency G'' hallmark is not actually exercised. The paper would be strengthened by reframing the simulation results as a validation of the numerical scheme and the Prony approximation, and by adding a direct test of Eqs. 5-6 in the αn < 1 regime, possibly with a broader set of Prony parameters that extends to shorter times."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Pedro — quick take on Azevedo & Rizzi 2506.05311. The paper is the numerical follow-up to Rizzi's NM-KVMH model (Ref. 6). What's genuinely new: a Prony-series approximation of the local NM-KV MSD, a GLE simulation scheme that propagates heterogeneous trajectories, and a clean derivation of the averaged MSD and D(τ) (Eqs. 11–12). The simulations reproduce those expressions well, and the van Hove distributions show the expected non-Gaussian tails. Credit where due: the Prony parameter extraction is careful, the limitations of the approximation are stated in the text, and the numerics appear reproducible.\n\nThe soft spot is where the paper overreaches. The abstract says the study 'provides an analytical way to characterize' heterogeneity — but that analytical way is Eqs. 5–6 from Ref. 6. The simulation, by the authors' own admission, does not test those equations directly. For p=0.7, Eq. 6 predicts D(τ)∝τ^{-2.5}, while the Prony-averaged Eq. 12 gives τ^{-0.7}; the simulated data follow the latter. The fit of Eq. 5 to the simulated MSD yields α≈3, but that α does not reproduce the D(τ) tail. The loss-modulus hallmark G″∝ω^{αn} also applies only for αn<1; the simulated case has αn≈1.5 and gives a linear G″. So the central quantitative predictions of the NM-KVMH model are not actually validated by these simulations. The qualitative features — non-exponential D(τ), non-Gaussian displacement distributions — do emerge, but those were already known. The gamma-distribution ansatz is inherited from Ref. 6 and not checked against independent local measurements.\n\nThat's the honest picture. Not a fatal flaw: the paper is transparent about the Prony approximation and its consequences. But a serious referee should ask for error bars, for a demonstration that the approximation can be pushed to shorter times, and for a quantitative comparison of predicted vs. experimental van Hove distributions. If those are addressable, the paper is a useful technical contribution for the microrheology community. As is, I'd send it to peer review, but I would not rely on its conclusions as independent confirmation of the NM-KVMH hallmarks. Not something I'd cite in my own work this year.","headline":"Transparent numerical follow-up to the NM-KVMH model, but the Prony approximation means the simulations validate the approximate equations, not the original hallmarks.","tokens_in":15409,"tokens_out":3382,"would_cite":false,"duration_ms":36714,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that a single fitted exponent α, extracted from particle-tracking data on semisolid materials, quantifies micro-heterogeneity and predicts the low-frequency loss modulus.","keywords":["microrheology","semisolid viscoelastic materials","non-Markovian dynamics","generalized Langevin equation","micro-heterogeneities","van Hove displacement distribution","time-dependent diffusion coefficient","Prony series"],"falsifier":"Measure the mean squared displacement and the linear shear moduli on the same gel, fit the MSD to Eq. (5) to extract $\\alpha$, and check whether the independently measured low-frequency loss modulus follows the predicted power law $G''(\\omega)\\propto\\omega^{\\alpha n}$ whenever $\\alpha n<1$; a gel whose MSD fit gives $\\alpha n<1$ but whose loss modulus stays linear in $\\omega$ would falsify the central claim. A more direct check is to map the local mechanical properties with a spatially resolved probe, such as AFM indentation or optical-trap compliance scans, and compare the measured distribution of local relaxation times with the generalized gamma distribution implied by the fitted $\\alpha$; a substantial mismatch would show that the single-parameter gamma ansatz is not an accurate description of the material's actual heterogeneity.","tokens_in":14194,"feed_emoji":"🔬","tokens_out":25006,"duration_ms":251101,"temperature":0.7,"pith_summary":"This paper argues that two physical ingredients in semisolid viscoelastic materials — non-Markovian memory in the response of the material's internal structures and micro-heterogeneities between different mesoscopic regions — each leave a distinct, identifiable mark in passive microrheology. The memory shows up as a power-law mean squared displacement at short times, $\\langle\\Delta r^2(\\tau)\\rangle\\propto\\tau^n$ with $n<1$, and as shear moduli growing like $\\omega^n$ at high frequencies; the micro-heterogeneities show up as a smooth crossover between the power-law and plateau regimes of the MSD, which makes the time-dependent diffusion coefficient decay as a power law, $D(\\tau)\\propto\\tau^{-(1+\\alpha n)}$, at later times instead of exponentially. Using generalized Langevin simulations in which each probe trajectory is trapped in a region with its own local viscoelastic properties, the paper validates this picture and shows that the displacement (van Hove) distributions become measurably non-Gaussian at short times. If the picture is right, fitting an experimental MSD to a single closed-form expression yields one heterogeneity parameter $\\alpha$ that also predicts the low-frequency loss modulus, giving a practical way to measure micro-heterogeneity from particle-tracking data alone.","feed_headline":"One fitted exponent quantifies gel micro-heterogeneity","feed_subtitle":"Fitting particle-tracking data yields one number α that predicts the low-frequency loss modulus.","key_machinery":"The argument is carried by a regional-average construction: the measured mean squared displacement is the integral over mesoscopic regions, $\\langle\\Delta x^2(\\tau)\\rangle=\\int_0^\\infty \\langle\\Delta x^2(\\tau)\\rangle_\\xi\\,\\rho(\\xi)\\,d\\xi$, in which each region is labeled by a single random variable $\\xi$ drawn from the generalized gamma distribution $\\rho(\\xi)=\\xi^{-(1-p)}e^{-\\xi}/\\Gamma(p)$. Heterogeneity enters through the rule that every local Prony relaxation time rescales as $\\gamma_{j,\\xi}=(p/\\xi)\\gamma^*_j$, so one parameter $p$ — whose counterpart in the closed-form model is $\\alpha$ — carries all the disorder. The average yields two closed-form results: the MSD $\\langle\\Delta x^2(\\tau)\\rangle=2k_BT/\\kappa-\\sum_{j=1}^{N+1}q^*_j\\,[1+\\tau/(p\\gamma^*_j)]^{-p}$ and its derivative $D(\\tau)$, the time-dependent diffusion coefficient. Numerically, the machinery is a Prony-series simulation scheme: the stretched-exponential local MSD with exponent $n=0.5$ is approximated by an eight-mode sum of exponentials, converted through the generalized Stokes–Einstein relation into a Prony memory kernel $\\mu_\\xi(\\tau)=\\mu_{0,\\xi}\\,\\delta(\\tau)-\\sum_j c_{j,\\xi}e^{-\\tau/\\Lambda_{j,\\xi}}$, and integrated through $N+1$ coupled stochastic differential equations, one trajectory per heterogeneity realization, with parameters recomputed per trajectory as in Appendix A. This last step is what makes it possible to interpret micro-heterogeneities as a distribution of local mobilities, i.e., local viscosities. Two acknowledged limits of this machinery are that the Prony approximation works only for $\\tau\\ge 0.1$ s and that Eqs. (11)–(12) are not strictly equivalent to Eqs. (5)–(6) of the closed-form model.","core_discovery":"The central claim is that the hallmark of micro-heterogeneities on probe particles in semisolids is a smooth crossover between the power-law and plateau regimes in the mean squared displacement, which leads to the more identifiable power-law behavior of the time-dependent diffusion coefficient at later times, $D(\\tau)\\propto\\tau^{-(1+\\alpha n)}$, where $\\alpha$ is a single parameter that characterizes the distribution of local viscoelastic properties. The paper further claims that this same parameter, extracted by fitting experimental MSD data to the non-Markovian Kelvin–Voigt model with micro-heterogeneities (NM-KVMH), predicts the low-frequency loss modulus, $G''(\\omega)\\propto\\omega^{\\alpha n}$ when $\\alpha n<1$, in contrast to the linear $G''\\propto\\omega$ of homogeneous viscoelastic models, and that heterogeneous response makes displacement distributions non-Gaussian at short times. To support these claims the authors build a simulation scheme that emulates a real microrheology experiment: the stretched-exponential local MSD of the NM-KV model (the same model without heterogeneities) is approximated by an eight-mode Prony series, a finite sum of exponentials, each simulated trajectory receives its own heterogeneity variable drawn from the generalized gamma distribution, and the overdamped generalized Langevin equation, a stochastic equation of motion with a memory kernel, is integrated through coupled stochastic differential equations. The simulated MSD, diffusion coefficient, van Hove distributions, and shear moduli agree with the analytical model and reproduce qualitatively the experimental behavior of polyacrylamide, $\\beta$-lactoglobulin, and colloidal gels. The paper itself notes, in Secs. II and IV, that the Prony approximation is accurate only for $\\tau\\ge 0.1$ s and that the resulting model is not strictly equivalent to the closed-form NM-KVMH at early and late times.","pith_inferences":["A decisive test of the model would map local mechanical properties directly, for instance by AFM indentation or optical-trap compliance scans across a gel, and compare the measured distribution of relaxation times with the gamma form implied by the fitted $\\alpha$; the previous validation of the gamma ansatz relied on displacement statistics, not on direct local mechanical measurements.","Materials with two coexisting sources of heterogeneity, such as a bimodal distribution of pore or cross-link sizes, would break the single-parameter assumption; the same averaging integral that produces Eq. (11) could host a mixture distribution and still give closed-form MSDs, offering a direct way to test whether one parameter ever suffices.","Because the Prony approximation and the closed-form NM-KVMH disagree at early and late times, the practical rule that follows from the paper's own comparisons is to fit experimental data with Eqs. (5)–(6) and then verify that the extracted $\\alpha$ also reproduces the measured $D(\\tau)$ tail and the low-frequency loss modulus.","The predicted short-time excess kurtosis is cheap to check: existing single-particle-tracking datasets on gels already contain full displacement histograms, so the predicted non-Gaussian signature and its decay can be tested on published data before any new experiment is run."],"forward_implications":["One fit of an experimental MSD to Eq. (5) delivers $\\alpha$, and the same $\\alpha$ fixes the late-time tail of the time-dependent diffusion coefficient, $D(\\tau)\\propto\\tau^{-(1+\\alpha n)}$, with no second measurement required.","For heterogeneous samples with $\\alpha n<1$, the low-frequency loss modulus should rise as $G''(\\omega)\\propto\\omega^{\\alpha n}$ rather than linearly, so conventional bulk rheology can confirm or challenge the heterogeneity parameter extracted from particle tracking.","Displacement (van Hove) distributions from the simulations show clearly non-Gaussian excess kurtosis at short times that decays at later times, meaning single-particle tracking histograms reveal heterogeneity directly, not only through averaged quantities.","The GLE scheme generates statistically faithful single-particle trajectories for a heterogeneous semisolid, so the same code can be applied to other observables of the model, such as two-time correlations or first-passage statistics, without new analytical work.","Because the heterogeneous response is equivalent to a distribution of local viscosities, the effective zero-shear viscosity of the heterogeneous model exceeds that of the homogeneous model at the same mean parameters, as the simulated moduli in Fig. 4 show."],"supporting_citations":[{"why":"Supplies the closed-form NM-KVMH model, Eqs. (5) and (6), whose hallmarks — the power-law tail in the time-dependent diffusion coefficient and the low-frequency power-law loss modulus — this paper sets out to validate.","marker":"Ref. 6"},{"why":"Introduces the KVMH averaging over the generalized gamma distribution of regional properties and the rescaled local relaxation times that the simulations inherit.","marker":"Ref. 4"},{"why":"Provides the discretization of the overdamped generalized Langevin equation into coupled stochastic differential equations, Eqs. (17) and (18), used for all simulations.","marker":"Ref. 32"},{"why":"Supplies the numerical routine that turns the simulated MSD-based compliance into the storage and loss moduli shown in Fig. 4.","marker":"Ref. 23"},{"why":"States the generalized Stokes–Einstein relation, Eq. (1), that links the mean squared displacement to the material compliance, the backbone of every quantity in the paper.","marker":"3"},{"why":"Supports the approximation of the stretched-exponential local MSD by a Prony series, the step that makes the non-Markovian simulations feasible.","marker":"Ref. 30"},{"why":"Provides the polyacrylamide gel master-curve data used in Fig. 1 to illustrate the hallmarks and motivate Eqs. (5) and (6) over Eqs. (3) and (4).","marker":"Ref. 12"},{"why":"Provides the beta-lactoglobulin gel data in Fig. 1, including the non-Gaussian displacement distributions and the non-exponential late-time diffusion coefficient.","marker":"Ref. 15"},{"why":"Provides the colloidal gel data in Fig. 1, connecting the model to dense particulate soft solids.","marker":"Ref. 20"}],"fun_headline_variants":["One fitted α from microrheology predicts gel loss modulus","Single α quantifies micro-heterogeneity in viscoelastic gels","Particle-tracking α reveals gel heterogeneity and loss modulus","Alpha from MSD crossover predicts semisolid loss modulus","Non-Markovian fits give α for gel micro-heterogeneity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire framework depends on the assumption that all micro-heterogeneity of a real semisolid is captured by a single random number per region, with every local relaxation time rescaled by the same fixed factor and those numbers drawn from one particular distribution the model chooses; if real materials distribute their local stiffnesses and viscosities in any other way, the fitted heterogeneity parameter $\\alpha$ and the predicted low-frequency loss modulus would be systematically wrong.","fun_headline_variants_meta":{"raw":{"variants":["One fitted α from microrheology predicts gel loss modulus","Single α quantifies micro-heterogeneity in viscoelastic gels","Particle-tracking α reveals gel heterogeneity and loss modulus","Alpha from MSD crossover predicts semisolid loss modulus","Non-Markovian fits give α for gel micro-heterogeneity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000684,"raw_usage":{"total_tokens":3162,"prompt_tokens":1065,"completion_tokens":2097,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":681,"completion_tokens_details":{"reasoning_tokens":2010}},"tokens_in":681,"tokens_out":2097,"duration_ms":16735,"temperature":1.0,"reasoning_tokens":2010,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:21:59.168030+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the mean squared displacement and the linear shear moduli on the same gel, fit the MSD to Eq. (5) to extract $\\alpha$, and check whether the independently measured low-frequency loss modulus follows the predicted power law $G''(\\omega)\\propto\\omega^{\\alpha n}$ whenever $\\alpha n<1$; a gel whose MSD fit gives $\\alpha n<1$ but whose loss modulus stays linear in $\\omega$ would falsify the central claim. A more direct check is to map the local mechanical properties with a spatially resolved probe, such as AFM indentation or optical-trap compliance scans, and compare the measured distribution of local relaxation times with the generalized gamma distribution implied by the fitted $\\alpha$; a substantial mismatch would show that the single-parameter gamma ansatz is not an accurate description of the material's actual heterogeneity.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the generalized Stokes–Einstein relation, Eq. (1), that links the mean squared displacement to the material compliance, the backbone of every quantity in the paper."}],"review_version":1}