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REVIEW 4 major objections 5 minor 69 references

Analysing gelation transition through fractional viscoelasticity and Mittag-Leffler-Prabhakar function

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper argues that continuity of the dynamic moduli and their derivative at the gel point forces one shared relaxation exponent and makes hyperscaling necessary.

desk verdict Useful new Prabhakar-function machinery, but the universal symmetry/hyperscaling claim rests on an unproven continuity postulate and partly circular scaling-input constraints. read the letter →

arxiv 2606.26656 v2 pith:GWNTZDD2 submitted 2026-06-25 cond-mat.soft

Yogesh M Joshi This is my paper · ORCID
classification cond-mat.soft MSC 76A1026A3382B27
keywords sol-geltransitioncriticalgelfractionalviscoelasticityMittag-Leffler-Prabhakarfunctionhyperscalingrelationpercolationdynamicmodulirheologicalfingerprint
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to show that the gelation transition is governed by a single symmetry: as a material crosses from liquid to solid, its viscoelastic response and the rate at which that response changes with crosslinking must both pass continuously through the critical gel point. If that continuity holds, the relaxation dynamics on the two sides cannot be independent; they share one exponent, and the classic hyperscaling relation becomes a logical requirement rather than an empirical pattern. The paper builds fractional viscoelastic models—one a chain of springpots and a dashpot, the other based on the three-parameter Mittag-Leffler (Prabhakar) function—that satisfy this constraint, and fits them to PDMS and PVA data. It also identifies the ratio of the changes in loss to storage modulus at the gel point as a frequency-independent fingerprint equal to tan((n-κ)π/2). A sympathetic reader would care because this gives one testable condition from which the major scaling laws of gelation follow.

What carries the argument

The load-bearing object is the continuity condition on the first derivative of the complex modulus with respect to the degree of crosslinking at the gel point: the left and right limits must match at every frequency. Combined with the power-law forms of the relaxation modulus on both sides, this identity forces the frequency exponents on either side to match, giving κ_S = κ_G. The Mittag-Leffler function and its three-parameter Prabhakar generalization carry the Laplace inversion that turns the asymptotic scaling constraints into explicit expressions for the relaxation modulus and dynamic moduli; the Prabhakar parameter is then pinned to the viscosity exponent s. The invariant (∂G''/∂G') at

What would settle it

Measure G'(ω,p) and G''(ω,p) on a system whose degree of crosslinking p can be tuned continuously and slowly through the gel point. At a fixed frequency, plot G'' against p; if the curve has a corner at p_c, with different slopes on the two sides, the continuity premise fails and with it the symmetry, hyperscaling, and fingerprint. Equivalently, if (∂G''/∂G') measured at the gel point is found to depend on frequency, the predicted fingerprint is wrong.

Watch

Extended reading notes

Core claim

The central claim is that continuity of the dynamic moduli and of their first derivatives with respect to the degree of crosslinking at the critical gel point is not a convenience but a physical law. From that law, model-agnostic consequences follow: the relaxation scaling exponents on the sol and gel sides are equal, the leading departure coefficients on both sides are equal, the hyperscaling relation n = z/(z+s) is a necessity, and (∂G''/∂G') at the gel point equals tan((n-κ)π/2). The paper constructs explicit fractional models realizing these relations. In the pre-gel Prabhakar model, the third parameter is forced to equal the viscosity divergence exponent s, making it experimentally meas

Load-bearing premise

The entire chain rests on the assertion that the first derivative of the complex modulus with respect to the degree of crosslinking is continuous at the gel point; the paper treats this as a physical necessity but does not derive it from percolation or any microscopic model.

Editorial extensions

If this is right

  • All gel-forming systems that satisfy the continuity premise must show symmetric divergence of the longest relaxation time on both sides, τ_max ~ ε^(−1/κ) with one shared κ.
  • The hyperscaling relation n = z/(z+s) becomes a necessary test; experimental deviations point either to off-critical measurements or to failure of the derivative-continuity premise.
  • In the Prabhakar-based pre-gel model, the shape parameter equals the viscosity divergence exponent s, turning a mathematical fitting parameter into an experimentally measurable physical quantity.
  • The frequency-independent ratio (∂G''/∂G') at the gel point provides a fingerprint of the critical state depending only on n and κ, useful for classifying the universality class of a gel.
  • The post-gel Prabhakar description is not more general than the parallel springpot-and-spring network; it is a constrained subset whose amplitude coefficients are tied together by the Prabhakar exponent.

Reading between the lines

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

  • If the derivative-continuity premise is accepted as universal, it imposes a selection rule on percolation classes: any measured triple (n, s, z) must satisfy n = z/(z+s), which could be tested systematically against published percolation exponents.
  • One could probe the continuity premise directly by measuring G' and G'' while sweeping the degree of crosslinking quasistatically through the gel point; a visible kink in G'' at fixed frequency would falsify the derivative-continuity condition.
  • The same analytical structure likely extends beyond rheology to dielectric or impedance spectroscopy, where a critical relaxation spectrum has an analogous mathematical form; continuity would then predict similar exponent relations in those spectroscopies.
  • The identification of the Prabhakar parameter with the viscosity exponent suggests that the shape of the relaxation modulus itself encodes the divergence of viscosity, so approximate models that omit a mechanical dashpot can still be thermodynamically complete if the Prabhakar shape is fixed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes fractional viscoelastic and Mittag-Leffler-Prabhakar models for the pre-gel and post-gel sides of the gelation transition. Pre-gel Model 1 (two springpots in series with a dashpot) gives a two-parameter Mittag-Leffler relaxation modulus but is restricted to n+κ<1; Pre-Gel Model 2 uses the three-parameter Prabhakar function, removes that restriction, and fixes the Prabhakar exponent γ_S to the viscosity exponent s. Post-gel Model 1 is a parallel spring/springpot network; Post-Gel Model 2 is its truncated Prabhakar version and is shown to be algebraically equivalent. The paper enforces continuity of G* and ∂G*/∂p at the critical gel point, claims this imposes κ_S=κ_G and hyperscaling n=z/(z+s), and proposes (∂G''/∂G')_{ε→0}=tan((n−κ)π/2) as a universal fingerprint. It fits time-domain PDMS and frequency-domain PVA data with the constrained models.

Significance. If the central claims are correct, the paper would establish a strong universality result: a symmetry between pre- and post-gel relaxation exponents and the hyperscaling relation as consequences of first-derivative continuity, together with an experimentally accessible frequency-independent fingerprint. The use of the Prabhakar function is mathematically motivated, the Laplace-transform manipulations are mostly checkable, and the identification γ_S=s is conceptually appealing. Credit is due for the explicit model construction, the clear treatment of complete monotonicity restrictions, and the simultaneous fits to two very different gel systems. However, because the decisive continuity condition is asserted rather than derived, and because the 'validation' of scaling relations largely restates model inputs, the universal claims are not established at the level asserted. The paper is best read as a consistency theorem for a class of fractional models conditional on Eq. (21).

major comments (4)
  1. [Table 1, Eq. (21); Continuity section] The universal symmetry κ_S=κ_G and the claimed 'necessity' of hyperscaling rest on Eq. (21), continuity of ∂G*/∂p at p_c. This is asserted as a physical necessity and attributed to the in-press [39], but no derivation from percolation theory, cluster statistics, or a general scaling ansatz is supplied. Continuity of G* at p_c does not imply continuity of its first derivative; in critical phenomena derivatives of response functions with respect to the control parameter can jump or diverge. If Eq. (21) fails, the exponent matching between Eqs. (58)-(59) and Eqs. (94)-(95) is not forced, so independent κ_S and κ_G are admissible. The supporting molecular argument using τ~ξ^v and ξ~ε^{-(z-1)} also assumes a common dynamic exponent v on both sides, which is the conclusion being derived. Please derive Eq. (21) or state the main theorem as conditional on it.
  2. [Eqs. (10)-(11), (38), (71)-(73)] The claimed validation of s=(1-n)/κ_S is circular. Table 1 lists s=(1-n)/κ_S as a target scaling law; Eq. (38) sets η(ε)=η* ε^{-s} and immediately invokes Eq. (10). The later result η0∼ε^{-(1-n)/κ_S} and identification γ_S=s (Eqs. (72)-(73)) restate this input. Similarly, G_e=G0 ε^z (Eq. (7)) is imposed in Eq. (83) before being recovered. Consequently, hyperscaling n=z/(z+s) is an algebraic consequence of κ_S=κ_G plus the assumed relations κ_S=(1-n)/s and κ_G=n/z. The paper should clearly separate assumed scaling laws from derived results and avoid the word 'validate' for these steps.
  3. [Figs. 3-4] The experimental fits are presented as validation of the symmetry/hyperscaling constraints, but the pre- and post-gel models are solved simultaneously with Eqs. (102)-(103) imposed, and ε is a fitted parameter for each dataset (Fig. 3 legend; Fig. 4 caption). Agreement with data therefore demonstrates internal consistency of a flexible model family, not an independent test of the constraints. A meaningful test would compare unconstrained fits (κ_S≠κ_G allowed) against constrained fits, report goodness-of-fit/residuals, or make a parameter-free prediction from independently measured exponents.
  4. [Eq. (24) and Fig. 5] The claimed model-agnostic fingerprint (∂G''/∂G')_{ε→0}=tan((n-κ)π/2) is derived from the specific fractional/Prabhakar ansätze (and from [39]), under the same continuity and scaling inputs. Fig. 5 plots the formula but contains no experimental data, so the fingerprint is not tested here. Please either provide an independent experimental test of Eq. (24) or restrict the claim to the model class considered.
minor comments (5)
  1. [Abstract and text] Typos and grammar: 'a present in variety' (Abstract); 'This assumption will be tasted below' should be 'tested'; 'The For springpots connected in series' is garbled.
  2. [Post-Gel Model 2, first paragraph] 'which we represented as Post-gel Model 2' should presumably be 'Pre-gel Model 2'.
  3. [Fig. 4 caption] Panel labels are wrong: the last pre-gel state is labelled '(f)' again; it should be '(i)'.
  4. [References] Ref. [16] has a corrupted DOI/URL ('2010.48550/arXiv.42603.11366', 'accessed 42026-48504-48515'); Ref. [39] is in press and cannot be checked. Please make [39] available or reproduce its decisive derivation.
  5. [Table 3] The row 'Post Gel, B_{G,2}' is unclear: only one post-gel amplitude is listed while the model in Eqs. (90)-(91) contains several B_{G,k}; clarify which coefficients are fixed and which are fit.

Circularity Check

3 steps flagged · score 6.0 of 10

Central 'universal necessity' claim rests on Eq. (21), asserted via self-citation [39], and the hyperscaling 'validation' is an algebraic restatement of imposed scaling relations (Eqs. (10)-(11)).

  1. self citation load bearing [Introduction; Table 1, Eq. (21); Section 'Continuity at the Critical Gel Point']
    "This condition, according to Joshi [39] is no longer an assumption but a strict requirement to validate continuity of dynamic moduli and its derivative as the material undergoes gelation transition."

    The paper's central universal claim—that continuity of the dynamic moduli and their first derivative across p_c forces κ_S=κ_G and thereby makes hyperscaling a 'theoretical necessity'—is derived by imposing Eq. (21), continuity of ∂G*/∂p at p_c. Eq. (21) is not derived from percolation theory, cluster statistics, or a general scaling ansatz; the only justification given is the author's own in-press reference [39], which asserts the same conclusion. Thus the load-bearing premise of the universality chain reduces to a self-citation rather than an independent physical derivation.

  2. self definitional [Table 1, Eqs. (10)-(11), (22)-(23); Section 'Continuity at the Critical Gel Point']
    "From the pre-gel analysis we have κ_S = (1−n)/s while from the post-gel analysis we have κ_G = n/z, consequently the equality κ_S=κ_G yields: n=z/(z+s), which is the hyperscaling relation tabulated in by Eq. (23) Table 1."

    The relations κ_S=(1−n)/s and κ_G=n/z are not derived in this paper; they are introduced as 'Scaling relation' constraints in Table 1 and used to construct the models. Given these definitions, n=z/(z+s) is an algebraic rearrangement of κ_S=κ_G. Therefore the abstract's claim that enforcing continuity 'further validates the hyper-scaling relation ... making it a theoretical necessity rather than an empirical coincidence' repackages the assumed scaling relations as a newly derived result.

1 more flagged steps
  1. fitted input called prediction [Eqs. (38) and (72)-(73), Pre-Gel Model 1 and Model 2]
    "We, therefore, set η(ε)=η∗ ε^{−s} ... that leads to pre-gel scaling η_0 ∼ ε^{−s} given by Eq. (6). Through, Eq. (10), the power law scaling exponent is given by: s=(1−n)/κ_S. ... Expression of viscosity given by Eq. (72) ... leads to the established expression for the power-law divergence of viscosity ... which on one hand validates the scaling relation s=(1−n)/κ_S"

    The viscosity divergence exponent s and the relation s=(1−n)/κ_S were imposed as inputs when constructing the model (Eq. (38); Table 1). Later, computing η_0∼ε^{−(1−n)/κ_S} and identifying the Prabhakar exponent γ_S=s is presented as 'validating' the same input relation. The predicted divergence is forced by the construction, so the validation is not independent.

full rationale

The paper's model development is internally consistent, and the fits to PDMS and PVA data are external empirical checks, so the work is not wholly circular. However, the headline claims of universality ('universally imposes', 'theoretical necessity rather than empirical coincidence') are conditional on Eq. (21), which is asserted as a physical necessity via an in-press self-citation [39] rather than derived from percolation or a first-principles scaling argument. Moreover, the 'validation' of the hyperscaling relation reduces to algebra once the scaling relations κ_S=(1−n)/s and κ_G=n/z are assumed as constraints, and the 'validation' of s=(1−n)/κ_S through the Prabhakar exponent γ_S restates an input of the model construction. The frequency-independent fingerprint C, and the symmetry condition κ_S=κ_G, are legitimately derived consequences of the assumed scaling and continuity structure, but the paper presents a conditional consistency theorem as a model-independent physical necessity. Overall, partial circularity: the central necessity claim depends on a self-cited premise and on scaling laws that were put in at the start.

Assumptions & free parameters 9 free parameters · 8 assumptions · 0 invented entities

The model family rests on three kinds of inputs: standard analytic facts about Mittag-Leffler/Prabhakar functions (2 items); empirical scaling laws of critical gelation taken from the literature or from the author's own prior work and imposed as constraints during model construction (items: scaling laws, Winter-Chambon forms, finiteness of eta_0, n>kappa); and the continuity-of-derivative postulate (Eq. 21) which is the genuine new premise carrying the symmetry and hyperscaling conclusions. The counts show that the paper's advertised results are consequences of these inputs plus algebra; the genuinely new movable part is the functional ansatz (1 item) and the fitted parameters (9 items). No new physical entities are postulated.

free parameters (9)
  • Critical relaxation exponent n = 0.45 (PDMS); 0.77 (PVA)
    Power-law exponent at the critical gel; fitted to the critical-gel data in Figs. 3-4 and Tables 2-3.
  • Relaxation scaling exponent kappa = 0.29 (PDMS); 0.266 (PVA)
    Exponent controlling tau_max ~ eps^{-1/kappa} and the frequency dependence of dlnG'/deps; fitted, then enforced equal on both sides by Eq. (102).
  • Critical gel stiffness S = 350.166 Pa s^n (PDMS); 1.184 Pa s^n (PVA)
    Amplitude of the Winter-Chambon power law; fitted.
  • Equilibrium modulus prefactor G_0 = 3677.22 Pa (PDMS); 287.9 Pa (PVA)
    Prefactor in G_e = G_0 eps^z; fitted. The exponent z is not free: z = n/kappa (1.55 and 2.89) via Eq. (11).
  • Leading departure amplitude B (=B_S=B_{G,1}) = 76.0 s^{-kappa} (PDMS); 10.555 s^{-kappa} (PVA)
    Coefficient in Eqs. (14)/(15); fitted under the equality constraint B_S = B_{G,1} (Eq. 103).
  • Pre-gel mode weights w_j and timescale factors vartheta_{S,j} = PDMS: w=(0.767,0.233), vartheta=(10.455,85.064) s^{-kappa}; PVA: single mode, vartheta=5.505 s^{-kappa}
    Parameters of the multimodal Prabhakar pre-gel model (Eq. 65); fitted to relaxation modulus / dynamic moduli data.
  • Post-gel Prabhakar exponent gamma_G = 52.8 (PDMS)
    Fitted post-gel Prabhakar exponent; the strikingly large value is rationalized phenomenologically as topological hindrance of trapped clusters. Not determined by any scaling constraint.
  • Reduced distance eps for each PDMS dataset = pre: 0.3454, 0.0751, 0.0076; post: 0.0046, 0.117
    The Fig. 3 legend states these are 'fitted values of eps' rather than values set by the arrested-crosslinking protocol; fitting the independent variable inflates apparent agreement.
  • Post-gel second-mode coefficient B_{G,2} (PVA) = 10.555 s^{-kappa}
    Listed in Table 3 with exactly the same value as B_{G,1}=B_S; either a typo or an unexplained constraint. With n=0.77, kappa=0.266, floor(n/kappa)=2, so the k=2 term is active.
assumptions (8)
  • standard math Prabhakar Laplace transform identity: L{t^{beta-1} E^gamma_{alpha,beta}(-lambda t^alpha)} = q^{alpha gamma - beta}/(q^alpha + lambda)^gamma
    Invoked in Eq. (62) to convert the Prabhakar relaxation modulus to the Laplace domain and to extract zero-shear viscosity.
  • standard math Complete monotonicity conditions 0<alpha<=1, 0<gamma<=beta/alpha for t^{beta-1}E^gamma_{alpha,beta}(-x), and analogous conditions for the two-parameter Mittag-Leffler function
    Used to derive the admissibility bound n+kappa<1 for Pre-Gel Model 1 and the extended admissibility for Model 2 (Eq. 63 and surrounding text).
  • domain assumption Conventional scaling laws: eta_0 ~ eps^{-s}, G_e ~ eps^z, tau_max ~ eps^{-1/kappa}, with s = (1-n)/kappa_S and z = n/kappa_G
    Taken from critical gelation theory (Table 1, Eqs. (6)-(11)) and imposed as constraints when the models are constructed, e.g. eta(eps) = eta* eps^{-s} in Eq. (38). The hyperscaling relation is then an algebraic consequence of these assumed relations.
  • domain assumption Winter-Chambon power-law forms of the critical gel state (Eqs. (1)-(5))
    The reference behavior recovered in the eps->0 limit of every model; empirically established for gel-forming systems.
  • domain assumption Continuity of the complex modulus and its first derivative with respect to the degree of crosslinking at the critical point (Eqs. (20)-(21))
    The load-bearing postulate of the paper, asserted as a physical necessity following the author's in-press self-citation [39], but not derived from percolation or statistical mechanics. It carries the symmetry kappa_S = kappa_G and the hyperscaling 'necessity' claim.
  • ad hoc to paper Functional ansatz G(t,eps) = S Gamma(1-n) t^{-n} E^gamma_{kappa,1-n}(-vartheta eps t^kappa) (pre-gel) and its post-gel analog with +vartheta eps t^kappa
    Equations (46), (64), and (97) postulate the Mittag-Leffler/Prabhakar forms; all scaling results for tau_max ~ eps^{-1/kappa} are structurally built into this ansatz via the argument -vartheta eps t^kappa.
  • domain assumption Finiteness of zero-shear viscosity for eps>0 in the pre-gel state
    Used to force gamma_S kappa_S = 1-n in Eqs. (71)-(73), uniquely fixing the Prabhakar exponent; physically reasonable for pre-gel liquids but it is this requirement, together with the assumed scaling relation, that produces gamma_S = s.
  • domain assumption Physical admissibility n > kappa
    Stated as 'values of n<kappa are unphysical' per Joshi [39]; restricts the fingerprint plot (Fig. 5) to kappa < n < 1 and ensures positivity of tan((n-kappa)pi/2).

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Pith. "Pith review of Analysing gelation transition through fractional viscoelasticity and Mittag-Leffler-Prabhakar function." pith.science (2026). https://pith.science/paper/GWNTZDD2

@misc{pith2026260626656,
  author       = {Pith},
  title        = {Pith review of: Analysing gelation transition through fractional viscoelasticity and Mittag-Leffler-Prabhakar function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWNTZDD2}},
  note         = {Machine review of arXiv:2606.26656}
}
read the original abstract

The gelation transition, a process that transforms a flowable liquid into an elastic solid, occurs in a variety of systems, ranging from colloidal to polymeric. During the gelation transition, a system passes through a critical gel state characterized by scale-free power-law viscoelasticity. Interestingly, the fractional calculus provides a natural mathematical language for such power-law viscoelasticity. In this work, we develop physically constrained fractional viscoelastic models, as well as those based on the three-parameter Mittag-Leffler-Prabhakar function, for both the pre-gel and post-gel regimes, ensuring consistency with the conventional scaling relations in each regime. While the fractional pre-gel model is observed to be valid only for a restricted subset of parameter values, the Prabhakar function-based model for relaxation modulus, which translates into Havriliak-Negami function based complex modulus, rigorously removes this limitation while simultaneously providing profound molecular significance. We enforce continuity of the dynamic moduli and their derivatives across the critical gel point, which universally imposes a symmetry in the relaxation dynamics on either side of the critical gel state. Such enforcement further validates the hyper-scaling relation connecting the critical exponents, making it a theoretical necessity rather than an empirical coincidence. We validate the proposed models against time- and frequency-domain experimental data. A model-agnostic, frequency-independent rheological fingerprint of the critical gel state, uniquely determined by two critical exponents, is also identified.

Figures

Figures reproduced from arXiv: 2606.26656 by the authors.

Figure 1
Figure 1. A series assembly of a single dashpot and 𝑁 springpots. It is a proposed fractional viscoelastic arrangement to model the pre-gel states. The dominant short-time term associated with 𝐺෨(𝑞) can be obtained by taking a limit of 𝑞 → ∞, which corresponds to 𝑡 → 0ା. For a material in the pre-gel states, therefore, lim ௤→ஶ 𝐺෨(𝑞) is expected to result in Laplace transform of the Winter–Chambon , , , [PITH_FULL_IMAGE:figur… view at source ↗
Figure 2
Figure 2. A parallel assembly of 𝑁 springpots and a spring, a fractional viscoelastic assembly proposed to model the postgel states. With this background, let us consider a general parallel fractional network assembly consisting of a single Hookean spring with modulus 𝐺௘(𝜀) in parallel with 𝑁 springpots as shown in [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Relaxation modulus 𝐺 [Pa] for a chemically crosslinked PDMS system is plotted as a function of reduced time 𝑡/𝑎் [s] at varying distances 𝜀 from the critical gel point. The experimental data is taken from Winter and coworkers [66, 67]. The symbols represent the transformed experimental data extracted from the time-temperature 10−4 10−3 10−2 10−1 100 101 102 100 101 102 103 104 Values of e Pre-gel states 0.3454 0.075… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Storage modulus 𝐺′(𝜔) and loss modulus 𝐺″(𝜔) for an aqueous poly(vinyl alcohol) (PVA) solution, as it undergoes gel-sol transition, are plotted as functions of angular frequency 𝜔, at various distances (𝜀) from the critical gel point: in the post-gel states (a) 0.4, (b…
Figure 5
Figure 5. Figure 5: Relative change in 𝐺 ᇱᇱ with respect to 𝐺 ᇱ as the system passes through the critical gel state (∂𝐺 ᇱᇱ/ ∂𝐺 ᇱ)ఌ→଴ is plotted as a function of the critical relaxation exponent 𝑛 for representative values of the relaxation scaling exponent 𝜅. It is important to note that …

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.