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REVIEW 4 major objections 4 minor 66 references

Thermodynamics of Flat 4D Einstein-Gauss-Bonnet Black Hole with R\'enyi Entropy: An RPST-like formalism

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Rényi entropy makes flat 4D-EGB black holes thermodynamically twin AdS black holes.

desk verdict A genuine but conditional extension of the Rényi-entropy RPST-like program to flat 4D-EGB black holes; the critical-point algebra holds, but the central rescaling is not stated correctly and the GTD comparison is asserted rather than shown. read the letter →

arxiv 2506.21917 v2 pith:5ORSBFAY submitted 2025-06-27 hep-th

Amijit Bhattacharjee , Prabwal Phukon This is my paper
classification hep-th
keywords Rényientropy4DEinstein-Gauss-BonnetgravityrestrictedphasespacethermodynamicsflatblackholesvanderWaalstransitiongeometrothermodynamicsthermodynamictopologyholographicduality
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

An asymptotically flat black hole carries no cosmological constant, and standard thermodynamics for it allows only Davies-type transitions. This paper claims that substituting Rényi entropy for Bekenstein-Hawking entropy, and treating the inverse Rényi parameter β as a free thermodynamic coordinate, produces a restricted-phase-space (RPST-like) structure for the flat 4D Einstein-Gauss-Bonnet black hole that reproduces the phase structure of its AdS counterpart: a new conjugate pair (β, ζ) analogous to central charge and chemical potential, van der Waals-like first-order transitions in both fixed-charge and fixed-potential ensembles, and identical GTD curvature and topological winding number (W=+1). If the claim is right, holographic thermodynamic structure is not exclusive to AdS spacetimes, and Rényi entropy becomes a bridge between flat-space black-hole thermodynamics and AdS/CFT.

What carries the argument

The load-bearing object is the Rényi-entropy-transformed, rescaled mass function $\tilde{M}(S, \tilde{Q}, \tilde{\alpha}, \beta)$, obtained from the flat 4D-EGB mass by writing the horizon radius in terms of Rényi entropy and applying the rescaling $\tilde{Q} = \kappa Q/\sqrt{G}$, $G = \kappa^2/\beta$, $\tilde{\alpha} = \alpha/\kappa^2$, and $\tilde{M} = \kappa M$. This function is homogeneous of degree one in $S$, $\tilde{Q}$, $\tilde{\alpha}$, and $\beta$, so the Euler relation $E = TS + \tilde{\phi}\tilde{Q} + \tilde{A}\tilde{\alpha} + \zeta\beta$ holds. The paper promotes $\beta$ to an independent coordinate with conjugate $\zeta = \partial\tilde{M}/\partial\beta$, which produces the RPST-like first law. Phase structure is extracted by solving the critical-point conditions on the temperature, and the comparison to AdS RPST is made through the GTD metric $g = S(\partial\tilde{M}/\partial S)(-\partial^2\tilde{M}/\partial S^2\,dS^2 + \partial^2\tilde{M}/\partial\tilde{Q}^2\,d\tilde{Q}^2)$ and the off-shell free energy $F = \tilde{M} - S/\tau$ with winding-number analysis.

What would settle it

Re-derive the mass function without the silent G=1 and κ=√β assumptions; if the resulting M~ still depends on G, the first-order homogeneity and the critical exponents will change. As a direct check of the phase transition, compute the on-shell free energy of the small and large black hole branches at T < T_C and verify that they cross exactly at the temperature predicted by the swallowtail of Fig. 1; if the crossing is absent, the van der Waals first-order transition is not real.

Watch

Extended reading notes

Core claim

Starting from the flat 4D-EGB black hole mass $M = r_+/(2G) + \alpha/(2r_+G) + Q^2/(2r_+)$ and the Rényi entropy $S = (1/\lambda)\ln(1+\lambda\pi r_+^2/G)$ with $\beta=1/\lambda$, the paper derives the rescaled mass $\tilde{M} = [\beta^2(e^{S/\beta}-1)+\pi\beta^2\tilde{\alpha}+\pi\tilde{Q}^2]/[2\sqrt{\pi\beta}\sqrt{e^{S/\beta}-1}]$ and posits the first law $d\tilde{M} = T\,dS + \tilde{\phi}\,d\tilde{Q} + \tilde{A}\,d\tilde{\alpha} + \zeta\,d\beta$. Solving $\partial T/\partial S=0$ and $\partial^2 T/\partial S^2=0$ in the fixed $\tilde{Q}$ ensemble yields $S_C = \beta\ln(2(\sqrt{3}-1))$, $\tilde{Q}_C = \beta\sqrt{(7-4\sqrt{3})/\pi - \tilde{\alpha}}$, and $T_C = 0.256236$; in the fixed $\tilde{\Phi}$ ensemble, $S_C$ is the same, $\tilde{\Phi}_C = \sqrt{1-\pi\tilde{\alpha}(7+4\sqrt{3})}$, and $T_C = 11.2121\tilde{\alpha}$. Below these critical values the $T-S$ curves are non-monotonic and the $F-T$ curves are swallowtails, indicating van der Waals-like first-order phase transitions. The paper then shows that the GTD scalar $R_{GTD}(S)$ for the Rényi flat black hole coincides with the AdS RPST black hole when the central charge $C$ is chosen so that the Rényi parameter $\lambda$ is inversely related to $C$, and that both systems carry total topological charge $W=+1$ with small and large branches stable and the intermediate branch unstable. The conclusion is that the Rényi-modified flat black hole belongs to the same thermodynamic class as the 4D-EGB AdS black hole in RPST.

Load-bearing premise

The construction stands on treating the inverse Rényi parameter β as an independent, fluctuating thermodynamic coordinate in the first law, together with a rescaling that silently sets Newton's constant G=1; if β is merely a fixed parameter of the entropy formula, or if the rescaling is not legitimate, the ζ–β pair and the claimed RPST-like structure are formal artifacts.

Editorial extensions

If this is right

  • If the central claim holds, asymptotically flat black holes with Rényi entropy exhibit genuine first-order phase transitions without any cosmological constant, so non-extensive entropy alone can replace an AdS background in generating critical phenomena.
  • The ζ–β duality gives the Rényi deformation parameter the status of a thermodynamic coordinate, which in the RPST-like dictionary plays the role of the central charge C; this makes β a candidate for a microscopic count of degrees of freedom in flat-space gravitational thermodynamics.
  • The coincidence of GTD curvature curves and of total topological charge W=+1 implies that the flat Rényi black hole and the AdS RPST black hole share the same stability pattern: small and large branches stable, intermediate branch unstable.
  • The λ–Λ relation β ≈ πl²/G suggests that flat-space Rényi thermodynamics can be reinterpreted as AdS thermodynamics with an effective AdS radius, making the flat-space results a probe of bulk/boundary correspondence.

Reading between the lines

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

  • Going beyond the paper, if the inverse proportionality between λ and C is universal, then a measurement of the Rényi parameter from flat-space black-hole data would directly fix the central charge of a putative dual field theory, giving a concrete holographic prediction.
  • A natural test of the framework's robustness is to repeat the analysis for rotating (Kerr-like) flat black holes with Rényi entropy; if the W=+1 topology and swallowtail behaviour persist without spherical symmetry, the correspondence is a property of Rényi statistics rather than of the 4D-EGB solution.
  • The paper leaves open whether the first law with β as a coordinate can be derived from a Hamiltonian or action principle; deriving dM~ from a variational principle with G explicitly varying would convert the formal β-duality into a mechanical one.
  • If the RPST-like equivalence holds, the flat/Rényi description may give a way to simulate AdS phase structure in laboratory analogues (e.g., optical or condensed-matter systems) that realize non-extensive entropy, since the cosmological constant is no longer needed.
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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 / 4 minor

Summary. The paper studies asymptotically flat charged 4D Einstein-Gauss-Bonnet black holes whose entropy is replaced by the Rényi entropy. The authors construct a rescaled mass function and propose a restricted-phase-space-like first law in which the inverse Rényi parameter β is promoted to a thermodynamic coordinate with conjugate potential ζ, analogous to the central charge/chemical potential pair of AdS RPST thermodynamics. They compute critical points in both fixed-charge and fixed-potential ensembles, report van der Waals-like first-order transitions, and compare the resulting GTD curvature and thermodynamic topology with those of the 4D-EGB AdS black hole in the standard RPST formalism. The central claim is that the flat Rényi system reproduces the thermodynamic, geometric, and topological features of the AdS RPST system.

Significance. If the construction is physically warranted, the paper would extend RPST-type holographic thermodynamics to asymptotically flat black holes and give evidence that Rényi non-extensive entropy can serve as a bridge between flat-space black hole physics and AdS holography. The critical-point algebra is internally consistent, and the paper makes an explicit, falsifiable prediction: the existence of van der Waals-like first-order phase transitions and a β-ζ conjugate pair in flat 4D-EGB black holes. The homogeneity check and Euler relation are useful explicit checks. However, the physical status of β as a fluctuating thermodynamic variable and the rescaling that produces the central mass function are not adequately justified, and the claimed equivalence with AdS RPST is largely qualitative.

major comments (4)
  1. [Section II, Eqs. (27)-(35)] The derivation of the rescaled mass \tilde M is not internally consistent as printed. Substituting r_+ from Eq. (29) into the flat mass Eq. (27) gives a numerator β(e^{S/β}-1) + πα/G + πQ^2, not β(e^{S/β}-1) + πα + πQ^2 as written in Eq. (30); the α term acquires an extra factor 1/G. If Eq. (30) is corrected in this way, the substitutions in Eq. (33) do indeed lead to Eq. (35) without any silent choice of κ. If, instead, the authors intend G=1 from the outset, then the substitution G→κ²/β in Eq. (33) is not a rescaling of the original flat EGB theory but a redefinition that introduces β as a variable. Either way, the manuscript must state the convention and show the intermediate algebra, because every subsequent quantity—T, ζ, the critical values, the GTD scalar, and the topological analysis—is computed from \tilde M in Eq. (35).
  2. [Section III, Eq. (39)] The first law promotes the inverse Rényi parameter β to an independent thermodynamic coordinate and defines ζ = ∂\tilde M/∂β. This is a postulate, not a consequence of the black hole solution: the Rényi parameter is fixed in the entropy functional, and varying it changes the statistical-mechanical definition of entropy rather than the black hole state. The Euler relation Eq. (40) then holds automatically because \tilde M in Eq. (35) is homogeneous of degree one in (S,\tilde Q,\tilde α,β), so the β-ζ duality is built in by construction. The paper should either justify β as a physical control parameter (for example, through an ensemble with variable non-extensivity) or explicitly present the formalism as a mathematical analogue. As it stands, the central claim that flat Rényi black holes have a holographic-like β-ζ pair is circular.
  3. [Section IV.1] The GTD comparison is not reproducible. The text states that the full expressions for R_GTD are omitted because of their substantial length, and only plots are shown (Fig. 5). Since the claim that the flat Rényi and AdS RPST geometries coincide is quantitative—the curves are said to match for specific C and β values—the authors should provide the explicit GTD scalar or at least a computational appendix. Without this, the geometric concordance claim cannot be independently checked.
  4. [Section IV.2 and Conclusions] The topological comparison does not establish a parameter map between the two systems. The paper computes winding numbers W=+1 for both the flat Rényi and AdS RPST black holes in both ensembles, but W=+1 is the generic topological class for many black hole families; the same result would hold for any two mass functions with three branches. The relation β≈πl²/G from Eq. (32) is not used to derive a quantitative correspondence C↔β. To support the claimed mimicry, the authors should show that the two mass functions are the same function under this map, rather than only that the topological charge coincides.
minor comments (4)
  1. [Section III, Eq. (39)] The last term in Eq. (39) is written ζ dλ, but throughout the paper the conjugate variable is β; λ was replaced by 1/β in Eq. (29). Please correct this.
  2. [Section III.2, Eqs. (47)-(49)] The symbols ρ and e in Eq. (48) are used before being defined; define ρ = \tilde Φ/\tilde Φ_C and e explicitly, and check the displayed expression for t, which is very hard to parse in its current form.
  3. [Figures] There are several typographical errors: 'iso-e-charge' should be 'iso-charge' or 'fixed-charge' in Fig. 1; 'modidfied' appears in figure captions; and Fig. 9's caption says 'Rényi modified flat black hole' in the AdS RPST subsection, which is likely a copy-paste error.
  4. [Section IV.1, Eq. (12)] The reduction from the general GTD metric in Eq. (12) to the specific metric with coordinates S and \tilde Q is not shown; a one-line explanation of the choice of potential and coordinates would help the reader reproduce the plots.

Circularity Check

3 steps flagged · score 7.0 of 10

The flat/AdS RPST resemblance is partly circular: the rescaled mass is fixed by hidden choices κ²=β, G=1, the GTD coincidence is a two-parameter fit, and the β–ζ duality is definitional.

  1. fitted input called prediction [Section II, Eqs. (32)-(35)]
    "Now, inspired by the relation in eqtn.(32) we use a rescaling constant κ to rewrite the Newton’s constant G, rescaled GB parameter ˜α and rescaled electric charge ˜Q as: ˜Q→ κ Q/√G, G→ κ²/β and ˜α→ α/κ². Using eq.(33) in eq.(30), the mass is rewritten as: M= β²(e^{S/β}−1)+πβ²˜α+π˜Q²/2√πκβ√(e^{S/β}−1) (34). And finally the rescaled mass ˜M=κM, is given by: ˜M= β²(e^{S/β}−1)+πβ²˜α+π˜Q²/2√πβ√(e^{S/β}−1) (35)."

    Using the paper's own rescaling (33) in (30) gives M = [βD + πκ²α~ + πQ~²/β]/(2κ√(πD)) with D=e^{S/β}−1. Multiplying numerator and denominator by β, equality with the printed (34) requires κ²=β, which also forces G=κ²/β=1. These silent choices are not stated. Therefore M~ in (35) is not the flat EGB mass (27) in general units; it is a specially normalized expression. Every subsequent result—T, ζ, the critical values (42)/(47), the GTD scalar, and the topological class—is computed from this chosen M~. The claimed VdW-like transitions and AdS RPST resemblance are thus built into the fitted rescaling rather than derived from the original flat EGB black hole.

  2. fitted input called prediction [Section IV.1, Fig. 5 caption and surrounding text]
    "We see that for C= 53 and β= 1/λ = 83.33 the thermodynamic geometry of the both the black holes coincide for the fixed ( ˜Q) ensemble as can be seen from the plot itself. ... We see that for C= 44 and β= 1/λ = 83 the thermodynamic geometry of the both the black holes coincide for the fixed ( ˜Φ) ensemble."

    The claimed coincidence of the GTD geometries is obtained by choosing free parameters: C=53 paired with β=83.33 in the fixed-Q~ ensemble, and C=44 paired with β=83 in the fixed-Φ~ ensemble. No equation determines C from β; the curves are made to overlap by a two-parameter fit. The further statement that C and the Rényi parameter λ are inversely proportional is inferred from these chosen pairs. The 'remarkably similar' thermodynamic geometry is therefore an input of the parameter choice, not an independent prediction.

1 more flagged steps
  1. self definitional [Section II, Eq. (31) and Section III, Eq. (39)]
    "Motivated by this parallel, we propose a formulation of the first law of black hole thermodynamics inspired by the RPST framework, extending it to non-AdS (asymptotically flat) black holes which is given as: dM=T dS+ ΦdQ+Adα+ζdβ, (31) where ... ζ= ∂M/∂β is the conjugate potential to β."

    The β–ζ 'thermodynamic duality' is introduced by decree: ζ is defined as ∂M~/∂β in the proposed first law, and it is repeated in rescaled variables in Eq. (39). Since M~ already depends on β through the Rényi entropy and the rescaling, any one-parameter entropy deformation automatically produces such a conjugate variable. The claimed analogy to the C–µ duality of AdS RPST is therefore a definitional consequence of the ansatz, not an emergent structural fact established from the flat EGB dynamics.

full rationale

The central object of the paper is the rescaled mass M~ in Eq. (35), which generates all of the claimed AdS-RPST phenomena. Substituting the paper's own rescaling (33) into the Rényi mass (30) yields a form that matches the printed (34) only under the hidden conditions κ²=β and G=1. Thus the mass whose temperature, ζ, critical points, GTD curvature, and topological charges are computed is a specially normalized expression, not the flat EGB mass (27) in general units. The 'prediction' of VdW-like first-order transitions and RPST-like structure therefore reduces to the chosen rescaling. The GTD comparison is likewise fitted: C=53 with β=83.33 and C=44 with β=83 are selected so that the curves coincide; no independent relation fixes these pairs. Finally, the β–ζ duality is defined by ζ=∂M~/∂β, so it holds for any β-dependent mass by construction. I also note that the paper explicitly omits the GTD scalar derivation ('we do not present here the explicit derivation or the full expressions'), making the curve comparison hard to verify independently. The internal computations after M~ is adopted—critical values, Euler relation, and W=+1 topology—are consistent, and the paper does not invoke a uniqueness theorem, so the paper is not wholly circular; hence score 7 rather than 8–10.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The main free parameters are the inverse Rényi parameter β, the undefined rescaling κ, and the AdS central-charge values chosen to make the GTD curves overlap. The axioms include the Rényi entropy formula, the 4D-EGB mass formula, the promotion of β to a thermodynamic variable, the imported GTD and topology methods, and the λ-Λ relation from ref. [51]. The invented entities are the β-ζ duality and the claimed equivalence, neither of which has independent evidence.

free parameters (4)
  • β (inverse Rényi parameter) = β = 83.33 in fixed-Q~ plots; β = 83 in fixed-Φ~ plots
    Promoted to a thermodynamic variable in eqs. (31) and (39). Its values are hand-picked for plots and for matching the AdS central-charge values in Fig. 5.
  • κ (rescaling constant) = not stated; consistency with eqs. (33)-(35) implies κ = √β and G = 1
    Introduced in eq. (33) to rewrite G, Q, and α; never defined. The derivation of eq. (34) from eq. (30) requires this additional unstated condition.
  • α~ (rescaled Gauss-Bonnet coupling) = 0.002 (fixed-Q~ plots), 0.01 (fixed-Φ~ plots)
    Input parameter from the 4D-EGB theory, not fitted, but its values are chosen for the plots and enter the critical values in eqs. (42) and (47).
  • C (AdS central charge in comparison) = C = 53 (fixed-Q~), C = 44 (fixed-Φ~)
    Chosen to make the AdS GTD scalar curves overlap the flat Rényi curves in Fig. 5; this is a curve fit, not a prediction.
assumptions (5)
  • domain assumption Rényi entropy formula S = (1/λ) ln(1 + λ S_BH)
    Adopted from refs. [40, 41] as the non-extensive entropy; the entire paper's thermodynamics depends on replacing Bekenstein-Hawking entropy with it (Section II, eq. 28).
  • domain assumption Flat 4D-EGB black hole mass formula M = r_+/2G + α/(2r_+G) + Q²/(2r_+)
    Taken from the d → 4 limit of EGB gravity, eqs. (24)-(27); if this regularization is not valid, the central results do not apply.
  • ad hoc to paper β is a legitimate thermodynamic variable in the first law
    Eqs. (31) and (39) add ζdβ to the first law without a statistical-mechanical derivation; the physical status of β as a charge-like variable is assumed.
  • standard math GTD and Duan phi-mapping topology methods are valid diagnostic tools
    The phase-space geometry of eq. (12) from refs. [52-54] and the topological current from refs. [55-58] are imported as background; phase transitions are identified with curvature singularities and winding numbers.
  • domain assumption Relation Λ ≈ ±3λπ/G from ref. [51]
    Used in eqs. (10)-(11) and (32) to motivate β ≈ πl²/G and the rescaling that maps flat Rényi variables to AdS RPST variables; this relation is itself an approximation from prior literature.
invented entities (2)
  • β-ζ conjugate pair as analogue of central charge and chemical potential
    purpose: Creates the RPST-like first law dM~ = ... + ζdβ and the claimed thermodynamic duality.
    β is a repurposed Rényi parameter and ζ is defined as ∂M~/∂β; no independent observable or falsifiable handle is provided outside the construction.
  • Flat Rényi / AdS RPST thermodynamic equivalence
    purpose: The central claim that the two black hole systems belong to the same thermodynamic class.
    The equivalence is demonstrated by curve matching with chosen C values and by the generic topological charge W = +1; no independent prediction is made.

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Cite this review

Pith. "Pith review of Thermodynamics of Flat 4D Einstein-Gauss-Bonnet Black Hole with R\'enyi Entropy: An RPST-like formalism." pith.science (2026). https://pith.science/paper/5ORSBFAY

@misc{pith2026250621917,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics of Flat 4D Einstein-Gauss-Bonnet Black Hole with R\'enyi Entropy: An RPST-like formalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ORSBFAY}},
  note         = {Machine review of arXiv:2506.21917}
}
abstract

We investigate the thermodynamics of asymptotically flat black holes in four-dimensional Einstein-Gauss-Bonnet (4D-EGB) gravity using R\'enyi entropy as a non-extensive generalization of the Bekenstein-Hawking entropy. The resulting thermodynamic structure, formulated within a restricted phase space-like (RPST-like) framework, reveals a striking resemblance to the thermodynamics of AdS black holes in the standard RPST formalism. In particular, we identify a thermodynamic duality between the R\'enyi deformation parameter $\beta$ and a conjugate response potential $\zeta$, analogous to the central charge and chemical potential in holographic theories. An extensive thermodynamic analysis in both fixed charge-$(\tilde{Q})$ and fixed potential-$(\tilde{\Phi})$ ensembles reveal Van der Waals-like first-order phase transitions which is an unexpected feature for asymptotically flat black holes. Furthermore, through the formalism of geometrothermodynamics (GTD) and thermodynamic topology, It is shown that the R\'enyi modified flat black hole mimics, in both its thermodynamic topology and geometry, the features of its counterparts in the 4D-EGB AdS black hole under RPST, reinforcing the structural similarity between these seemingly different systems. Our findings point to a deeper correspondence between non-extensive entropy and holographic thermodynamics, suggesting that R\'enyi entropy may serve as a natural bridge between flat-space black hole thermodynamics and AdS holography.

Figures

Figures reproduced from arXiv: 2506.21917 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The GTD scalar versus entropy plot for the RPST like formalism of the Renyi modified flat black hole(Blue) [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Thermodynamic topology for the R´enyi modified flat 4D-EGB black hole in fixed ( [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Thermodynamic topology for the AdS RPST 4D-EGB black hole in fixed ( [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Thermodynamic topology for the R´enyi modified flat 4D-EGB black hole in fixed ( [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Thermodynamic topology for the AdS RPST 4D-EGB black hole in fixed ( [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

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