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REVIEW 6 major objections 4 minor 72 references

Probing Quantum Gravity through Chaotic Orbits and Strong-Field Effects in Kerr Black Holes Embedded in Perfect Fluid Dark Matter

T0 review · 6 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper claims that adding quantum-gravity and dark-matter corrections to a rotating black hole makes photon trajectories increasingly chaotic as the correction strengths grow.

desk verdict The abstract says the quantum and PFDM parameters drive photons toward chaos, but the paper's own Lyapunov and KS-entropy curves decay to regular motion, and the printed formulas fail standard Schwarzschild limits – the central claim is contradicted by the paper's own evidence. read the letter →

arxiv 2607.14170 v1 pith:KDR6W4L7 submitted 2026-07-15 gr-qc

Shubham Kala , Sara Saghafi , M. Yousaf , Hemwati Nandan , Ahmadjon Abdujabbarov , Chengxun Yuan , G. Mustafa This is my paper · ORCID
classification gr-qc MSC 83C5783C1037D45
keywords quantum-improvedblackholesperfectfluiddarkmatterphotonchaosnullgeodesicsLyapunovexponentsKolmogorov-SinaientropyweightedBirkhoffaveragesholeshadow
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

The paper constructs a rotating black hole metric that combines renormalization-group quantum corrections (parameter ω̃) with a perfect-fluid dark matter environment (parameter ζ), then studies how these parameters affect the motion of light rays. It claims that increasing either parameter transforms photon dynamics from regular, quasi-periodic orbits into chaotic motion: the invariant tori in phase space fragment, chaotic bands spread, and indicators such as Lyapunov exponents, KS entropy, and weighted Birkhoff averages all confirm stronger sensitivity to initial conditions. If true, the result matters because it links two speculative additions to gravity — quantum improvement and dark matter halos — to observable optical signatures like black hole shadows and lensing, and to timing variability in accretion flows.

What carries the argument

The machinery is a two-parameter rotating metric (Eqs. 21–22) with mass function m(r)=GM r²/(r²+ω̃G)−(ζ/2)ln(r/|ζ|), which reduces to Kerr when ω̃=ζ=0. The logarithmic PFDM term and the RG-improved coupling G(r)=G/(1+ω̃G/r²) modify the effective potential for photons. The paper uses four complementary diagnostics to classify orbits: Poincaré sections, the largest Lyapunov exponent and fast Lyapunov indicator, Kolmogorov–Sinai entropy (via the sum of positive Lyapunov exponents), and the weighted Birkhoff average with its difference-based chaos indicator (DIC), which provides a fast, quantitative map of regular and chaotic regions in phase space.

What would settle it

Integrate the full non-equatorial geodesic equations for the metric (21)–(22) at the parameter values used in Figs. 2–4 and compute the full Lyapunov spectrum or a frequency analysis (e.g., spectral entropy) on a dense grid of initial conditions; if the scattered Poincaré points disappear when the section is taken with a different projection or when the Hamiltonian is numerically verified to be integrable, the chaos claim fails. More directly: check whether ∆(r)=r²+a²−2m(r)r has a positive root for, say, ω̃=0.8, ζ=0.4, a=0.4—if no event horizon exists for the plotted parameters, the orbits are

Watch

Extended reading notes

Core claim

The paper's central claim is that in the quantum-improved Kerr metric surrounded by perfect fluid dark matter, the two extra parameters ω̃ and ζ act as chaos-inducing controls on null geodesics. Starting from a spherically symmetric seed metric with lapse function f(r)=1−2GM/(r+ω̃G)+ζ/r ln(r/|ζ|), the authors apply a standard non-complexification rotation prescription to obtain a Kerr-like rotating metric whose mass profile is m(r)=GM r²/(r²+ω̃G) − (ζ/2) ln(r/|ζ|). They then analyze the photon phase space with Poincaré sections, Lyapunov exponents, Kolmogorov–Sinai entropy, and weighted Birkhoff averages. Their numerical results show that raising ζ or ω̃ deforms and fragments the invariant t

Load-bearing premise

The whole argument rests on the unverified assumption that the two-parameter metric obtained by applying the rotation algorithm to the quantum-improved dark-matter seed is a genuine rotating black hole—with a real event horizon for the scanned parameters—whose geodesic flow is non-integrable, so the plotted scattered points are physical chaos rather than projection artifacts.

Editorial extensions

If this is right

  • If the central claim holds, black hole shadows and lensing images computed from this metric will show parameter-dependent signatures: increasing ζ or ω̃ shifts the photon sphere radii and changes the instability timescales of null circular orbits.
  • The chaotic transition implies that photon trajectories in strong-field regimes can become unpredictable over long integration times, with implications for ray-tracing and accretion-disk imaging codes.
  • Because the equatorial motion is integrable (as the paper notes), the chaos is a genuinely non-equatorial effect; any observational test would need to access off-equatorial photon paths.
  • The KS-entropy growth implies faster loss of predictability in the photon flow, which could appear as decorrelation timescales in high-frequency variability from black hole accretion environments.
  • The paper explicitly suggests QPO-related observables: parameter-dependent shifts in photon orbital frequencies could show up in quasi-periodic oscillation timing from accreting black holes.

Reading between the lines

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

  • A concrete extension would be to compute the shadow boundary and lensing deflection angles from the metric (21)–(22) and compare them with Event Horizon Telescope-like observations to place upper bounds on ω̃ and ζ; the paper's chaos analysis does not itself yield an observational prediction, but it identifies the parameter range where such signatures would be visible.
  • There is a quantitative tension between the decreased Lyapunov exponent of the circular orbits (Sec. IV) and the overall growth of phase-space chaos claimed in the abstract; separating these two notions—local circular-orbit instability versus global geodesic chaos—would sharpen the physical interpretation.
  • If the chaotic regions are as large as the DIC maps suggest, long-time geodesic integration in this spacetime is genuinely unpredictable, which could affect numerical relativity and plasma-photon interaction simulations that assume stable photon rings.
  • The non-integrability of the full four-dimensional flow is asserted but not proved; a Melnikov-type or frequency-analysis calculation on the non-equatorial reduction would settle whether the scattered points are real chaos rather than artifacts of projecting a quasi-periodic higher-dimensional flow onto a two-dimensional surface.
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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

6 major / 4 minor

Summary. The paper constructs a rotating 'quantum-improved Kerr' metric surrounded by perfect fluid dark matter by applying the Azreg-Aïnou algorithm to a static seed with RG-improved Newton constant, and then studies null geodesic dynamics. The central claim is that increasing the quantum-improvement parameter ω̃ and the PFDM parameter ζ drives photon motion from regular to chaotic, with quantitative support claimed from Poincaré sections, Lyapunov indicators (LLE, FLI), Kolmogorov–Sinai entropy, and weighted Birkhoff averages (WBA/DIC). The paper also derives expressions for the null circular orbit radii, critical impact parameter, and null Lyapunov exponent, and argues for observational implications for shadows and lensing.

Significance. The topic is timely, and the use of multiple independent diagnostics (Poincaré sections, LLE/FLI, KS entropy, WBA/DIC) is in principle a strength. If the claimed regular-to-chaotic transition with ω̃ and ζ were established for a validated black hole spacetime, it would be of interest to the strong-field gravity and quantum-gravity phenomenology communities. However, the manuscript has load-bearing errors: the paper's own Lyapunov and KS-entropy curves indicate convergence to regular motion, contradicting the abstract; the null Lyapunov formula (44) and critical impact parameter (39) fail simple Schwarzschild limits; the rotating spacetime's horizon structure is not checked for the plotted parameters; the chaos diagnostics are based on an under-specified system; and the 'independent' KS entropy is computed from the same largest Lyapunov exponent. These issues collectively invalidate the central claims as presented.

major comments (6)
  1. [Abstract; Secs. IV, VI, VII] The abstract claims a 'transition from regular to chaotic motion' and that ω̃ and ζ 'enhance the sensitivity of photon trajectories to initial conditions.' The paper's own quantitative sections state the opposite: Sec. IV says both parameters 'reduce the instability of black hole spacetime'; Sec. VI reports that the LLE curves 'gradually mov[e] towards the monotic decay' and that 'the similar pattern of gradually monotonic decay in the asymptotic dynamics regime is observed'; Sec. VII describes KS entropy oscillations that 'gradually decrease as the time increases, indicating the transition towards weak chaotic and regular motion.' A Lyapunov exponent decaying to zero and an entropy damping to zero are diagnostics of regular/quasi-periodic motion, not of growing chaos. This is a direct contradiction of the central claim, not a wording issue.
  2. [Eq. (44), Fig. 1] The null Lyapunov formula (44) fails the Schwarzschild test. Setting a=0, ω̃=ζ=0, m=M, r_c=3M gives 2∆_c − r_c²∆_c'' = −12M² and denominator 2r_c²(r_c²)² = 1458M⁶, hence λ² = −12/(1458M⁴) < 0 with the wrong dimension. The standard Schwarzschild photon-sphere Lyapunov exponent is λ² = 1/(27M²). Therefore Eq. (44) cannot be the expression used to produce the positive λ² values in Fig. 1; either the derivation from Eq. (43) is wrong or the plotted quantity is not the stated one. This invalidates the circular-orbit stability analysis.
  3. [Eq. (39)] The critical impact parameter formula (39) is also incorrect in the Schwarzschild limit. For a=0 it reduces to |b_c| = r_c√(2Mr_c∆_c)/(2Mr_c−r_c²), which at r_c=3M gives |b_c|=3√2 M, whereas the exact Schwarzschild photon-sphere impact parameter is 3√3 M. The correct circular-orbit relation is b_c = a ± 4r_c√∆_c/∆'_c. Since b_c enters Eq. (44), the numerical Lyapunov values and the photon-sphere plots are affected by this error as well.
  4. [Sec. II, Eqs. (21)–(23); Fig. 1] The rotating metric is generated from the effective mass (22) without verifying that it solves the Einstein equations with the PFDM energy-momentum tensor, and without checking the horizon condition ∆(r)=0 for the scanned parameters. For the right panel of Fig. 1, M=1, a=0.4, ζ=0.2, ω̃=0.4, numerical evaluation of Eq. (22) gives ∆(r)>0 for all r>0, i.e. no event horizon; for ζ=0.2, ω̃=0.2 a horizon may exist, but the plotted range r_c∈[0.6,2.0] anyway extends below the classical Kerr horizon r₊≈1.92. The authors must restrict to genuinely black-hole parameter sets, state the horizon radii, and show that the spacetime is a valid solution before drawing physical conclusions.
  5. [Secs. III, V; Eq. (36)] The chaos analysis is under-specified. The Poincaré sections are described as constructed at θ=π/2 with pθ>0 and as using 'the equations of motion given in Eqs. 34-36,' but Eq. (36) is the equatorial radial equation and no equation for θ̇ or the Carter constant is provided. The equatorial reduction of Sec. III is Liouville integrable (one radial degree of freedom with two conserved quantities), so any claim of chaos in the (r,ṙ) sections must come from a full four-dimensional geodesic system that the paper never defines. Without the θ dynamics and the surface-of-section definition, the scattered points in Figs. 2–4 cannot be interpreted as physical chaos.
  6. [Sec. VII, Eq. (47); Sec. VI, Eq. (45)] The abstract calls the KS entropy 'an independent measure of dynamical complexity,' but Sec. VII explicitly computes h_KS from the largest finite-time Lyapunov exponent via the Pesin form (47). This is the same number relabeled, not independent confirmation. It is also not a correct application of Pesin's theorem, which requires the sum over all positive Lyapunov exponents. Additionally, Eq. (45) defines the LLE from the coordinate-space separation d(τ); the text claims this formula is coordinate-independent, but no such invariance is established for two-particle separations in curved spacetime, and standard results (e.g. Motter 2003) show that naive coordinate-dependent Lyapunov exponents are not invariant.
minor comments (4)
  1. [Eq. (7)] There is an algebraic error in the second equality of Eq. (7): with G(r)=G/(1+ω̃G/r²), one has 2G(r)M/r = 2GM r/(r²+ω̃G), not 2GM/(r+ω̃G). The later mass function (10) uses the correct form, but the displayed equality in (7) is inconsistent and should be corrected.
  2. [Throughout] Numerous typos: 'quantum-impoved' (Sec. II heading), 'PDFM' (Figs. 5 and 7 captions), 'quantum-proved' (Sec. VI), 'monotic decay' (Sec. VI), 'Weyl conformal gravity' (Sec. VII, should refer to the present spacetime). These should be fixed.
  3. [Fig. 1] The y-axis in Fig. 1 is labeled λ²_null, but Eq. (44) gives negative values for the plotted parameter ranges; the figure caption should state explicitly whether the plotted quantity is |λ²|, a corrected formula, or something else.
  4. [Sec. VIII, Eq. (49)] The WBA observable f(r,p_r)=sin(2π(r+p_r)) mixes a coordinate and a momentum with possibly different physical dimensions; in geometrized units this may be acceptable, but the units should be stated.

Circularity Check

1 steps flagged · score 5.0 of 10

The 'independent' KS-entropy confirmation is the same finite-time Lyapunov observable relabeled via Pesin's theorem.

  1. self definitional [Abstract; Sec. VII, Eq. (47) and surrounding text]
    "Abstract: 'The KS entropy provides an independent measure of dynamical complexity and confirms the growth of chaotic behavior with increasing quantum and PFDM corrections.' Sec. VII: 'In this study, the KS entropy is calculated based on the largest finite-time Lyapunov exponent, which is determined by analyzing the divergence of initially nearby null geodesics [44, 45]' and Eq. (47): 'hKS = X λi>0 λi.'"

    The paper presents KS entropy as an independent confirmation of the Lyapunov-based chaos growth, but Eq. (47) defines h_KS via Pesin's theorem as the sum of positive Lyapunov exponents, and the text states that it is 'calculated based on the largest finite-time Lyapunov exponent.' Therefore the 'independent measure' is the same finite-time divergence observable used for the LLE diagnostic in Sec. VI, relabeled as entropy; it cannot independently confirm the Lyapunov result. Poincaré sections and WBA/DIC remain separate diagnostics, so the circularity is partial rather than total.

full rationale

The metric construction (Eqs. 1-22) is a direct substitution of an RG-running Newton constant into a PFDM seed metric followed by a standard Azreg-Ainou rotation procedure; the null geodesic equations and the null-circular Lyapunov formula (Eqs. 36-44) are derived from the metric without fitted parameters, so no circularity appears there. No curve-fitting is performed, and the paper does not invoke an imported uniqueness theorem or a load-bearing self-citation chain: the self-citations are mostly methodological (Poincaré, Lyapunov, WBA tooling) or used for a parameter range, not as the argument that forces the chaos claim. The one concrete circular step is the KS-entropy 'independent measure': because h_KS is computed from the largest finite-time Lyapunov exponent via Pesin's theorem (Eq. 47), the abstract's claim that KS entropy independently confirms the Lyapunov results is the same number relabeled. This does not destroy the entire paper—Poincaré sections and WBA/DIC maps are genuinely separate diagnostics—but it removes the stated independence of one confirmatory method. Separately, the manuscript contains a consistency problem (the abstract claims a transition to chaos, while Sec. IV says both parameters 'reduce the instability of black hole spacetime' and Secs. VI-VII describe asymptotic monotonic decay toward regular motion); that is a scientific-contradiction issue rather than a circular-derivation issue and therefore does not further raise the circularity score.

Assumptions & free parameters 4 free parameters · 8 assumptions · 1 invented entities

The paper's contribution rests on a model metric assembled from literature ingredients plus hand-scanned parameters; no step is machine-checked and no field-equation verification of the rotating metric is given. The main burdens: validity of the RG-improved+PFDM rotating metric, the unproven non-integrability of its geodesic flow, and the unvalidated chaos diagnostics (no Kerr baseline, KS entropy relabeled from the LLE).

free parameters (4)
  • ω̃ (quantum-improvement parameter) = scanned 0.0–0.8 (Planck units); O(0.2–0.4) in most runs
    Quantum-improvement parameter in the RG-running G(r)=G/(1+ω̃G/r²); taken from asymptotic-safety literature, not constrained by data in this paper. The claimed chaos transition is driven by hand-picked values.
  • ζ (PFDM parameter) = scanned 0.1–0.6; |ζ| ≲ M stated as physical bound
    PFDM strength in the logarithmic metric term; free model parameter scanned to vary the background; no fit.
  • a (rotation parameter) = 0.2–0.99 across figures (0.2–0.4 in Poincaré/DIC runs)
    Standard Kerr spin parameter; its scanning is routine and not constrained by data here.
  • E, L (photon conserved quantities) = E = 0.94–0.97, L = 3–4 (figure-dependent)
    Initial-condition/energy-surface choices for the Poincaré and DIC scans; the paper notes E is 'the primary parameter governing the global extent' but does not scan it systematically with the chaos diagnostics.
assumptions (8)
  • domain assumption RG improvement via G(r) = G/(1+ω̃G/r²) (Eq. 6) captures the leading quantum-gravity correction to the spacetime.
    Imported from asymptotic-safety literature [67–70]; applied here heuristically inside the seed metric before rotation (Eqs. 7–10).
  • domain assumption PFDM is described by T^μ_ν = diag(−ρ,−ρ,P,P) with P/ρ = 1/2, yielding the log term in f(r) (Eqs. 2–5, 7).
    Standard phenomenological PFDM ansatz [64–66]; assumed to survive the rotation procedure as the matter content of Eq. (21).
  • ad hoc to paper The Azreg-Aïnou algorithm produces a valid rotating metric that solves the field equations with the same m(r) (Eqs. 15–22).
    The paper does not check the Einstein/PFDM field equations for the rotating metric, nor horizon existence for the plotted parameter ranges; the validity of (21) is asserted.
  • standard math Null geodesic motion restricted to θ=π/2 yields the radial equation (Eq. 36).
    Standard reduction; used for the effective potential and circular-orbit analysis, but the chaos sections use pθ>0 without writing the corresponding equations.
  • domain assumption Two-nearby-trajectory divergence defines coordinate-independent Lyapunov exponents (Sec. VI, Eqs. 45–46).
    Standard method [34–36]; the deviation-vector propagation is stated but never written down.
  • domain assumption Pesin's theorem with only the largest finite-time Lyapunov exponent approximates the KS entropy (Eq. 47).
    h_KS = Σ_{λ_i>0} λ_i; the paper retains only λ_max, so the 'independent' chaos measure is a relabeling of the LLE.
  • domain assumption WBA/DIC convergence distinguishes regular from chaotic orbits with the chosen observable f = sin(2π(r+p_r)) (Eqs. 49–53).
    Standard WBA method [47–54]; the observable selection is arbitrary and no convergence validation is reported.
  • ad hoc to paper A (r, ṙ) Poincaré section with pθ>0 at fixed E, L faithfully represents the 4-D null geodesic flow (Sec. V).
    For a 4-D flow a 2-D section produces fuzzy projections even for regular quasi-periodic orbits; the paper does not distinguish torus-projection structure from true chaos, and calls the orbits 'equatorial' while taking pθ>0.
invented entities (1)
  • Quantum-improved Kerr–PFDM background (Eqs. 21–22)
    purpose: Composite effective spacetime carrying the RG-improvement parameter ω̃ and the PFDM parameter ζ; all dynamics and chaos claims live in this geometry.
    No observable unique to this composite (shadow radius, lensing deflection, QPO frequency) is computed; the metric is a heuristic assembly of components from [25,26,64–66] without a field-equation verification.

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Cite this review

Pith. "Pith review of Probing Quantum Gravity through Chaotic Orbits and Strong-Field Effects in Kerr Black Holes Embedded in Perfect Fluid Dark Matter." pith.science (2026). https://pith.science/paper/KDR6W4L7

@misc{pith2026260714170,
  author       = {Pith},
  title        = {Pith review of: Probing Quantum Gravity through Chaotic Orbits and Strong-Field Effects in Kerr Black Holes Embedded in Perfect Fluid Dark Matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KDR6W4L7}},
  note         = {Machine review of arXiv:2607.14170}
}
abstract

We study the nonlinear photon dynamics in quantum improved rotating black hole surrounded by perfect fluid dark matter (PFDM) using several methods of analysis, including Poincar\'e sections, Lyapunov exponents, Kolmogorov-Sinai (KS) entropy and weighted Birkhoff averages (WBA). In particular, we examine the effect of quantum-improved parameter $(\tilde{\omega})$ and PFDM parameter $(\zeta)$ on the null geodesic motion hence stability of circular orbit. Poincar\'e sections illustrate the transition from regular to chaotic motion as these parameters increase, characterized by the deformation and fragmentation of invariant tori and the emergence of scattered chaotic regions in phase space. The stability properties of null circular orbits are quantified through the Lyapunov indicators, revealing that both the quantum improvement parameter and the PFDM parameter enhance the sensitivity of photon trajectories to initial conditions. The KS entropy provides an independent measure of dynamical complexity and confirms the growth of chaotic behavior with increasing quantum and PFDM corrections. Additionally, the WBA method offers a robust quantitative criterion for distinguishing regular and chaotic orbits and allows a detailed mapping of the phase-space structure. The results demonstrate that the combined effects of quantum gravity corrections and PFDM significantly modify the effective potential governing photon motion, leading to a rich mixed phase-space structure with coexisting regular and chaotic regions. These findings underscore the crucial role of quantum and dark matter contributions in shaping photon dynamics near rotating black holes and suggest possible observational implications on black hole shadows and gravitational lensing in strong-field regimes.

Figures

Figures reproduced from arXiv: 2607.14170 by the authors.

Figure 1
Figure 1. FIG. 1: Variation of Lyapunov exponent as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Poincar´e sections of null geodesic motion around the quantum-improved Kerr black hole surrounded by PFDM, constructed [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Poincar´e sections of null geodesic motion around the quantum-improved Kerr black hole surrounded by PFDM, constructed [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Poincar´e sections of null geodesic motion around the quantum-improved Kerr black hole surrounded by PFDM, constructed [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Logarithmic Lyapunov exponent log [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Logarithmic Lyapunov exponent log [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The FLI versus proper time for Quantum improved Kerr black hole surrounded by PFDM, shown for different values of the [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The FLI versus proper time for Quantum improved Kerr black hole surrounded by PFDM, shown for different values of the [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Time evolution of the Kolmogorov–Sinai (KS) entropy [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Time evolution of the Kolmogorov–Sinai (KS) entropy [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: presents the Dynamical Indicator of Chaos (DIC) maps in the (r, pr) phase space for null geodesics around the quantum-improved Kerr black hole surrounded by PFDM, for different values of the PFDM parameter ζ, with fixed parameters a = 0.4, ˜ω = 0.2, E = 0.95, and L = …
Figure 12
Figure 12. Figure 12: FIG. 12: DIC maps in the ( [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

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