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A black hole's ringdown under dilute perfect-fluid accretion carries a clean, time-independent frequency-ratio signal that determines the fluid's equation-of-state parameter and a second signal that measures the accretion rate.

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For odd-parity ringdown on a slowly accreting Schwarzschild black hole, the ratio ω_I/ω_R relative to Schwarzschild is time-independent, proportional to the accretion rate, and encodes the fluid equation-of-state parameter w.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection A serious, honest numerical study of ringdown around an accreting BH whose central observable is not yet connected to an asymptotically defined measurement. the 3 major comments →

arxiv 2607.03231 v2 pith:STLD3S2T submitted 2026-07-03 gr-qc astro-ph.HEhep-th

Odd-parity ringdown gravitational waves of a spherically symmetric black hole with perfect fluid accretion

Rikuto Ohashi , Yasutaka Koga , Hiroyuki Nakano , Kota Ogasawara , Chul-Moon Yoo This is my paper
classification gr-qc astro-ph.HEhep-th
keywords black hole ringdownquasinormal modesperfect fluid accretionodd-parity perturbationsequation of stategravitational wave spectroscopytime-domain double-null simulation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Black holes in astrophysical settings are rarely isolated: matter accreting onto them makes the spacetime dynamical, so ringdown frequencies drift in time and the standard vacuum quasinormal-mode picture is only approximate. This paper shows that a dimensionless ratio built from the extracted frequency — the imaginary-to-real part ratio relative to Schwarzschild, called Ξ — cancels both the redshift and the slow secular mass growth, leaving a signal that is constant in the fundamental-mode-dominated part of the waveform and proportional to the accretion rate A. On the dilute, steady, spherically symmetric perfect-fluid accretion model, this makes Ξ/|A| a direct probe of the fluid's equation-of-state parameter w, with a one-to-one mapping in the physically relevant range 0

Core claim

The central claim is that for a Schwarzschild black hole accreting a dilute steady perfect fluid, the odd-parity ringdown is governed by a purely tensorial master equation, because the odd-parity fluid perturbation decouples and can be set to zero. Solving this equation in the time domain and extracting the instantaneous complex frequency, the relative deviation Ξ = (ω_I/ω_R)/(ω_I^(Sch)/ω_R^(Sch)) − 1 is constant once the fundamental mode dominates and scales linearly with the accretion rate A. The ratio construction removes the uniform redshift and the slow mass growth of the hole, so Ξ isolates the surrounding matter. For 0<w≤1 the background has no free parameter beyond w (at fixed sign o

What carries the argument

The load-bearing object is the gauge-invariant odd-parity master variable ψ_lm on a Schwarzschild background corrected to first order in the accretion rate, evolved in double-null coordinates with a second-order characteristic scheme that adaptively redefines the null coordinate to prevent near-horizon grid blow-up. The background fluid profile is fixed by the equation-of-state p=wρ and the steady-accretion equations, parameterized by accretion rate A, EoS parameter w, and an integration constant F. Two observables are extracted: Ξ (the ratio deviation, which cancels redshift and mass growth) and \tilde A (the accretion-rate estimator from the frequency's time drift). The argument that makes

Load-bearing premise

If waves generated near the black hole scatter off the distant region where the dilute-fluid approximation breaks down (roughly beyond 10^2 M0 for the chosen parameters) and return into the observed signal, the extracted Ξ and \tilde A are contaminated and the clean mapping to (w, F) fails; the paper assumes such backscattering is highly suppressed rather than proving it.

What would settle it

Run a high-resolution simulation with a Schwarzschild black hole embedded in a finite, smoothly truncated perfect-fluid accretion region — with the transition matched so that the background is exactly Schwarzschild outside — and measure Ξ at small and large l. If the cutoff region backscatters radiation so that Ξ becomes time-dependent in the fundamental-mode window, or if Ξ/|A| enters the Vaidya interval for some w>−1, the paper's central claim fails. A simpler diagnostic: check whether the time-averaged Ξ/|A| changes when r_obs is moved from 20M0 to 100M0 in the cutoff model; the paper's App

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • A single measurement of Ξ/|A| in the fundamental mode of a dilute-accreting Schwarzschild black hole fixes the equation-of-state parameter w whenever 0<w≤1, with no other free parameter in that regime.
  • The frequency-drift estimator \tilde A recovers the input accretion rate to first order and is essentially independent of observer location, giving a practical way to measure accretion rates from ringdown data.
  • At fixed l and sign of A, the observer-location dependence of Ξ/|A| is independent of the fluid parameters, so environment and geometry effects can be separated.
  • In the large-l limit, all weak-energy-condition perfect-fluid backgrounds give Ξ/|A| values outside the Vaidya interval, providing a null test that can distinguish accreting perfect-fluid environments from null-dust or vacuum models.
  • Differences in Ξ/|A| between low-l and high-l modes carry extra information about (w,F) in multi-parameter regimes (w≤0 or w>1), beyond the single-mode measurement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the outer-region backscatter suppression is confirmed, the same ratio-Ξ construction should generalize to other spherically symmetric accretion models, giving a generic way to measure the local environment's equation of state.
  • The cutoff experiments in Appendix C imply that real accretion regions of finite extent will show deviations between \tilde A and A of order A, so future data analysis might use \tilde A − A as a probe of matter at intermediate radii rather than treating it as noise.
  • The forbidden Vaidya interval offers a clean observational test: a ringdown measurement falling inside it would rule out steady spherical perfect-fluid accretion with weak energy condition, pointing instead to anisotropic flows, modified gravity, or non-steady accretion.
  • Extending the computation to κ∼ε (comparable accretion strength and perturbation amplitude) would require second-order perturbation theory; the paper's mechanism suggests Ξ's cancellation property, not its specific numerical value, is what survives.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper studies odd-parity gravitational-wave perturbations of a Schwarzschild black hole growing by steady, spherically symmetric accretion of a perfect fluid with equation of state p = wρ. Working to first order in the accretion rate, the authors derive a master equation in double-null coordinates, show that the odd-parity fluid perturbation decouples and can be set to zero, and evolve the Regge–Wheeler-type field on the accreting background. They define two observables: Ξ, the relative deviation of the ratio ω_I/ω_R from its Schwarzschild value, and ~A, a time-domain estimator of the accretion rate. The numerical results show that Ξ/|A| is constant in the fundamental-mode-dominated epoch to the achieved accuracy and is independent of the magnitude of A at O(κ), while ~A reproduces the input accretion rate. The paper maps Ξ/|A| as a function of (w, F, l, r_obs), argues that for 0 < w ≤ 1 the value of w can be uniquely determined, and reports a 'forbidden interval' for Ξ/|A| for any w > −1 background based on a numerically observed inequality against the Vaidya value.

Significance. If correct, the paper provides a novel and valuable step toward environmental BH spectroscopy: a concrete, non-Vaidya accreting-background model with a first-order-in-accretion time-domain computation, an observable insensitive to redshift and mass growth, and a demonstration that the accretion rate can be recovered independently. The derivation is careful, the numerical scheme is described in detail with convergence checks at two resolutions and two accretion rates, and the paper is transparent about limitations (e.g., l = 2 tail contamination, the failure of the dilute condition, the r_obs dependence). These strengths make the paper a useful contribution even if the final interpretation requires qualification. However, the central claim that the extracted Ξ/|A| is a local property of the accreting background is not established against backscattering from the outer, non-dilute region, and the claimed universal inequality goes beyond the numerical evidence.

major comments (3)
  1. [Sec. VI A, Eq. (39), Tables II–III, Appendix C] The central observable is extracted at r_obs = 20M0, while the dilute condition (39) fails already at r ~ 30–100M0 for |A| = 3×10^-5. The paper asserts (Sec. VI A) that the influence of these outer regions is 'expected to be highly suppressed,' but the only test offered is the A-scaling of Ξ/|A| (Tables II–III), which cannot distinguish local from backscattered contributions because both scale linearly in A. Appendix C directly contradicts the suppression assumption: introducing a cutoff in the outer region produces large, persistent changes in Ξ and ~A (Figs. 23, 25, 27, 28), and a deviation of O(A) in ~A/A0, even when using the smoother Eq. (84) evolution. Although the cutoff model violates the perfect-fluid Einstein equations in the transition zone, it demonstrates that the extraction is not causally inert to the geometry outside the formally valid domain. Until this is addressed (e.g
  2. [Sec. V B, Eq. (89), Tables IV–V] The paper shows that Ξ depends significantly on the observer radius r_obs (e.g., for l=10, Vaidya: -1.186 at 10M0, -1.918 at 20M0, -2.231 at 30M0), and that this dependence is not universal across w (the values shift by different amounts for different w). No extrapolation to r_obs → ∞ is provided; the correction formula used for asymptotically flat spacetimes (Refs. [36,37]) is stated to be inapplicable because the background is not asymptotically flat. Since the measurable waveform is the one at (large) distance, the mapping from Ξ/|A| to (w,F) is incomplete and observer-dependent. The paper acknowledges this, but the abstract and conclusion present the determination of w as a completed result. A prescription for converting finite-radius extractions to infinity, or a demonstration that the r_obs-dependence is itself a useful observable, is needed before the central claim can be accepted
  3. [Sec. VI C, Eqs. (101)–(102)] The inequality Ξ/|A| ≤ −|Ξ/A|^(Vaidya) for all w > −1 in the large-l limit is presented as a general result ('for any BG satisfying the weak energy condition'), but it is supported only by numerical exploration of a finite set of (w,F) values and l up to 15/20. No analytical argument (e.g., eikonal approximation, WKB) is given to justify 'always holds' and the 'cannot be explained' statement. Given that this inequality defines a forbidden interval that is a headline result, the authors should either prove it in the geometric-optics limit or explicitly restrict the claim to the parameter range actually computed, with a discussion of the risk that unexplored regions (e.g., extreme F, negative w near −1) may violate it.
minor comments (5)
  1. [Eq. (87)] The symbol A is used both for the accretion rate and for the amplitude of the initial Gaussian pulse. This is confusing; please use a different symbol (e.g., a0) for the pulse amplitude.
  2. [Sec. VI A] The sentence 'the approximation is shown to be valid for |A|=3×10^-5 in Subsec. VI B' is misleading: the A-scaling test shows convergence in A, not validation against an external exact result. Rephrase to avoid implying a stronger statement.
  3. [Fig. 2 caption] The term 'ingoing linear Vaidya' is used without defining 'linear'. Please clarify that it means the first-order-in-A truncation of the Vaidya metric.
  4. [Sec. VI C] The statement that Ξ/|A| 'exhibits a jump across w=−1' is imprecise; it is the sign of Ξ/|A| that flips, while the magnitude may be continuous. Please rephrase.
  5. [Appendix C, Eq. (C11)] The potential in the cutoff region is shown to develop peaks (Figs. 25–26). It would be helpful to state explicitly which terms in the master equation become discontinuous or stiff for the cutoff model, to aid the reader in assessing the reliability of the numerical results there.

Circularity Check

0 steps flagged

No significant circularity: Ξ and à are extracted observables, not fitted; the derivation is self-contained and the paper's own caveats are validity limitations, not circular reductions.

full rationale

The derivation chain is self-contained. Sections II–IV construct the master equation and source term from the background Einstein equations and fluid equations of motion; Section V defines Ξ and à purely as extraction rules from the time-domain waveform, with no free parameter fitted to enforce the reported values. The background parameters (A, w, F) are inputs, and Tables II–III verify Ξ/|A| and Ã/A at two accretion rates, as well as the vacuum and Vaidya limits; the Vaidya background is recomputed in the same double-null code rather than assumed from the authors' prior work. The only self-citations ([21] for the Vaidya benchmark, [33–35] for the steady perfect-fluid equations) are contextual: the fluid equations are re-derived in Eqs. (44)–(50), and the Vaidya comparison is regenerated numerically. No uniqueness theorem, fitted ansatz, or calibrated constant is imported from those references. The paper explicitly flags its main physical limitation in Sec. VI A ('the influence of such regions is expected to be highly suppressed') and Appendix C, which shows that introducing a cutoff creates potential peaks and contaminates both Ξ and Ã; this is a correctness/validity caveat about outer-region backscatter, not a circular reduction, because the claimed output is not equivalent to any input by construction. There is therefore no circular step to report.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The central claim rests on the linear-in-accretion background construction, the perfect-fluid steady-accretion model, the choice to set the odd-parity fluid perturbation to zero, the adiabatic frequency extraction, and the untested locality assumption about the outer dilute-breaking region. The free parameters are the physical inputs (A, w, F) plus the observer radius and analysis choices; none are fitted to the target observable. No new physical entities are postulated.

free parameters (6)
  • A (accretion rate) = 3e-5 and 3e-6 (both signs)
    Central model input; sets the rate of background mass growth. The paper verifies Ξ ∝ A using two values; not fitted to the output.
  • w (equation of state, p = wρ) = sampled over -2 ≤ w ≤ 2, e.g. 1.1, 1/3, 0, -1/3, -1.5
    Fluid equation-of-state parameter; the mapping Ξ/|A| vs w is the paper's main result, and the allowed F ranges depend on w.
  • F (accretion integration constant) = F_saddle, 1/4, 0.02, 0.25, 1, etc.
    Determines the radial density/velocity profile of the accreting fluid; the paper scans F at fixed w and reports the resulting Ξ/|A|.
  • r_obs (observer extraction radius) = 10, 20, 30 M0
    The reported Ξ/|A| depends significantly on r_obs in a way that is EoS-independent but not extrapolated to infinity; this choice is required to quote numbers and is a limitation for observational use.
  • Initial Gaussian pulse parameters (amplitude, center, width) = A=0.1, r_c=2.5M0, s=0.1M0
    Initial data for the GW perturbation; standard ringdown extraction should be insensitive to them, but the paper does not demonstrate that insensitivity for all reported cases.
  • Time-averaging window for Ξ and à = eight earliest peaks with V > V_peak0 + 100M0, with N→N/2 Richardson extrapolation
    Analysis choice used to define the quoted time averages; the window length and number of peaks affect the error estimates.
axioms (6)
  • domain assumption The accreting background is adequately described to first order in the accretion rate κ; O(κ²) terms do not affect the leading ringdown (Sec. III A, Eq. 26).
    Standard perturbation scheme borrowed from Babichev et al. [22]; the paper never computes O(κ²) corrections, so the regime of validity ends where the dilute condition fails.
  • domain assumption The accretion is steady and spherically symmetric, with constant A = M,V and EoS p = wρ; the background is obtained from solving Eqs. (50) and (48) (Sec. III C).
    The central results (Ξ ∝ A, à ≈ A) assume steady accretion; the paper explicitly notes that globally steady accretion is required and that otherwise à ≠ A.
  • ad hoc to paper The odd-parity fluid perturbation δu3 is set to zero on the initial surface and remains zero because its equation of motion is homogeneous (Eq. 66, Sec. IV B).
    Choice of initial data; the paper does not analyze excited fluid perturbations, only the purely tensorial mode.
  • domain assumption The ringdown frequency varies slowly enough that instantaneous ω_R and ω_I can be read off from adjacent peaks via Eq. (88).
    Standard adiabatic extraction for time-dependent backgrounds; not rigorously justified for all parameter choices, but supported by the observed slow drift.
  • ad hoc to paper The influence of the region where the dilute condition fails (r ≳ 10^2 M0 for |A|=3e-5) on the waveform at r_obs is negligible (Sec. VI A).
    Load-bearing locality assumption; Appendix C shows cutoffs create potential peaks that contaminate Ξ and Ã, so the assumption is non-trivial and untested.
  • domain assumption The hierarchy 1 ≫ κ ≫ ε permits dropping ε² and higher-order terms (Sec. III A).
    Standard double-expansion; the paper notes at the end that the more practical regime κ∼ε would require second-order terms (Refs. [40,41]).

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of Odd-parity ringdown gravitational waves of a spherically symmetric black hole with perfect fluid accretion." pith.science (2026). https://pith.science/paper/STLD3S2T

@misc{pith2026260703231,
  author       = {Pith},
  title        = {Pith review of: Odd-parity ringdown gravitational waves of a spherically symmetric black hole with perfect fluid accretion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/STLD3S2T}},
  note         = {Machine review of arXiv:2607.03231}
}
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read the original abstract

The ringdown waves from a black hole offer a clean probe of strong-field gravity, but a matter distribution that may be present around a realistic black hole renders the background spacetime dynamical and the ringdown frequencies time-dependent. We study the odd-parity ringdown of a Schwarzschild black hole that grows through the dilute, steady, spherically symmetric accretion of a perfect fluid. Working to first order in the accretion rate, we compute the ringdown waveform directly in the time domain on this dynamical background. Since the odd-parity matter perturbation decouples from the metric perturbation, the wave mode can be described by a purely tensorial mode on the accreting background. In particular, the ratio of the imaginary to the real part of the frequency cancels both the secular variation caused by the growth of the black hole and the redshift factor, so that its deviation from the Schwarzschild value purely reflects the surrounding environment. The time dependence of the frequency, on the other hand, reflects the accretion rate and allows us to define a second observable tied to it. We argue that measuring these observables across multiple modes may provide significant information to constrain the surrounding environment of the black hole.

Figures

Figures reproduced from arXiv: 2607.03231 by Chul-Moon Yoo, Hiroyuki Nakano, Kota Ogasawara, Rikuto Ohashi, Yasutaka Koga.

Figure 1
Figure 1. Figure 1: FIG. 1. Contour plots of [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. BG functions [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Frequencies extracted from the waveform at [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Time dependence of Ξ at [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Time dependence of [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Behavior of the shift in the vacuum case: varying the number of grid points at [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Layout of grid points. The blue-shaded region in the left panel represents the computational domain. Grid points [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Specifically, after obtaining x(U, Vn) from x(U, Vn−1) and x(U0, Vn) for the n-th step V = Vn, we redefine the U coordinate by r(U, Vn) = rmax(Vn) − (rmax(Vn) − rmin) U − U0 Umax − U0 . (A10) Here, rmax is taken on U = U0 as in [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Schematic of the [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: plots Ξ/|A| for w > 1. In this case, ρ∞ is monotonically decreasing in F, and the dilute condition is always satisfied for F ≥ Fsaddle (see [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Magnified view of a portion of Fig. 14 for [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p030_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20 [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Schematic showing how the cutoff is implemented. The accretion region with [PITH_FULL_IMAGE:figures/full_fig_p031_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. BG functions [PITH_FULL_IMAGE:figures/full_fig_p033_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Time dependence of Ξ [PITH_FULL_IMAGE:figures/full_fig_p033_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Potential-like factor [PITH_FULL_IMAGE:figures/full_fig_p034_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. Difference [PITH_FULL_IMAGE:figures/full_fig_p034_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. Difference [PITH_FULL_IMAGE:figures/full_fig_p035_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27. Time dependence of Ξ [PITH_FULL_IMAGE:figures/full_fig_p035_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: FIG. 28. Time dependence of [PITH_FULL_IMAGE:figures/full_fig_p036_28.png] view at source ↗

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Reference graph

Works this paper leans on

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    In general, when solving an equation with the form of Eq

    Double null formalism The numerical method used in this work, the DNF, is known as a scheme that can solve the master equation on DN coordinates with second-order accuracy [27]. In general, when solving an equation with the form of Eq. (67), it is sufficient to have the initial values ofxonU=U 0, V0 ≤V≤V max andV=V 0, U0 ≤U≤U max, together with a method t...

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    Result As before, we solve the master equation numerically. The parameters are A0 :=A (old) =±3×10 −5 , (Mfin, τM ) = (1.0075M0,0.0015M 0), robs = 100M0 , (U0, Umax) = (V0, Vmax) = (0,400M 0), (C8) with the remaining parameters following Eq. (87). First, the numerical solutions of the BG functions at the initial time are presented in Fig. 22. Next, the nu...

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This paper was first reviewed by deepseek-v4-flash on August 2, 2026.