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Two non-asymptotic bounds govern CARD conditional diffusion: the Wasserstein-2 gap between generated and target conditionals is bounded by integrated score error plus endpoint mismatch, and the score error has an explicit Taylor rate.

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CARD's generated conditional distribution is shown to converge in Wasserstein distance to the true conditional distribution, with a separate score-estimation error bound controlled by network resolution and distribution smoothness.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection The W2 reduction for CARD is a real but narrow adaptation of Kwon et al.; the paper's central claim that Y|f_phi(X) approximates Y|X is unsupported, and Theorem 2 rests on a missing lemma. the 3 major comments →

arxiv 2508.10944 v1 pith:KPPPOCS2 submitted 2025-08-13 stat.ML cs.LG

Non-asymptotic convergence bound of conditional diffusion models

Mengze Li This is my paper
classification stat.ML cs.LG MSC 49Q2260H1060J6062G07
keywords conditional diffusion modelsnon-asymptotic convergenceWasserstein distancescore function estimationCARDFokker-Planck equationstochastic differential equationsTaylor approximation
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

The paper presents non-asymptotic convergence guarantees for CARD, a classification-and-regression diffusion model that conditions generation on a pretrained network $f_\phi(x)$ estimating the conditional mean $E[Y\mid X]$ — the first such bounds for this model. The headline result, Theorem 1, bounds the second-order Wasserstein distance between the true and generated conditional distributions by a time integral of the score-estimation error $H(t)$ plus the mismatch between endpoint noise distributions; when the score loss vanishes and the endpoints agree, the generated distribution provably converges to the target. Theorem 2 then controls $H(t)$ itself: under Hölder smoothness, a light-tailed conditional density, and truncation of both the data range and low-density regions, the integrated squared error between the network score and the true conditional score has an explicit rate in the Taylor-grid size $N$, the dimension $d$, and the smoothness $\beta$. A reader should care because these are quantitative statements of the working belief that 'diffusion works when the score is learned well,' delivered without log-concavity or strong smoothness of the target — and the price is visible: the bounds control $Y\mid f_\phi(x)$, and the intended $Y\mid X$ statement rests on the pretrained mean being nearly sufficient.

Core claim

The paper's central claim: CARD's sampling error is a score term plus an endpoint term. Theorem 1 gives $W_2(q(y_0\mid f_\phi(x)),p(y_0\mid f_\phi(x)))\le \int_0^T \beta_t M(t)\sqrt{H(t)}\,dt+M(T)W_2(q(y_T\mid f_\phi(x)),p(y_T\mid f_\phi(x)))$, with $M(t)$ a Grönwall factor, $H(t)$ the mean squared score error; vanishing loss plus endpoint gap implies convergence. Theorem 2 bounds $H(t)$: for a $\beta$-Hölder light-tailed conditional density with bounded $f_\phi(x)$ and truncated domains, the integrated squared score error is $O\big((\sqrt{\log\varepsilon^{-1}}+1)(N^{-\beta}/\gamma_t^{d/2}+N^{-d}/\sigma_t^d+N^{-d-\beta}/\sigma_t^d)\big)$, with $N$ grid divisions per dimension.

What carries the argument

Two mechanisms carry the argument. First, the Fokker–Planck equations of the forward and reverse CARD SDEs become continuity equations on probability space, and the time derivative of the Wasserstein-2 distance between their solutions is bounded by the optimal-transport contraction formula; the one-sided Lipschitz constants of drift and score estimator feed the Grönwall factor $M(t)=\exp(\int_0^t(l_1(s)+l_2(s)\beta_s)ds)$, yielding Theorem 1. Second, Theorem 2 approximates the diffused conditional density — a Gaussian convolution of the original conditional density, with mean $\gamma_t y_0+(1-\gamma_t)f_\phi(x)$ and variance $\sigma_t^2 I$ — by piecewise Taylor polynomials of the density and

Load-bearing premise

The load-bearing premise is that conditioning on the predicted mean $f_\phi(x)$ is as informative as conditioning on $x$ once $f_\phi(x)\approx E[Y\mid X]$; for a response whose conditional mean is not a sufficient statistic — a symmetric two-mode mixture, say — that identification can fail even with a perfect score estimator, and every bound here is stated for $Y\mid f_\phi(x)$, not $Y\mid x$.

What would settle it

Two checks. (1) On a target $Y\mid X$ that is a symmetric two-mode mixture (so the conditional mean sits between modes), train CARD with an accurate score estimator for $Y\mid f_\phi(X)$: if generated samples miss a mode while $L_1\to 0$, the Assumption 4 bridge — not Theorem 1's arithmetic — failed, since the theorem controls $Y\mid f_\phi(x)$, not $Y\mid x$. (2) Equation (20) fixes the slope of $\log W_2$ vs $\log L_1$ at $1/2$; on a heavy-tailed conditional violating the light-tail/truncation assumptions, a flattened slope or violated bound would localize where Theorem 2's rate breaks.

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

If this is right

  • If Theorem 1 is correct, CARD's generated conditional distribution converges to the target in Wasserstein-2 whenever the score-matching loss $L_1(\phi,\theta,\lambda)$ and the endpoint gap $W_2(q(y_T\mid f_\phi(x)),p(y_T\mid f_\phi(x)))$ both vanish — generation quality is governed by the two quantities the model actually trains.
  • Corollary 1 upgrades the bound to $\log W_2 \le \tfrac12\log(2\int_0^T\beta_t M^2(t)\,dt)+\tfrac12\log L_1$ when $\lambda(t)=\beta_t$; the paper's simulations on Moon, Circle, Gaussian, and Gaussian-mixture data confirm the upper bound stays above the measured Wasserstein distances throughout training.
  • Because the Fokker-Planck derivation carries two covariates $(y_t,f_\phi(x))$, the bound covers a generalized score function with two covariates, extending the theory from scalar to multivariate conditional-generation settings.
  • Theorem 2's rate $O\big((\sqrt{\log\varepsilon^{-1}}+1)(N^{-\beta}/\gamma_t^{d/2}+N^{-d}/\sigma_t^d+N^{-d-\beta}/\sigma_t^d)\big)$ makes the score error explicit in the grid size $N$, the Hölder smoothness $\beta$, and time $t$; since $\sigma_t\to 0$ as $t\to 0$, the bound is only meaningful away from time zero, and the paper itself notes a threshold step $t_0$ is required.
  • The approximating score is realized by a concrete ReLU network (the clipped ratio $h_4/(\sigma_t h_{1,\mathrm{clip}})$), so the non-asymptotic guarantee applies to an architecture that can be trained, not only to an idealized estimator.

Where Pith is reading between the lines

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

  • My inference: the $Y\mid X$ framing in the abstract is an extrapolation from what is proven; the theorems only control $Y\mid f_\phi(x)$, and on symmetric-multimodal conditionals — where $E[Y\mid X]$ destroys mode information — the bounds could hold while generation conditioned on $x$ badly misses modes. A sufficient-statistic surrogate (learned representation, quantiles) is the natural fix to tes
  • My inference: the slope-$1/2$ law $\log W_2 \le \text{const}+\tfrac12\log L_1$ is falsifiable per dataset; measuring the empirical slope on heavy-tailed conditionals (where the truncation assumptions fail) would show where the theory's rate degrades, giving practitioners a cheap diagnostic.
  • My inference: the paper leaves optimality untouched; comparing the $N^{-d-\beta}$ term against known minimax rates for conditional density estimation under Hölder classes would reveal whether the Taylor construction is sharp in dimension.
  • My inference: the H1/H2 truncation terms depend on the radius $R\sim c\sqrt{\log\varepsilon^{-1}}$ and the threshold $\epsilon_{\mathrm{low}}$; tracking $H_1+H_2$ separately during training on outlier-prone data would show whether the bound is dominated by out-of-domain or by low-density errors — a split the paper's simulation plots do not isolate.
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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. The paper studies Classification and Regression Diffusion Models (CARD), where a pre-trained conditional-mean estimator f_phi(x) is injected into the forward and reverse diffusion processes. The main theoretical contributions are: (i) an SDE/Fokker-Planck formulation of CARD; (ii) Theorem 1, a non-asymptotic bound on the second-order Wasserstein distance W2(q(y0|f_phi(x)), p(y0|f_phi(x))) in terms of an integrated conditional score-estimation error H(t) plus an endpoint mismatch term; (iii) Corollary 1, a Cauchy-Schwarz reformulation in terms of the score-matching loss; and (iv) Theorem 2, a non-asymptotic bound on the conditional score-estimation error under Hölder smoothness, light-tailedness, and truncation assumptions, proved via local Taylor polynomials. Numerical experiments on four 2-D datasets plot the Wasserstein distance against the loss to illustrate the bound. The derivation of Theorem 1 is structurally standard, but the paper's advertised claim about generating from Y|X is not actually proved, because all theoretical statements are about the surrogate conditioning variable f_phi(x). The proof of Theorem 2 also rests on an unproved truncation assumption and contains a time-dependence inconsistency.

Significance. If Theorem 1 is read literally as a statement about the conditional distribution given f_phi(x), it is a clean and potentially useful reduction: it separates the score-estimation error, measured by H(t), from the endpoint error, and it does not require log-concavity or other strong structural assumptions on the target. The attempt in Theorem 2 to control H(t) under Hölder smoothness and light-tail assumptions is also a legitimate direction, and the authors correctly identify that the practical loss L2 differs from the theoretical L1. However, the central advertised result --- that CARD generates from the true conditional distribution Y|X --- is not established. Assumption 4 only gives convergence of f_phi(X) to E[Y|X], and the conditional mean is not a sufficient statistic in general. Thus the Wasserstein bound in Eq. (14) controls the wrong conditioning variable relative to the abstract's claim. The proof of Theorem 2 is also not self-contained: Assumption 9 is asserted rather than proved, and the final bound in Eq. (19) has the opposite time dependence from what the text claims. These issues are load-bearing, so the manuscript requires substantial revision before the results can

major comments (3)
  1. [Section 3.2, Assumption 4 and Theorem 1 (Eq. 14)] Assumption 4 only asserts that f_phi(X) converges to E[Y|X] in probability. The paragraph following it states that when f_phi(X) performs well, Y|f_phi(X) 'closely approximates' Y|X, but no theorem or lemma bounds W2(q(y|x), q(y|f_phi(x))) in terms of ||f_phi(x) - E[Y|X]||. This implication is false in general because the conditional mean is not a sufficient statistic. For example, let X=(X1,X2) with X2~Bernoulli(1/2), X1 independent of Y, and Y|X2=1 ~ N(0,25), Y|X2=0 ~ N(0,1). Then E[Y|X]=0 for every x, so Assumption 4 holds exactly with f_phi(X)=0, yet q(y|X2=1)=N(0,25) while q(y|f_phi(X)=0)=0.5 N(0,1)+0.5 N(0,25), and the two distributions have strictly positive Wasserstein distance. Consequently, Theorem 1 and Corollary 1 bound only W2(q(y0|f_phi(x)), p(y0|f_phi(x))), not W2(q(y0|x), p(y0|f_phi(x))). A triangle-inequality term for the covariate conditioning gap is missing; without it
  2. [Appendix B.2, Assumption 9 (Eqs. 49-50)] Assumption 9 asserts that the integrals of q(y0|f_phi(x)) phi(yt|y0,f_phi(x)) and of the gradient-weighted versions over the complement of the truncation domain D0 are bounded by epsilon. This assumption is load-bearing for the proof of Theorem 2: it is used in Lemma 3 (Appendix B.3.2) to control the score-function bound, and it ultimately enters the H1 and H2 estimates in Appendix B.3.3. However, it is introduced with the sentence 'From the inspiration of the Lemma A.8 in XXX, we may as well obtain Assumption 9', and no proof or reference is supplied. The text in Section B.2 says the error 'could be controlled' but gives no argument. Since Theorem 2's upper bound in Eq. (19) depends critically on these inequalities, the score-estimation result is not self-contained. The authors need either to prove Eqs. (49)-(50) from the preceding assumptions or to provide a complete reference.
  3. [Theorem 2, Eq. (19), and Appendix B.4.4] The claimed upper bound in Eq. (19) contains the term N^{-beta} / gamma_t^{d/2}, where gamma_t = e^{-beta t/2}. For fixed N and beta>0, this term equals N^{-beta} e^{beta t d/4}, which grows exponentially in t. Thus the right-hand side of Eq. (19) diverges as t increases, for example as t -> infinity. The surrounding text states that 'the approximation error increases as time t decreases' and that 'we consider the error upper bound when t is large', which is the opposite of what the displayed formula implies. At a minimum, the theorem and proof must specify a quantitative regime for t (for example, an upper bound on t or a lower bound on gamma_t) for the bound to be meaningful, and the discussion must be corrected. Without this, the stated non-asymptotic convergence guarantee in Theorem 2 is not supported.
minor comments (5)
  1. [Corollary 1, Eqs. (16)-(17)] The displayed formulas contain corrupted symbols such as '⌟roo⟪⟪op' and the square-root notation is mangled. Please regenerate these equations and verify the typesetting.
  2. [Appendix A.1, reverse SDE derivation] The derivation of the reverse SDE is written in a way that is hard to follow: the quantities HH1, HH2, HH3 are introduced without clear definitions, and several limit steps are stated informally. A cleaner presentation with explicit definitions of the remainder terms would help the reader check the argument.
  3. [Section 5] The text says that the loss used in simulations is L2 (Eq. (18)), but the plots are described as showing L1(phi,theta,lambda). Please clarify which quantity is actually computed and plotted, and make the notation consistent.
  4. [Notation and typos] There are numerous typos and inconsistencies: 'Futhermore', 'Firstlt', 'detrive', 'fractional function' for score function, and inconsistent use of f_phi(x) vs fphi(x). A careful proofreading pass is needed.
  5. [Assumptions section] Assumption 9 is stated in the appendix rather than in the main Assumptions section (Section 4.1). Since it is used in the proof of Theorem 2, it should be listed in the main assumptions or explicitly cross-referenced from Section 4.1.

Circularity Check

0 steps flagged

No significant circularity: the main bounds reduce sampling error to score-estimation error under explicit assumptions; weaknesses are correctness gaps, not circular reductions.

full rationale

Theorem 1 (Eq. 14) is a genuine reduction: it bounds W2(q(y0|f_phi(x)), p(y0|f_phi(x))) by ∫ β_t M(t)√H(t) dt plus an endpoint mismatch term, where H(t) is the mean squared score-estimation error with respect to the forward conditional density. This is not a tautology: H(t) is a training objective that must be controlled, and the bound has independent content as a standard Fokker–Planck/optimal-transport argument adapted to the CARD drift. Theorem 2 then gives an approximation-theoretic bound on H(t) under explicit Hölder, light-tail, truncation, and density-lower-bound assumptions. No fitted parameter is renamed as a prediction, no load-bearing self-citation chain appears (the paper cites external prior work such as Kwon et al.), and the construction of the network estimator in Eq. (72) is a standard existence/approximation argument rather than a circular fit. The paper's abstract-level claim that Y|f_phi(X) approximates Y|X when f_phi(X) approximates E[Y|X] is not proved by any theorem — the theorems concern q(·|f_phi(x)), not q(·|x) — but this is an unsupported bridge, not a circular derivation. Similarly, Assumption 9 is introduced as an explicit tail/truncation assumption (with an unnamed 'Lemma A.8 in XXX' reference) and the proof uses it as a hypothesis; this is an assumption, not an equation that reduces the theorem to itself. There is also an apparent technical mismatch in the proof of Theorem 1 (H(t) is defined with E_{π_t} in Eq. 37 but with E_{q(y_t|f_phi(x))} in the theorem statement), but that is a correctness/proof gap, not circularity. Overall, the derivation chain is not circular; the paper's central claims either follow from the stated assumptions or are explicitly conditional on them.

Axiom & Free-Parameter Ledger

5 free parameters · 10 axioms · 0 invented entities

No new particles, forces, or mediators are introduced. The quantitative load is carried by the error parameters listed above, plus the technical assumptions that restrict the data distribution, the score estimator, and the truncation domains.

free parameters (5)
  • Truncation radius R
    Chosen in Assumption 7 to clip yt to an l_infinity ball; all tail integrals in Theorem 2 depend on R, and the proof only requires R > 1.
  • Density threshold epsilon_low
    Introduced in Assumption 8 to keep q(yt|fphi(x)) away from zero; controls H2 and Lemma 5.
  • Condition truncation radius Rf
    Chosen in Assumption 9 and Lemma 6 to clip fphi(x); controls error from unbounded conditions.
  • Truncation error epsilon
    Small parameter in (0,1/e) controlling clipped integrals (Assumption 9) and Taylor remainder orders; chosen by the proof.
  • Taylor grid resolution N
    Number of intervals per dimension in the conditional diffused local Taylor polynomial (Appendix B.4.3); it is the network-size parameter in the Theorem 2 rate.
axioms (10)
  • standard math Ito's lemma and Fokker-Planck equation for SDEs with drift -beta/2(y-fphi(x)) and diffusion sqrt(beta)
    Used in Appendix A.2 to derive Eq. (8).
  • standard math Wasserstein contraction inequality: d/dt W2^2(qt,pt) = 2 E_pi_t[(yt-zt)(v[pt](yt)-v[pt](zt))]
    Eq. (35), quoted from optimal transport (Ambrosio et al. 2005, Santambrogio 2015), foundation of Theorem 1.
  • domain assumption Lipschitz drift, bounded diffusion coefficient, one-sided Lipschitz score estimator (Assumptions 1-3)
    Needed for existence/uniqueness of SDE solutions and for the Gronwall-type integration in Theorem 1.
  • ad hoc to paper Assumption 4: fphi(X) -> E[Y|X] implies Y|fphi(X) approximates Y|X
    States the bridge from covariate conditioning to conditional-mean conditioning; no proof or reference given, and it fails for general distributions.
  • domain assumption Assumption 5: q(y0|fphi(x)) is Holder beta and light-tailed sub-Gaussian
    Used in Lemmas 1, 2 and the Taylor approximation error (Appendix B).
  • domain assumption Assumption 6: distribution of fphi(x) is light-tailed with ||fphi(x)|| <= c
    Bounds condition variable fphi(x) and enables truncated expectations over fphi(x) (Lemma 7).
  • domain assumption Assumptions 7-8: truncation of yt to radius R and density threshold epsilon_low
    Define domains D1, D2 used to split H(t) into H1+H2+H3.
  • ad hoc to paper Assumption 9: integrals over truncation domain bounded by epsilon (Eqs. 49-50)
    Load-bearing control for the score bound; introduced in Lemma 3 proof as 'obtained' from a missing 'Lemma A.8 in XXX', not derived in the paper.
  • domain assumption Constant diffusion coefficient beta in Gaussian representation gamma_t = exp(-beta t/2), sigma_t^2 = 1 - exp(-beta t)
    Appendix B.1 derives q(y|y0,fphi) assuming beta is constant; CARD's beta_t is time-dependent, so Theorem 2 either needs this extra assumption or a re-derivation.
  • domain assumption Sub-Gaussian tail bound P(||yt||_infinity >= m) <= c6 exp(-m^2/(2 sigma_t^2))
    Introduced in Lemma 7 proof to bound moments of yt; not derived from Assumptions 5-6.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of Non-asymptotic convergence bound of conditional diffusion models." pith.science (2026). https://pith.science/paper/KPPPOCS2

@misc{pith2026250810944,
  author       = {Pith},
  title        = {Pith review of: Non-asymptotic convergence bound of conditional diffusion models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPPPOCS2}},
  note         = {Machine review of arXiv:2508.10944}
}
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read the original abstract

Learning and generating various types of data based on conditional diffusion models has been a research hotspot in recent years. Although conditional diffusion models have made considerable progress in improving acceleration algorithms and enhancing generation quality, the lack of non-asymptotic properties has hindered theoretical research. To address this gap, we focus on a conditional diffusion model within the domains of classification and regression (CARD), which aims to learn the original distribution with given input x (denoted as Y|X). It innovatively integrates a pre-trained model f_{\phi}(x) into the original diffusion model framework, allowing it to precisely capture the original conditional distribution given f (expressed as Y|f_{\phi}(x)). Remarkably, when f_{\phi}(x) performs satisfactorily, Y|f_{\phi}(x) closely approximates Y|X. Theoretically, we deduce the stochastic differential equations of CARD and establish its generalized form predicated on the Fokker-Planck equation, thereby erecting a firm theoretical foundation for analysis. Mainly under the Lipschitz assumptions, we utilize the second-order Wasserstein distance to demonstrate the upper error bound between the original and the generated conditional distributions. Additionally, by appending assumptions such as light-tailedness to the original distribution, we derive the convergence upper bound between the true value analogous to the score function and the corresponding network-estimated value.

Figures

Figures reproduced from arXiv: 2508.10944 by Mengze Li.

Figure 1
Figure 1. Figure 1: Sample distribution of four datasets. The four subfigures exhibit the sample distributions of various datasets. The leftmost subfigure portrays the Moon dataset, modeling a quintessential nonlinear binary classification challenge. The second subfigure introduces the Circle dataset, where two concentric circles represent two distinct classes and are inherently nonlinearly separable in the feature space. The… view at source ↗
Figure 2
Figure 2. Figure 2: Generating sample distributions of four datasets. In each subfigure, the blue sample points represent the original sample distribution, and the green sample points represent the corresponding sample distribution generated by CARD. (a) Moon (b) Circle (c) N(0, 0.1I) (d) N((±0.5,±0.5) T , 0.01I) [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Log-plots about Wasserstein distances and loss function of four datasets. The horizontal axis represents the logarithmic form of the Wasserstein distance between the given sample and the final generated sample, denoted by log W2(q(y0∣fϕ(x)), p(y0∣fϕ(x))). The vertical axis represents the logarithm of the loss function during training, denoted by log L1(ϕ, θ, λ). The red line represents the theoretical uppe… view at source ↗
Figure 3
Figure 3. Figure 3: Product term of Wasserstein distances for four datasets. In each subfigure, the hor￾izontal axis represents the diffusion time step, and the vertical axis represents the change in variable M(t)W2(q(yt∣fϕ(x)), p(yt∣fϕ(x))). 16 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Trends in four datasets with respect to Wasserstein distance and its product term. The horizontal axis shows the diffusion time step t, and the vertical axis illustrates the evolution of Wasserstein distance W2(q(yt∣fϕ(x)), p(yt∣fϕ(x))), as well as the product of Wasserstein distance with the integral factor M(t), written as M(t)W2(q(yT ∣fϕ(x)), p(yt∣fϕ(x))). (a) Moon (b) Circle (c) N(0, 0.1I) (d) N((±0.5,… view at source ↗
Figure 6
Figure 6. Figure 6: Log Wasserstein Distances plots for Four Datasets. The subfigures from left to right repre￾sent Moon, Circle, zero-mean Gaussian, four-class Gaussian datasets in turn. The figures above illustrate the evolution of the logarithmic Wasserstein distance log W2(q(yt∣fϕ(x)), p(yt∣fϕ(x))) as the training iterations progress for various four datasets. (a) Moon (b) Circle (c) N(0, 0.1I) (d) N((±0.5,±0.5) T , 0.01I… view at source ↗
Figure 7
Figure 7. Figure 7: Log loss function plots for Four Datasets. The subfigures from left to right represent Moon, Circle, zero-mean Gaussian, four-class Gaussian datasets in turn. The figures above illustrate the evolution of the logarithmic Wasserstein distance log L1(ϕ, θ, λ) as the training iterations progress for various four datasets. 45 [PITH_FULL_IMAGE:figures/full_fig_p045_7.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.