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REVIEW 4 major objections 5 minor 73 references

Learning collective variables that respect permutational symmetry

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Sorting atom descriptors before manifold learning and autoencoding yields collective variables that respect permutational symmetry, and the reduced-model committor used as a reaction coordinate reproduces brute-force transition rates.

desk verdict Useful, honest engineering for permutationally symmetric CVs, but the headline overclaims: the LJ8 3D success skips the autoencoder step and the rate agreement is mainly kA. read the letter →

arxiv 2507.00408 v1 pith:HY726WVN submitted 2025-07-01 physics.chem-ph cs.NAmath.NA

Jiaxin Yuan , Shashank Sule , Yeuk Yin Lam , Maria Cameron This is my paper · ORCID
classification physics.chem-phcs.NAmath.NA PACS 82.20.Db05.10.Gg
keywords collectivevariablespermutationalsymmetrycoarse-grainingautoencodersdiffusionmapsLennard-Jonesclusterscommittorforwardfluxsampling
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 tries to establish a workable route from raw atomic coordinates to low-dimensional collective variables for clusters of identical particles, where swapping atoms leaves the system unchanged. Its proposal is to sort a symmetry-invariant descriptor—pairwise distances squared or coordination numbers—so that every permutation of a configuration maps to the same feature vector, learn the residence manifold in that feature space, and train an autoencoder whose loss enforces the Legoll–Lelièvre orthogonality condition between the collective-variable level sets and that manifold. The reduced model's committor, the hitting probability of one metastable state before the other, is then lifted back to atomic coordinates and used as the reaction coordinate for forward flux sampling. The paper reports that for Lennard-Jones-7 in 2D and Lennard-Jones-8 in 3D the resulting transition rates and residence times match brute-force all-atom simulation, which matters because it offers a way to estimate rare-event rates from cheap reduced models and to visualize metastable states in two dimensions.

What carries the argument

The load-bearing identity is the orthogonality condition (3.19), $D\xi(x)\nabla V_1(x) = 0$, which demands that the level sets of the collective variable be orthogonal to the residence manifold where the invariant measure concentrates; Legoll and Lelièvre (2010) and Duong et al. (2018) proved that this condition suppresses the relative-entropy bound (3.17) measuring the gap between the reduced dynamics and the projected original dynamics. The paper turns this analytic condition into a training term in the autoencoder loss (4.6), alongside reconstruction and linear-independence terms. The complementary mechanism is the rate-versus-committor identity of Zhang, Hartmann, and Schütte (2016): the transition rate in any collective-variable space is never smaller than the true rate, with equality when the collective variable is the committor, which is why the reduced-model committor, not the raw learned variables, is lifted to atomic coordinates and used as the reaction coordinate for forward flux sampling and for the control $2\beta^{-1}\nabla\log q(x)$ for sampling transition paths.

What would settle it

Extend the brute-force all-atom estimates with about ten times more statistics in the channel where the paper already reports the worst agreement—the back-rate $k_B$ for LJ7 at $\beta = 7$ with the learned CVs, where the FFS and brute-force means differ by roughly a factor of two, and the $\nu_{AB}$ channel for LJ8 at $\beta = 20$, where the FFS mean is about three times smaller than the brute-force mean with comparatively tight error bars. If either discrepancy persists beyond three standard deviations at full statistics, the claim that reduced-model rates agree with brute force is false for that channel.

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Extended reading notes

Core claim

The paper's central claim is that Algorithm 1—sort-based featurization, residence-manifold learning by diffusion maps and a diffusion net, and an autoencoder whose loss contains an orthogonality term—produces collective variables that respect translational, rotational, and permutational symmetry, and that the committor of the reduced model in those variables, composed with the learned features to give $q(x) = \tilde{q}(\xi(\varphi(x)))$, is a good reaction coordinate for forward flux sampling and for the stochastic control used to sample transition paths. For both case studies the reported escape rates $k_A$ and the transition rates $\nu_{AB}$ obtained with the learned variables agree with brute-force all-atom results within statistical error; the standard coordination-moment variables $(\mu_2, \mu_3)$ fare comparably, while linear-discriminant-analysis variables show notably larger rate errors. The paper further claims that its proposed feature map, the sorted vector of coordination numbers $\mathrm{sort}[c_s]$, separates the relevant metastable basins in both systems, whereas the sorted squared-distance map $\mathrm{sort}[d^2]$ works for LJ7 in 2D but fails to separate the two main minima of LJ8 in 3D.

Load-bearing premise

The theoretical guarantee behind the training loss is proven only for smooth feature maps, while the sorted feature maps actually used are only piecewise smooth, so the orthogonality condition the loss enforces is not rigorously justified for the true inputs, a limitation the paper itself acknowledges in Section 4.1.

Editorial extensions

If this is right

  • Rate estimation for permutationally symmetric clusters can be done with forward flux sampling driven by a two-dimensional reduced-model committor instead of an all-atom committor, and the reported $k_A$ and $\nu_{AB}$ values match brute-force all-atom estimates within the combined statistical error bars at essentially every temperature tested.
  • The $\mathrm{sort}[c_s]$ feature map separates the free-energy basins of the relevant minima in both LJ7 in 2D and LJ8 in 3D, so the learned collective variables can serve both to define metastable sets and to visualize where transition trajectories concentrate.
  • The same lifted committor defines an approximately optimal control for the transition path process, allowing the bias vector $2\beta^{-1}\nabla\log q(x)$ to be added to the drift so that reactive trajectories can be sampled in bulk and their probability density estimated by binning in the CV space.
  • Because the reduced-model rates follow Arrhenius behavior across $\beta = 5,7,9$ for LJ7 and $\beta = 10,15,20$ for LJ8, in agreement with brute force, the framework inherits the temperature dependence of the underlying dynamics rather than imposing an ad hoc one.

Reading between the lines

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

  • The sorting trick is generic: any permutation-invariant ordering of a symmetric descriptor collapses the symmetry group, so the same pipeline should transfer to other symmetric clusters, and the paper's own named next target, LJ38 with its double-funnel landscape, is a direct test of whether $\mathrm{sort}[c_s]$-style features separate competing funnels as cleanly as they separate the LJ8 minima.
  • The piecewise-smoothness gap suggests a discriminating experiment the authors did not run: replace $\mathrm{sort}[d^2]$ and $\mathrm{sort}[c_s]$ with a smooth permutation-invariant embedding, for instance an equivariant graph-network descriptor, and re-run the pipeline; if rates stay the same, the orthogonality loss's theoretical basis is incidental to the empirical success, and if they change, th
  • A weaker reading of the paper's message is that the orthogonality loss does not have to preserve rates directly; it only has to produce collective variables whose level-set foliation lets the lifted committor approximate the true committor closely enough for interface-crossing probabilities, and measuring the overlap of the reduced-model committor's level sets with true isocommittor surfaces for L
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a framework, Algorithm 1, for learning collective variables that are invariant under translation, rotation, and permutation of identical particles, by combining sorted feature maps, diffusion-map/diffusion-net residence manifold learning, and an autoencoder whose loss enforces the Legoll–Lelièvre orthogonality condition. The reduced-model committor is then used as a forward-flux-sampling reaction coordinate and as a stochastic control for sampling transition paths. Two Lennard-Jones case studies are presented: LJ7 in 2D and LJ8 in 3D, with rates obtained by FFS compared against brute-force all-atom simulations. The paper reports that the reduced-model rates agree with brute-force rates, and it makes code and detailed implementation data available.

Significance. If the method works as claimed, it is a useful contribution to collective-variable discovery for systems with permutational symmetry, an important class that is poorly served by traditional physically motivated CVs. The manuscript is particularly strong in its engineering: it provides reproducible code, detailed appendices on neural-network architectures, Jacobian derivatives of the sorted feature maps, and extensive FFS versus brute-force comparisons. However, the theoretical grounding is weakened by the acknowledged non-smoothness of the sorted features, and the 3D case study does not actually execute the autoencoder/orthogonality step that is the paper's main algorithmic novelty. The rate agreement is also more mixed than the abstract suggests. The 2D LJ7 demonstration is credible, but the central 3D claim currently overstates what has been tested.

major comments (4)
  1. [Section 6.1] The LJ8 'ML CV–sortrcs' case does not execute Algorithm 1. After computing the target-measure diffusion map, the authors state: 'Therefore, there is no need to learn the confining potential V1 and the CVs: ψ1pϕpxqq and ψ2pϕpxqq are the desired CVs in this case.' These ψ's are diffusion-map eigenvectors, not outputs of the autoencoder trained with the orthogonality loss (4.6). Consequently, the conclusion's statement that the LJ8 CVs were 'learned by Algorithm 1' (Section 8) is not supported by the reported workflow. The distinctive component of the proposed framework—learning CVs by enforcing Dξ∇V1 = 0—remains untested on the 3D example used to support the abstract's rate-agreement claim.
  2. [Sections 3.3 and 4.1] The orthogonality condition (3.19) is motivated by the relative entropy bound (3.17), which is derived under a regularity assumption (Duong et al., Assumption 2.3). Section 4.1 explicitly acknowledges that the sorted feature maps 'sortrd2s and sortrcs are continuous but only piecewise smooth, which violates the regularity assumption.' Since the CV ξ(ϕ(x)) and its Jacobian Dϕξ pass through sorting boundaries, the theoretical justification for the orthogonality-based loss does not directly apply to the feature maps actually used. The paper should either provide a relaxation of the regularity condition for sorting maps, or supply quantitative evidence (e.g., orthogonality residuals and sensitivity of the computed rates to small perturbations that change sorting order) that the lack of smoothness does not materially affect the bound.
  3. [Section 6.2, Tables 5–6] The abstract's claim that 'the transition rates and residence times computed with the aid of the reduced models agree with those obtained via brute-force methods' is stronger than the data support. For the LJ8 ML CV–sortrcs case at β = 10, Table 6 reports FFS νAB = 3.1 ± 2.2 × 10⁻² versus brute-force νAB = 6.1 ± 0.1 × 10⁻², a factor-of-two discrepancy that is not within one standard deviation, and kB = 3.4 ± 2.6 × 10⁻² versus 7.2 ± 0.2 × 10⁻², also a factor of about two. The LJ7 kB estimates in Tables 1 and 3 likewise deviate by more than one standard deviation at several temperatures. The kA estimates generally agree well, but the overall rate and residence-time claim needs to be qualified, with relative errors reported systematically.
  4. [Section 5.2] The successful LJ7 result with the sortrd2s feature map uses the minimum-energy-path arc-length constraint (5.1), which is not part of Algorithm 1. The unmodified Algorithm 1 is therefore validated only by the LJ7 sortrcs case and by the LJ8 diffusion-map-eigenvector case, which, as noted above, skips the autoencoder step. This limitation should be stated in the conclusion so that the reader does not attribute the sortrd2s success to the baseline algorithm.
minor comments (5)
  1. [Section 6.1] The label 'ML CV' for the LJ8 sortrcs CVs is misleading because the CVs are diffusion-map eigenvectors, not outputs of the machine-learned autoencoder. A name such as 'diffusion-map CV' would be clearer.
  2. [Appendix D] Table 2 appears to be an exact duplicate of Table 1. This duplication should be removed or the two tables should be given distinct content.
  3. [Throughout] There are numerous typos and garbled expressions: 'vice versus' in Table 1, 'Mimimum 2' in Section 7.3, 'committtor' in several Appendix headings, and corrupted figure labels such as 'vs , ML CV, , LJ8log(kA) β 𝗌𝗈𝗋𝗍[c]' in Figure 24. These should be corrected in a careful revision.
  4. [Sections 4.4 and 5.4] The quantities ρA and ρB are last-hit probabilities, not residence times in the usual sense of mean residence times in the metastable sets. The paper's use of 'residence times' should be defined or replaced with 'last-hit probabilities' to avoid confusion.
  5. [Equation (4.6)] In the reconstruction loss, the decoder's target is Ψ_DNet(ϕ), the diffusion-net embedding, rather than the original feature vector ϕ. This is an unusual choice and should be clearly motivated, since the autoencoder is then not reconstructing the input feature space but a lower-dimensional embedding of it.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the reduced-model rates are compared with independent brute-force all-atom benchmarks, though the LJ8 'ML CV' success bypasses the autoencoder step.

full rationale

The central rate claims are not circular. FFS rates computed with the reduced-model committor as reaction coordinate are compared against brute-force unbiased all-atom simulations for the same redefined sets A and B; neither rate is fitted to the other, so the agreement is an independent empirical check. The orthogonality loss (4.6) is motivated by the external results of Legoll-Lelievre and Duong et al., not by a self-citation, and the disclosed use of the MEP arc-length constraint (5.1) or the post-hoc redefinition of A and B as committor level sets is a modeling choice rather than a tautological input. Self-citations such as [63] and [71] describe the framework and FFS/control recipes, but they do not carry the load-bearing empirical claim, which rests on brute-force benchmarks. Two ancillary concerns are flagged as correctness/attribution gaps rather than circularity: Section 4.1 admits that sortrd2s and sortrcs are only piecewise smooth, so the Duong et al. regularity assumption behind (3.17) and (3.19) is not strictly satisfied; and Section 6.1 states that for LJ8 with sortrcs, 'there is no need to learn the confining potential V1 and the CVs: ψ1pϕpxqq and ψ2pϕpxqq are the desired CVs,' meaning the 3D 'ML CV–sortrcs' case uses diffusion-map eigenvectors directly and does not execute the autoencoder/orthogonality step. Neither of these makes the reported rate agreement a consequence of the fitted inputs.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on several hand-chosen hyperparameters and on assumptions about the validity of the effective dynamics theory under non-smooth feature maps, the adequacy of the sampled dataset, and the transfer of the reduced-model committor to the atomic space. No new physical entities are postulated; the method is a numerical pipeline built from existing theories and algorithms.

free parameters (6)
  • Diffusion map bandwidth epsilon and kNN count k = epsilon=4.0, k=256 (LJ7 sortrd2s); epsilon=1.0, k=4160 (LJ7 sortrcs); epsilon=0.5, k=4160 (LJ8 sortrcs)
    Chosen by hand to make the residence manifold smooth and two-dimensional; different choices change the diffusion map coordinates and hence the learned CVs and rates.
  • Confining potential loss weights and off-manifold sampling threshold = a1=1, a2=0.002, r=0.005, bounding box [-0.02,0.02]^3
    Ad hoc tuning of V1; the zero-level set defines the residence manifold used in the orthogonality loss.
  • Metastable region parameters for A and B = Free energy thresholds F_Omega and ellipse parameters in Tables S2-S3 and SI S5
    Hand-defined per beta from the free energy landscape; rates and committor depend on these domains.
  • Metadynamics biasing parameters = sigma=0.02 (LJ7) or 0.01 (LJ8), h0=0.02 or 0.01, gamma=1, Nskip=500, Nbumps=50000
    Used to generate the training dataset; not fit to the rate target but control which configurations are sampled.
  • FFS thresholds epsilon_A, epsilon_B and milestone counts = epsilon_A from 1e-3 to 1e-2, epsilon_B from 1e-2 to 1e-3 depending on case; m=19 interfaces
    Define the redefined A and B as committor level sets; chosen per temperature.
  • Neural network sizes and learning hyperparameters = e.g., encoder [7,30,30,2], Adam learning rate 1e-3 to 5e-3, 1000-2000 epochs
    Hand-chosen; architectures in Table S1; not fitted to data but affect the learned CV.
assumptions (5)
  • domain assumption The overdamped Langevin SDE (3.1) with the Lennard-Jones potential and a restraining spring is an adequate model of the cluster dynamics.
    All sampling, FFS, and brute-force rates are generated from this SDE; the spring prevents evaporation and is assumed not to distort the transition region.
  • standard math The effective dynamics theory of Legoll-Lelievre and Duong et al. (relative entropy bound and orthogonality condition) applies to the feature space and CVs.
    The method is built on (3.17)-(3.19); however, the sorted feature maps used in practice are only piecewise smooth, violating the theory's regularity assumption (Section 4.1).
  • ad hoc to paper The residence manifold is well approximated by a 2D manifold in the span of the top three diffusion map eigenvectors for the chosen kernel bandwidth.
    The choice of d=3 and the retention of psi1, psi2, psi3 is validated only by visual inspection of the point clouds (Figures 5, 8, 16).
  • ad hoc to paper The committor of the reduced model, lifted through the learned CV, is a sufficiently good reaction coordinate for FFS in the original atomic space.
    Section 5.4 states that the orthogonality condition does not ensure small discrepancy between atomic and CV dynamics; the transfer is assumed and tested empirically, not derived.
  • domain assumption The dataset generated by well-tempered metadynamics biased by (mu2,mu3) provides adequate coverage of the transition region for learning CVs.
    The training data are collected by binning a biased trajectory in (mu2,mu3); if this sampling misses relevant configurations, the learned CVs and manifold will be incomplete.

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Cite this review

Pith. "Pith review of Learning collective variables that respect permutational symmetry." pith.science (2026). https://pith.science/paper/HY726WVN

@misc{pith2026250700408,
  author       = {Pith},
  title        = {Pith review of: Learning collective variables that respect permutational symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HY726WVN}},
  note         = {Machine review of arXiv:2507.00408}
}
read the original abstract

In addition to translational and rotational symmetries, clusters of identical interacting particles possess permutational symmetry. Coarse-grained models for such systems are instrumental in identifying metastable states, providing an effective description of their dynamics, and estimating transition rates. We propose a numerical framework for learning collective variables that respect translational, rotational, and permutational symmetries, and for estimating transition rates and residence times. It combines a sort-based featurization, residence manifold learning in the feature space, and learning collective variables with autoencoders whose loss function utilizes the orthogonality relationship (Legoll and Lelievre, 2010). The committor of the resulting reduced model is used as the reaction coordinate in the forward flux sampling and to design a control for sampling the transition path process. We offer two case studies, the Lennard-Jones-7 in 2D and the Lennard-Jones-8 in 3D. The transition rates and residence times computed with the aid of the reduced models agree with those obtained via brute-force methods.

Figures

Figures reproduced from arXiv: 2507.00408 by the authors.

Figure 1
Figure 1. The disconnectivity graph for the Lennard-Jones-7 cluster in 2D and descriptions [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The disconnectivity graph for the Lennard-Jones-8 cluster in 3D and descriptions [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The diffusion map of a set of 30467 representative configurations of LJ7 in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: The free energy landscape of LJ7 in 2D in the second and third central moments [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: The learned residence manifold in the space spanned by the three dominant [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: The free energy with respect to the learned CVs with the [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: The diffusion matrix with respect to the learned CVs with the [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: The learned residence manifold in the space spanned by the three dominant [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: The free energy for LJ7 in 2D with respect to the ML CV [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: The diffusion matrix with respect to the learned CVs with the [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: The chosen sets A and B surround the hexagon and the trapezoid minima respectively. The committor in the learned CV space is computed using the finite element method at β “ 9 and then approximated by a neural network (NN). The feature maps are: the sorted vector of pa…
Figure 12
Figure 12. Figure 12: Comparison of FFS escape rate kA and brute force kA for three temperatures in log scale. Standard deviations are labeled in figures as error bar (left) or shaded regions (right). 25 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: The probability density of reactive trajectories at [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]
Figure 14
Figure 14. Figure 14: The free energy for LJ8 in 3D in pµ2, µ3q with the projections of the potential energy minima (left) and representative configurations (right). HEre and in all further “balls-and-sticks” depictions of the configurations of LJ8 in 3D, any two atoms, i, and j are connec…
Figure 15
Figure 15. Figure 15: The diffusion matrix for LJ8 in 3D in pµ2, µ3q. 3D manifold of LJ8 via diffusion map ϕ1 ϕ2 ϕ3 ψ1 ψ2 ψ3 [PITH_FULL_IMAGE:figures/full_fig_p028_15.png]
Figure 16
Figure 16. Figure 16: The residence manifold for LJ8 in 3D with the feature map [PITH_FULL_IMAGE:figures/full_fig_p028_16.png]
Figure 17
Figure 17. Figure 17: The free energy for LJ8 in 3D with respect to the CV learned by Algorithm [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]
Figure 18
Figure 18. Figure 18: The diffusion matrix for LJ8 in 3D with respect to the CV learned by Algo [PITH_FULL_IMAGE:figures/full_fig_p029_18.png]
Figure 19
Figure 19. Figure 19: The sortrcs data for LJ8 in 3D projected onto the span of the three dominant eigenvectors of the linear discriminant analysis generalized eigenvalue problem. LDA3 LDA2 LDA3 LDA2 [PITH_FULL_IMAGE:figures/full_fig_p030_19.png]
Figure 20
Figure 20. Figure 20: The free energy for LJ8 at β “ 10 in the CV (LDA2,LDA3) with the projections of the potential energy minima (left) and representative configurations (right). 30 [PITH_FULL_IMAGE:figures/full_fig_p030_20.png]
Figure 21
Figure 21. Figure 21: The free energy for LJ8 at β “ 10 in the CV (LDA1,LDA2) with the projections of the potential energy minima (left) and representative configurations (right). LDA1 LDA2 [PITH_FULL_IMAGE:figures/full_fig_p031_21.png]
Figure 22
Figure 22. Figure 22: The diffusion matrix for LJ8 in 3D at β “ 10 in (LDA2,LDA3) (left) and (LDA1,LDA2) (right). 31 [PITH_FULL_IMAGE:figures/full_fig_p031_22.png]
Figure 23
Figure 23. Figure 23: The committors for LJ8 in 3D at β “ 10 in pµ2, µ3q (top left), CVs learned by Algorithm 1 (top right), (LDA2,LDA3) (bottom left), and (LDA1,LDA2) (bottom right). 33 [PITH_FULL_IMAGE:figures/full_fig_p033_23.png]
Figure 24
Figure 24. Figure 24: The escape rates kA from sets A to sets B for LJ8 in 3D obtained using the forward flux sampling (FFS) the committors as the reaction coordinates and brute force sampling (BF). The sets A and B are defined and the committor problem is solved in the following CV spaces…
Figure 25
Figure 25. Figure 25: The probability density of transition trajectories for LJ8 in 3D at [PITH_FULL_IMAGE:figures/full_fig_p035_25.png]

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Pith tools

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