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Causal Discovery in Symmetric Dynamic Systems with Convergent Cross Mapping

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Pith's one-line read Two-fold rotational symmetry can make convergent cross mapping report the wrong causal direction, and segmenting the shadow manifold restores the true bidirectional link.

desk verdict A real mechanism for a known CCM failure plus an empirically promising but unproven k-means fix; deserves a careful referee. read the letter →

arxiv 2505.04815 v1 pith:42BK725G submitted 2025-05-07 math.DS

Yiting Duan , Yi Guo , Jack Yang , Ming Yin This is my paper · ORCID
classification math.DS MSC 37M1037D4537C8062H30
keywords convergentcrossmappingcausaldiscoverysymmetricchaosshadowmanifolddelayembeddingk-meansclusteringLorenzsystemrotationsymmetry
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 argues that when a chaotic system's attractor is symmetric under a half-turn rotation, the standard causal-discovery tool convergent cross mapping (CCM) systematically misreports bidirectional causal links as one-way links. The reason is that the delay-coordinate map built from the symmetry-invariant coordinate is two-to-one: it glues the two mirror halves of the attractor together, so the reconstructed shadow manifold is no longer an embedding. The paper proposes segment convergent cross mapping (sCCM), which splits the symmetric shadow manifold into its two fundamental domains with k-means clustering, runs CCM on each half, and averages the resulting scores. On Lorenz63 and many other C2-symmetric systems, this restores the true bidirectional causality without using information from a third variable. The paper also reports that the same step recovers causality in four- and five-dimensional symmetric systems, while noting that the method does not transfer to attractors of higher cyclic symmetry.

What carries the argument

The load-bearing object is the shadow manifold, the delay-coordinate or differential reconstruction of a chaotic attractor from one observed time series. For a two-fold rotation symmetric system, the reconstructed manifold from the invariant coordinate has even parity under the symmetry and therefore identifies points in opposite fundamental domains, making the reconstruction mapping two-to-one rather than an embedding. The mechanism that carries the argument is the induced non-injective projection from the non-symmetric shadow manifold to the symmetric one, which is what makes CCM's Pearson-correlation score converge in only one direction. The corrective device is k-means clustering with $k=2$ applied to the symmetric shadow manifold, whose two clusters are taken to be the two fundamental domains; the time indices of those clusters split the other shadow manifold, and CCM is run on the two resulting segment pairs.

What would settle it

Take a two-fold symmetric system whose fundamental domain is known analytically, run sCCM, and compare the k-means cluster labels with the analytic domain labels: if the two disagree on a positive-measure set of points yet sCCM still reports high bidirectional scores, the explanation in Proposition 1 is not the whole story; conversely, a system where the cluster boundary demonstrably crosses a fundamental domain should make sCCM fail to recover bidirectionality.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Proposition 1: for a C2-equivariant system with invariant variable $x_n$, the differential or delay-coordinate map $F_{x_n,n}$ from the attractor to the shadow manifold $\mathcal{M}_{x_n}$ has even parity and is generically two-to-one, not injective. Because bidirectional causality would require a homeomorphism between the shadow manifolds, the non-injectivity turns that homeomorphism into a non-injective projection, and CCM reports $x_n \Rightarrow x_i$ when the truth is $x_i \Leftrightarrow x_n$. The proposed remedy is to partition the symmetric shadow manifold into the two covers of the quotient by the symmetry—the fundamental domain and its reflected image—using k-means clustering with $k=2$, then to segment the invariant-variable shadow manifold by the same time indices and cross-map each pair separately. On each segment the restriction of the reconstruction map is one-to-one, so the cross-map scores converge high in both directions. The paper validates this on low- and high-dimensional rotation-symmetric systems and under added noise, and explicitly notes that the clustering-based segmentation no longer recovers the map for four-fold symmetric attractors.

Load-bearing premise

The method assumes that a k-means split of the symmetric shadow manifold into two clusters coincides with the two fundamental domains of the half-turn symmetry, so that each cluster contains exactly one copy of the attractor; if the cluster boundary cuts through a fundamental domain, the two-to-one mixing persists and the true bidirectional score is not recovered.

Editorial extensions

If this is right

  • For any C2-symmetric chaotic system in which one variable is invariant under the rotation, a CCM output of one-way causation from that variable should be checked by segmentation before being read as true causality.
  • Applying sCCM converts the two-to-one reconstruction into two one-to-one restrictions, so the bidirectional link is recovered without appealing to a third variable's time series.
  • The same correction works in higher-dimensional Lorenz-like systems, including the four- and five-dimensional cases tested in the paper, provided the single-variable embedding retains sufficient observability.
  • The method is robust to moderate Gaussian observational noise: at a noise level of $\sigma = 1$, the recovered cross-map scores remain high while plain CCM still shows a large asymmetry.
  • For $k$-fold symmetric attractors with $k > 2$, sCCM as written does not restore the one-to-one map, so a refined segmentation is required.

Reading between the lines

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

  • The paper leaves implicit that the failure is a quotient phenomenon: any symmetry whose quotient map is non-injective on the observed coordinate will bias CCM, so the same segmentation idea should extend to reflection or glide symmetries as long as the fundamental domains can be separated.
  • A natural testable extension is to compare the k-means labels against the analytic symmetry map on systems where the fundamental domain is known; a large label mismatch would predict sCCM failure even when the reported scores look high.
  • The C4 example suggests a boundary case: when covers of the original attractor degenerate in the shadow manifold, no 2-cluster partition can separate the overlapping copies, so clustering into exactly two segments is not a general remedy.
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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

3 major / 5 minor

Summary. The paper studies how rotational symmetry of a chaotic attractor can make convergent cross mapping (CCM) misreport bidirectional causality as unidirectional. The proposed mechanism is that when a dynamical system is equivariant under a C2 group and the measurement function is invariant under that symmetry, the differential or delay-coordinate embedding of the invariant variable becomes two-to-one rather than one-to-one, so the shadow manifold of that variable is not diffeomorphic to the original attractor. Proposition 1 formalizes this for differential embeddings. To repair the inference, the authors propose sCCM: k-means clustering with k=2 is applied to the symmetric shadow manifold Mx, the time indices of the two clusters are used to split the invariant shadow manifold Mz into two sub-manifolds, and CCM is run separately on each pair before averaging the scores. The method is tested on 13 three-dimensional C2-symmetric systems and 3 high-dimensional C2-symmetric systems, plus a noise-robustness study on Lorenz63, and uniformly recovers bidirectional X⇔Z in the tables. A C4 example is discussed as a limitation.

Significance. If the proposed mechanism and repair are correct, the paper makes a useful contribution by connecting attractor symmetry to a specific, previously underappreciated failure mode of CCM and by offering a practical correction that does not use information from other variables. The mechanism is derived independently of the experiments and yields a falsifiable prediction: the failure should occur only for the invariant variable whose shadow manifold quotients out the symmetry, and segmenting by the symmetry domains should restore one-to-one cross mapping. The benchmark coverage is broad across Lorenz-like systems, and the honest discussion of the C4 failure in Section 5.3 is a strength. The main limitation is that the central repair step, k-means segmentation, is asserted rather than proven or diagnosed, and the empirical validation, while wide, is presented without code, data, error bars, or convergence curves.

major comments (3)
  1. [§4.3, Algorithm 1; §5.3] The load-bearing assumption of sCCM is that k-means with k=2 on the inversion-symmetric shadow manifold Mx partitions it exactly into the two fundamental domains D and R·D. Section 4.3 provides no theorem, diagnostic, or validation for this, and the citation [40] concerns segmentation of remote-sensing datasets, not dynamical shadow manifolds. If the k-means boundary cuts across a fundamental domain, then Mz|[ti] still mixes points from both symmetric copies, the two-to-one degeneracy survives, and the bidirectional result is not recovered. The paper's own discussion in Section 5.3 shows that the same segmentation idea fails for a C4 system because of cover degeneration, and Section 4.3 notes that the two domains can have very different point densities. Since every sCCM score in Tables 2–4 inherits the k-means labels, the method needs at least a diagnostic (e.g., cluster-purity against known symmetry labels for benchmark systems, per-segment scores and sizes, or a stability analysis over k-means initializations) to support the claim that the clusters coincide with the fundamental domains.
  2. [§4.2 vs. §5] Proposition 1 is proved for the differential mapping Fh,n, while all experiments in Section 5 use delay-coordinate mappings Fh,τ,n. The only bridge is the statement in Section 2 that for a suitable lag τ the delay-coordinate mapping is affinely equivalent to the differential mapping, but no proof or selection criterion is given, and the text merely says that τ was 'optimized'. The parity-inheritance argument and the two-to-one conclusion must be shown for delay-coordinate embeddings themselves, or at least stated as a required assumption with supporting evidence, because the causal inference in CCM is implemented with delay coordinates. As written, the central theorem does not cover the experimental setting on which the validation rests.
  3. [§5, Tables 2–4] The empirical validation consists of single Pearson-correlation values with no error bars, no confidence intervals, no repeated initial conditions or noise realizations, and no displayed convergence curves or library lengths. The text states that the scores converge, but the tables do not show the convergence behavior that CCM's logic requires. Moreover, no code or data are provided, so the uniform success in Tables 2–4 cannot be checked or reproduced. Given that the paper's only support for the k-means alignment assumption is this empirical success, the absence of reproducibility material and statistical detail is a substantive gap rather than a presentation issue.
minor comments (5)
  1. [Algorithm 1] Line 2 of Algorithm 1 hardcodes the embedding dimension as 3 in Fx,τ,3 and Fz,τ,3, even though the algorithm's input includes n and Section 5.2 uses n=4 and n=5; the pseudocode should use Fx,τ,n and Fz,τ,n.
  2. [§5.2, Eq. (5.3)] The five-dimensional system in Eq. (5.3) is written in variables (x,y,z,u,v), but the symmetry map and Table 4 refer to a variable W in the rows Z⇒W and W⇔Z; either the equation or the table uses inconsistent notation and this should be corrected.
  3. [Introduction and references] There are several typos: 'Ganger causality' should be 'Granger causality', 'Taken's theorem' should be 'Takens's theorem', 'Bulter' should be 'Butler', and the references to 'Appendix 6' should be to Appendices A and B.
  4. [§5.1, Table 3] The noise-robustness results report single trials for each σ with no signal-to-noise ratios, no repeated noise realizations, and no error bars; a claim of robustness would be stronger with multiple trials and a summary of the spread.
  5. [§4.1, Eq. (4.5)] The displayed definition of the derivative in Eq. (4.5) appears to omit the denominator t in the limit; this is likely a typesetting issue but should be fixed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Proposition 1 follows from the system's C2 equivariance and parity of the invariant coordinate, and the sCCM results are empirical scores rather than fitted or definitionally forced outputs.

full rationale

The paper's central explanatory claim is Proposition 1, which shows that for a C2-symmetric system the differential mapping built from the invariant coordinate is noninjective because the measurement function and all its derivatives inherit even parity. This argument is self-contained: equations (4.5)-(4.6) establish parity preservation, and the conclusion that F_{xn,n} is two-to-one follows directly without any fitted parameter or desired causal direction being used as input. The subsequent sCCM method is an algorithmic intervention, not a renaming of the target result: k-means partitions the symmetric shadow manifold, the partition indices are applied to the other shadow manifold, and the CCM scores are then computed by the usual nearest-neighbor cross-mapping procedure. These scores are not constructed to equal the claimed bidirectional causality; they are empirical outputs that the paper reports in Tables 2-4. The cited Cross theorem [38] is external prior work and is not used in a way that reduces the present claim to a self-citation. No load-bearing self-citation appears in the derivation chain. The acknowledged limitation in Section 5.3, where sCCM fails to reveal the one-to-one mapping for the C4 system, further supports the noncircular character of the method: the authors explicitly concede conditions under which their proposed fix does not work rather than defining success into the method. The concern that k-means cluster boundaries may not coincide with the fundamental domains is a correctness or robustness risk, not a circularity, because no equation or definition forces the cluster-boundary assumption to be true. Accordingly, the appropriate finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on standard embedding theorems, on the parity behavior of invariant observables under C2, and on two ad hoc assumptions: (1) the delay-coordinate embeddings used in experiments inherit the parity structure of differential embeddings, and (2) k-means segmentation aligns with the fundamental domains. There are no fitted constants and no invented entities.

free parameters (2)
  • k (number of k-means clusters) = 2
    Fixed by the assumed order-2 rotation symmetry; not fitted to data. The paper chooses k=2 for all C2 systems.
  • Embedding lag tau and dimension n per system = e.g., tau=9, n=3 for Lorenz63; values for all systems in Section 5
    Selected by mutual information / false nearest neighbors according to Section 2.1; results for Lorenz63 are shown to be stable across tau and n in Figure 4, so these are standard CCM inputs rather than method-specific tuned parameters.
assumptions (4)
  • standard math Takens' embedding theorem and the Whitney embedding theorem (Theorems 1 and 2 of the paper)
    Foundation for shadow-manifold reconstruction and the assumption that delay-coordinate mappings are embeddings for generic h and generic dynamics.
  • ad hoc to paper The chosen delay-coordinate mapping is affinely equivalent to the differential mapping for the selected tau, preserving parity/symmetry properties
    The parity argument in Proposition 1 is developed for differential mappings F_{h,n}, but experiments use delay-coordinate mappings F_{h,tau,n}. The paper asserts this equivalence by citing [14] without verifying it for each system.
  • domain assumption The attractor is connected and a single trajectory densely visits both fundamental domains of C2, so the two-to-one structure and both clusters are observed in finite time series
    The paper explicitly excludes reflection-symmetric 'kissing' attractors (Section 4.1) and notes density imbalances between domains (Section 4.3). If the trajectory only visits one domain, the shadow manifold is not two-to-one in practice and sCCM has nothing to segment.
  • ad hoc to paper k-means with k=2 on the inversion-symmetric shadow manifold returns clusters aligned with the fundamental domains
    Stated in Section 4.3 with a citation to a remote-sensing paper [40]; no formal proof or diagnostic is provided, and Section 5.3 admits the approach fails for k-fold (k>2) symmetry.

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Cite this review

Pith. "Pith review of Causal Discovery in Symmetric Dynamic Systems with Convergent Cross Mapping." pith.science (2026). https://pith.science/paper/42BK725G

@misc{pith2026250504815,
  author       = {Pith},
  title        = {Pith review of: Causal Discovery in Symmetric Dynamic Systems with Convergent Cross Mapping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/42BK725G}},
  note         = {Machine review of arXiv:2505.04815}
}
read the original abstract

This paper systematically discusses how the inherent properties of chaotic attractors influence the results of discovering causality from time series using convergent cross mapping, particularly how convergent cross mapping misleads bidirectional causality as unidirectional when the chaotic attractor exhibits symmetry. We propose a novel method based on the k-means clustering method to address the challenges when the chaotic attractor exhibits two-fold rotation symmetry. This method is demonstrated to recover the symmetry of the latent chaotic attractor and discover the correct causality between time series without introducing information from other variables. We validate the accuracy of this method using time series derived from low-dimension and high-dimensional chaotic symmetric attractors for which convergent cross mapping may conclude erroneous results.

Figures

Figures reproduced from arXiv: 2505.04815 by the authors.

Figure 1
Figure 1. Sequential figures representing the changes in reconstructed shadow manifold [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. First line: Bidirectional causation X⇔Y . (a). Induced homeomorphism f between Mx and My. (b). Induced homeomorphism f ′ between Ax and Axy. Last line: Unidirectional causation X⇒Y when f exists. (c). Induced noninjective projection Πy,x between My and Mx. (d). Induced noninjective projection Π˜ y,x when Πy,x exists. Based on the above illustration, Sugihara et al. proposed the CCM algorithm to infer causal links be… view at source ↗
Figure 3
Figure 3. True causality versus CCM’s result. a. Graphical model of causal relations within the Lorenz63 system. b. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Influence of embedding parameters. Left. The influence of lag value [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: x-z plane projections of attractors. a. Chen [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: A symmetric pair of attractor "kissing" of the reflection equivariant system, when a = 0.7 and two initial values [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Induced projection Πxixn : Mxi → Mxn , homeomorphism f ′ : Axixn → Axn and f˜ : Mxi → Axn . Next, we use the Lorenz63 and the Burke & Shaw systems to show concrete examples. Both systems are two-fold symmetric system under the cyclic group C2 = {e, Rz(π)}. For the Lore…
Figure 8
Figure 8. Figure 8: Left: Strange attractors generated by the Lorenz63 system (3.1) and Burke & Shaw system (4.14). Right: [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: A schematic depicting the details for implementation of sCCM between [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: From left to right: Two-wing butterfly chaotic attractors in the [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: Projections of the attractor generated by equation (5.3) a. (v,z) plane projection. b. (y,u) plane projection. c. [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]
Figure 12
Figure 12. Figure 12: Attractors of equation (5.4). a. (x, y) plane projection of the four-fold Burke & Shaw Attractor. b. (x, y) plane projection of the shadow manifold Mx. c. (x, y) plane projection of the shadow manifold My. Equations CCM Four-fold Burke & Shaw System X ⇔ Y (ρxy = 0.981…
Figure 13
Figure 13. Figure 13: Left: Chaotic attractor produced by nine-dimensional Lorenz system. Right: Reconstructed shadow manifold by delay-coordinate mapping. the reconstructed shadow manifold exhibits a butterfly attractor similar to the Lorenz63 system rather than reflecting the more comple…
Figure 14
Figure 14. Figure 14: Left: Chaotic attractor produced by the Rössler system. Middle: Reconstructed shadow manifold Mz. Right: Reconstructed shadow manifold Mz′ . shadow manifold Mz ′ . The lag value is τ = 40 for both shadow manifolds, and the embedding dimension is n = 3. It can be obser…
Figure 15
Figure 15. Figure 15: Distance plot using the Euclidean distance and corresponding trajectories of Lorenz63 attractor. This system is [PITH_FULL_IMAGE:figures/full_fig_p036_15.png]
Figure 16
Figure 16. Figure 16: Left: Trajectories of Eq. (6.10). Right: Causal links obtained by CCM. [PITH_FULL_IMAGE:figures/full_fig_p036_16.png]

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