{"id":"f3b85837-27e8-452d-b563-4ea2c79308ee","arxiv_id":"2505.04815","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For two-fold rotationally symmetric chaotic attractors, convergent cross mapping reports false unidirectional causality from the invariant variable, and a k-means segmentation of the symmetric shadow manifold restores the correct bidirectional links.","lead":"The paper shows that when a chaotic system has two-fold rotational symmetry, convergent cross mapping can mistake bidirectional causality for one-way causality because the measured variable's shadow manifold glues together symmetric copies. The authors offer a k-means based splitting fix that recovers the correct bidirectional links in simulations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The k-means segmentation step, not the symmetry mechanism, is the weakest link: Algorithm 1 assumes cluster boundaries equal the C2 fundamental domains with no diagnostic, and Section 5.3 shows the same segmentation idea can fail.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: sCCM's correctness hinges on k-means clusters coinciding with the C2 fundamental domains, and this is not established. I agree with that assessment. I considered alternative concerns, such as the gap between the differential-embedding proof and the delay-embedding experiments, or the lack of error bars, but those are secondary and addressable; the symmetry mechanism itself is well-supported because an even measurement function makes the delay embedding exactly invariant under the symmetry, so non-injectivity is real. The unresolved issue is the recovery of the quotient label. k-means is a heuristic with no guarantee of finding the symmetry domains, and the paper provides no diagnostic to verify that the segmentation is correct before averaging CCM scores. The admitted failure in Section 5.3 for C4, where segmentation cannot recover the one-to-one map, strengthens the concern that the method's success on C2 benchmarks may be tied to the specific geometry of those attractors rather than to a general principle. This does not overturn the empirical demonstrations, but it does justify the reader's CONDITIONAL verdict; no verdict adjustment is needed.","tokens_in":21756,"tokens_out":7446,"duration_ms":84337,"concrete_test":"For every C2 system in Table 2 and the high-dimensional cases in Table 4, compute ground-truth fundamental-domain labels from the sign of any coordinate odd under the symmetry (e.g., sign(x_t) for Lorenz63), then compare them with the k-means labels assigned to Mx in Algorithm 1. Report the misclassification rate and check whether the two cluster centers are approximate negatives under the inversion symmetry. Then rerun sCCM using exact labels instead of k-means labels. If the exact-label and k-means scores agree and the misclassification rate is zero, the segmentation assumption holds on these benchmarks; if exact labels materially improve rho_xz or any misclassification occurs, the central claim should be conditional on a reliable symmetry-domain segmentation criterion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 1 (Section 4.2) is not the problem: for an even-parity measurement h, the delay/differential map F_{xn,n} satisfies F_{xn,n}(p)=F_{xn,n}(R·p), so the invariant-variable shadow manifold genuinely quotients out the C2 symmetry and cannot be a diffeomorphic copy of the attractor. The weak point is the proposed fix. sCCM (Algorithm 1) assumes that k-means with k=2 on the inversion-symmetric shadow manifold Mx partitions exactly the two fundamental domains D and R·D. No theorem or diagnostic supports this; Section 4.3 only cites [40] for k-means on symmetric data and relies on empirical success. If the k-means boundary cuts across a fundamental domain, then Mz|[ti] still mixes points from both symmetric copies and the two-to-one degeneracy survives, so the bidirectional result is not recovered. The paper itself notes that the two domains can have very different point densities (Mx|[t2] is visibly denser than Mx|[t1]), and Section 5.3 concedes that for a C4 system the same segmentation approach cannot reveal the one-to-one mapping because of cover degeneration. Since every sCCM result in Tables 2-4 inherits the k-means labels, a single C2 system where k-means splits lobes instead of domains would invalidate the claimed general fix. No code or data is provided to rule this out.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22103,"tokens_out":7779,"duration_ms":83024,"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":[{"comment":"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.","section":"§4.3, Algorithm 1; §5.3"},{"comment":"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.","section":"§4.2 vs. §5"},{"comment":"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.","section":"§5, Tables 2–4"}],"minor_comments":[{"comment":"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.","section":"Algorithm 1"},{"comment":"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.","section":"§5.2, Eq. (5.3)"},{"comment":"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.","section":"Introduction and references"},{"comment":"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.","section":"§5.1, Table 3"},{"comment":"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.","section":"§4.1, Eq. (4.5)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest about the C4 limitation in Section 5.3, which is a point in its favor, but that same section underscores that the segmentation mechanism is not robust enough to be assumed without diagnostics. I would encourage the editor to require code and data deposition as part of the revision, since the empirical tables are the only evidence for the k-means alignment assumption and the paper's central claim depends on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe key thing to know: this paper nails the mechanism behind a known CCM failure and offers a simple fix that looks effective on the cases they test. The mechanism is real. The fix is heuristic, and the paper has not shown it is safe beyond the examples.\n\nWhat's new: Proposition 1 gives a clean group-theoretic explanation: when the attractor is invariant under an order-two rotation and you measure the invariant coordinate, the differential embedding is two-to-one; CCM then mistakes bidirectional causality for one-way. This unifies examples (Lorenz z, Chen, Burke-Shaw) that were known but treated as counterexamples. The sCCM method — partition the symmetric shadow manifold into the two symmetry domains via k-means, run CCM on each half — is a genuinely simple idea and it works in every C2 example in the paper, including three higher-dimensional systems.\n\nWhere it gets soft. First, the proof of Proposition 1 is for differential embeddings, but all experiments use delay-coordinate embeddings. The bridge ('small lag approximates the derivative') is standard but not proven, and symmetry can be sensitive. Second — and this is the real weakness — Algorithm 1 assumes k-means with k=2 cuts the symmetric shadow manifold exactly along the two fundamental domains. There is no diagnostic or theorem for that. In Fig. 9 the two domains have visibly different densities, so a generic clustering algorithm could easily assign the boundary differently. The C4 case in Section 5.3 shows the segmentation approach stops working when covers degenerate, so this isn't an abstract worry. Third, no code or data is provided and none of the tables have error bars, so the empirical claims cannot be stress-tested.\n\nThe stress-test note about k-means is right. That's the load-bearing assumption and it is too casually treated.\n\nWho it's for: people who apply CCM to symmetric or nearly symmetric attractors, and anyone building state-space methods for causality. The explanation alone is worth publishing. The sCCM method deserves a proper referee, but the paper needs revisions: provide code and data, add uncertainty quantification, justify or diagnose the k-means segmentation, and test at least one C2 system where the symmetry domains are not linearly separable. I would accept it for peer review with major revision.","headline":"A real mechanism for a known CCM failure plus an empirically promising but unproven k-means fix; deserves a careful referee.","tokens_in":22571,"tokens_out":5830,"would_cite":true,"duration_ms":58035,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M10","37D45","37C80","62H30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two-fold rotational symmetry can make convergent cross mapping report the wrong causal direction, and segmenting the shadow manifold restores the true bidirectional link.","keywords":["convergent cross mapping","causal discovery","symmetric chaos","shadow manifold","delay embedding","k-means clustering","Lorenz system","rotation symmetry"],"falsifier":"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.","tokens_in":21564,"feed_emoji":"🦋","tokens_out":5054,"duration_ms":46780,"temperature":0.7,"pith_summary":"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.","feed_headline":"Splitting the shadow manifold restores true causal links","feed_subtitle":"For two-fold symmetric chaotic systems, convergent cross mapping hides bidirectional causality until each mirror half is analyzed…","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces convergent cross mapping, the method whose symmetric-system failures this paper diagnoses and repairs.","marker":"[4]"},{"why":"Takens' embedding theorem is the theoretical basis for reconstructing the shadow manifold from a single observed time series.","marker":"[12]"},{"why":"Formalizes unidirectional and bidirectional causality as projection and homeomorphism between shadow manifolds, the framework Proposition 1 builds on.","marker":"[20]"},{"why":"Cross and Gilmore's theorem that differential reconstruction preserves at most two-fold symmetry explains why only the C2 case is treated.","marker":"[38]"},{"why":"Supplies the claim that k-means clustering performs well on symmetric datasets, justifying the segmentation step.","marker":"[40]"},{"why":"Establishes recurrence and autopredictability as prerequisites for CCM, which the proposed pipeline checks before segmenting.","marker":"[9]"},{"why":"Provides the recurrence-based indicator used to detect whether a shadow manifold has inversion symmetry before sCCM is applied.","marker":"[39]"}],"fun_headline_variants":["Symmetry tricks CCM into one-way links; k-means restores the truth","K-means fixes CCM's one-way mistake on mirror-symmetric chaos","Mirror symmetry fools CCM; clustering unmasks bidirectional causality","Split symmetric shadow manifold to recover hidden causal arrows","Two-fold rotation symmetry misleads CCM until you cluster"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry tricks CCM into one-way links; k-means restores the truth","K-means fixes CCM's one-way mistake on mirror-symmetric chaos","Mirror symmetry fools CCM; clustering unmasks bidirectional causality","Split symmetric shadow manifold to recover hidden causal arrows","Two-fold rotation symmetry misleads CCM until you cluster"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00074,"raw_usage":{"total_tokens":3280,"prompt_tokens":898,"completion_tokens":2382,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":2290}},"tokens_in":514,"tokens_out":2382,"duration_ms":16067,"temperature":1.0,"reasoning_tokens":2290,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:21:03.654565+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Sugihara, R","cited_arxiv_id":null,"evidence_quote":"Introduces convergent cross mapping, the method whose symmetric-system failures this paper diagnoses and repairs."},{"cited_title":"Cummins, T","cited_arxiv_id":null,"evidence_quote":"Formalizes unidirectional and bidirectional causality as projection and homeomorphism between shadow manifolds, the framework Proposition 1 builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cross and Gilmore's theorem that differential reconstruction preserves at most two-fold symmetry explains why only the C2 case is treated."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the claim that k-means clustering performs well on symmetric datasets, justifying the segmentation step."},{"cited_title":"Butler, G","cited_arxiv_id":null,"evidence_quote":"Establishes recurrence and autopredictability as prerequisites for CCM, which the proposed pipeline checks before segmenting."},{"cited_title":"Marghoti, T","cited_arxiv_id":null,"evidence_quote":"Provides the recurrence-based indicator used to detect whether a shadow manifold has inversion symmetry before sCCM is applied."}],"review_version":1}