{"id":"5af0e6df-e548-4198-9f2c-ed669e9563f3","arxiv_id":"2607.07286","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":2,"one_line_summary":"Structural synergy, defined as the excess predictive power of a joint model over the best additive model, isolates non-additive interaction mechanisms from dependency-driven statistical synergy in complex systems.","lead":"The paper introduces 'structural synergy' — a measure of how much predictive power is lost when two information sources are forced to contribute additively to a target, rather than being allowed to interact. This lets researchers tell apart statistical synergy (an artifact of correlated sources) from genuine non-additive interaction mechanisms, which matters for interpreting complex systems like brains and climates.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Interaction terms can proxy non-polynomial additive effects when sources are correlated, inflating Ss as a false positive for HOMs in the empirical applications.","rationale":"The reader correctly identified the polynomial basis misspecification as the weakest assumption, but framed it generically. I sharpen it to a specific mechanism: correlated interaction terms acting as proxies for non-polynomial additive structure, which the surrogate test cannot distinguish from genuine non-additive mechanism effects. This is a real concern for the empirical applications but does not affect the analytical results, which are exact and correct. The theoretical contribution (the dissociation of HOBs and HOMs, the masked structural synergy regime) stands on its own merits. The practical implementation has a identifiable failure mode that the authors' caveats partially but not fully address. The verdict remains ACCEPT because: (1) the analytical framework is sound, (2) the concept of structural synergy is well-defined and correctly derived, (3) the limitations are honestly acknowledged, and (4) the concern I raise is a known type of model misspecification issue common to all polynomial regression approaches, not a fundamental flaw in the framework. The concern would become more serious if the authors had claimed direct mechanistic identification from Ss alone, but they explicitly disclaim this. The proposed concrete test would clarify the practical scope of the false-positive risk and could strengthen the paper if included as a robustness check.","tokens_in":22651,"tokens_out":3564,"duration_ms":175694,"concrete_test":"Generate data from a purely additive but non-polynomial model: Y = sin(πX₁) + cos(πX₂) + U, with X₁, X₂ jointly Gaussian and varying r₁₂ ∈ {0, 0.3, 0.5, 0.7}. Compute Ss using polynomial orders p ∈ {2,3,4} and apply the surrogate significance test. If Ss is declared statistically significant for any r₁₂ ≠ 0 and any p, the framework produces false positives for HOM in the presence of non-polynomial additive mechanisms with correlated sources, weakening the empirical claims.","verdict_should_be":"ACCEPT","load_bearing_attack":"The central analytical claim (masked structural synergy) is correct and well-derived. The concern is with the practical implementation. When sources are correlated (r₁₂ ≠ 0), interaction monomials X₁ᵏX₂ʲ are correlated with individual sources. If the true generative model is additive but non-polynomial (Y = h₁(X₁) + h₂(X₂) + U with smooth non-polynomial h₁, h₂), the polynomial additive model (Eq. 22) approximates h₁, h₂ imperfectly. The full model (Eq. 21) can then use interaction terms to capture the residual non-polynomial additive structure — because X₁X₂ carries information about X₁ when r₁₂ ≠ 0. This yields Ss > 0 and a significant surrogate test (since the interaction terms genuinely improve prediction), but the mechanism is additive, so this is a false positive for HOM. The analytical examples avoid this because they use polynomial functions (linear or multiplicative). The simulations use p=2 matching the quadratic generative term. The empirical applications (climate, EEG) have unknown functional forms and correlated sources, so this confound is live. The authors' caveat that Ss is 'evidence of non-additive predictive structure rather than direct identification of the underlying generative mechanism' partially addresses this, but the applications still frame significant Ss as evidence compatible with HOMs (e.g., 'masked structural synergy regime' in the EEG application).","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The manuscript introduces a predictability-based framework to distinguish high-order behaviours (HOBs) — observed synergistic or redundant statistical dependencies among variables — from high-order mechanisms (HOMs) — structural or dynamical rules containing genuine non-additive interaction terms. The core quantity, structural synergy Ss, is defined as the excess predictive power of the full joint model over the best additive model (Eq. 19). The framework is validated through analytical derivations for linear Gaussian and multiplicative non-linear systems, stochastic autoregressive simulations, and two empirical applications (climate indices and source-reconstructed EEG). The central conceptual contribution is the identification of a 'masked structural synergy' regime, where redundancy-dominated HOBs (Δ < 0) coexist with a genuine non-additive mechanism (Ss > 0).","tokens_in":23434,"tokens_out":1473,"duration_ms":369294,"significance":"The dissociation between HOBs and HOMs is a conceptually important contribution to the high-order interactions literature. The analytical derivations are clean and verifiable: the linear Gaussian case correctly yields Ss = 0 (Eq. 23), and the multiplicative case derivation in the Appendix is rigorous, using Isserlis' theorem for Gaussian moments to obtain the closed-form Ss = c²(1−r₁₂²)²/(1+r₁₂²) (Eq. 26). The masked structural synergy regime is a concrete, falsifiable theoretical prediction. The authors provide publicly available MATLAB code, which supports reproducibility. The two empirical applications are illustrative and connect to existing information-theoretic literature on the same datasets.","major_comments":[{"comment":"The practical implementation relies on polynomial regression (Eqs. 20–22), where the additive model (Eq. 22) and the full model (Eq. 21) share the same polynomial basis for individual source terms. When sources are correlated (r₁₂ ≠ 0), interaction monomials X₁ᵏX₂ʲ are correlated with individual source variables. If the true generative model is additive but non-polynomial (Y = h₁(X₁) + h₂(X₂) + U with smooth non-polynomial h₁, h₂), the polynomial additive model approximates h₁ and h₂ imperfectly. The full model can then use interaction terms to capture residual non-polynomial additive structure — because X₁X₂ carries information about X₁ when r₁₂ ≠ 0 — yielding Ss > 0 and a significant surrogate test even though the mechanism is genuinely additive. This is a false positive for HOMs. The analytical examples avoid this confound because they use polynomial functions (linear or multiplicable","section":null},{"comment":"The surrogate test (Methods, 'Statistical assessment of structural synergy') permutes the interaction terms while leaving additive terms unchanged. This tests whether interaction terms provide additional predictive contribution beyond the additive model, but it does not test whether that contribution reflects a genuine non-additive mechanism versus an artifact of basis misspecification. A more informative null would involve fitting the additive model with a richer or alternative basis (e.g., splines, kernels) and comparing against the polynomial interaction model. The authors should either (a) add a simulation where the true model is additive but non-polynomial (e.g., Y = sin(X₁) + cos(X₂) + U with correlated Gaussian sources) and show whether Ss exhibits false positives, or (b) explicitly state in the Methods and Applications that Ss is defined relative to the polynomial basis and that,","section":null},{"comment":"In the EEG application (Fig. 4), the results are framed as evidence of a 'masked structural synergy regime' compatible with non-additive mechanisms. Given the confound above, and given that EEG source-reconstructed signals have unknown functional forms and weak but positive source correlations (~0.1), the interpretation should be more cautious. The authors' caveat in the Conclusions ('structural synergy should be interpreted as evidence of non-additive predictive structure rather than as a direct identification of the underlying generative mechanism') partially addresses this, but the application sections still use language like 'compatible with a non-additive predictive component' and 'masked structural synergy regime' that may overstate what the polynomial-regression-based Ss can establish. A brief additional caveat in the application sections, referencing the basis-dependence of Ss,","section":null}],"minor_comments":[{"comment":"Eq. (21): the double summation notation 'p−1 Σ_{k=1} p−k Σ_{j=1}' is correct but dense; explicitly stating that the constraint is k+j ≤ p in the equation line would prevent confusion.","section":null},{"comment":"Table I: the 'Dependency-driven synergy' and 'Dependency-driven redundancy' rows list ΔA = Δ, which is correct, but the column header 'HOM' lists 'Absent' for these rows. This is a consequence of Ss = 0, but a brief footnote clarifying that 'Absent' means 'no non-additive mechanism detectable by this framework' rather than 'no mechanism of any kind' would improve precision.","section":null},{"comment":"Fig. 2: the y-axis scales differ substantially across panels (e.g., panel d ranges from −3 to 0, panel a from 0 to ~0.04). A note in the caption or shared scaling for comparable panels would aid interpretation.","section":null},{"comment":"Fig. 3: the asterisks denoting significant increases in Ss when increasing polynomial order are described in the caption but their placement relative to specific bars is ambiguous; clarifying which polynomial orders are being compared would help.","section":null},{"comment":"Climate application: the Spearman correlation values in Fig. 3 (top row) appear to be plotted on a scale from −0.15 to 0.2, but the text does not state whether the indices were standardized or detrended before correlation. A brief preprocessing note would be appropriate.","section":null},{"comment":"EEG application: the Bonferroni correction across 10 trials is mentioned, but it is unclear whether correction was also applied across the 3 ROI triplets or the 3 polynomial orders. Clarification of the full multiple-comparison structure would be appropriate.","section":null},{"comment":"Ref [22] (Caprioglio et al., 2026) and Ref [1] (Peixoto et al., 2026) have 2026 dates; verify these are correct and not placeholder dates.","section":null},{"comment":"The phrase 'high-order mechanisms (HOMs) as intended in the present work' (Introduction, final paragraph) is slightly awkward; rephrasing to 'as defined in the present work' would improve readability.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The skeptic's concern about interaction terms proxying non-polynomial additive effects is valid and is the most substantive issue I identified. However, it does not invalidate the analytical contributions (which are exact) or the simulation results (where the generative model is known and polynomial). It primarily affects the interpretation of the empirical applications. I believe this can be addressed with an additional simulation and revised language, hence minor revision rather than major revision. The paper is a solid contribution to the HOB/HOM distinction literature."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The core contribution is the decomposition of interaction predictability into Δ (observed HOB) and Δ_A (additive benchmark), with structural synergy Ss = Δ − Δ_A isolating non-additive contributions. The analytical derivations are solid: the linear Gaussian case correctly gives Ss = 0 (Eq. 23), and the multiplicative model derivation in the Appendix is rigorous, using Isserlis' theorem properly. The 'masked structural synergy' regime — where Δ < 0 but Ss > 0, meaning a genuine non-additive mechanism hides behind redundancy-dominated observed behavior — is a real conceptual advance. The autoregressive simulations (Fig. 2) reproduce the expected regimes cleanly, and the authors ship code and data on GitHub. Credit earned on all of this. The HOB/HOM distinction from Rosas et al. is built on rather than merely cited, and the framework genuinely adds something the prior literature doesn't have: a practical, regression-based criterion for separating dependency-driven effects from mechanism-driven ones. The reader's take (ACCEPT, HIGH confidence) is well-calibrated on the analytical and simulation portions. Now the soft spot. The stress-test concern about polynomial basis misspecification in the empirical applications is real and not fully addressed. When sources are correlated, interaction monomials X₁ᵏX₂ʲ carry information about individual sources. If the true generative model is additive but non-polynomial, the full model (Eq. 21) can use interaction terms to mop up residual additive structure that the polynomial additive model (Eq. 22) fails to capture — yielding Ss > 0 as a false positive for HOMs. The analytical examples dodge this because they use polynomial functions. The simulations dodge it because p=2 matches the quadratic generative term. The climate and EEG applications have unknown functional forms and correlated sources, so this confound is live there. The authors' caveat that Ss is 'evidence of non-additive predictive structure rather than direct identification of the underlying generative mechanism' partially addresses it, but the applications still frame significant Ss as evidence 'compatible with HOMs' and label it 'masked structural synergy regime' in the EEG analysis. That's slightly overselling what the method can establish with unknown functional forms. This is a proportionate concern, not a load-bearing flaw. The analytical framework is correct, the simulations are controlled, and the code is public. The empirical interpretations just need more careful hedging — or ideally, a sensitivity analysis showing how Ss behaves under alternative additive bases (splines, kernels) for the real data. The paper is for researchers working on high-order interactions in complex systems who need to distinguish statistical synergy from mechanistic synergy. It deserves a serious referee who can push on the empirical claims and the basis-sensitivity issue.","headline":"Clean framework for separating dependency-driven synergy from genuine non-additive mechanisms, with one live confound in the empirical applications","tokens_in":23627,"tokens_out":648,"would_cite":true,"duration_ms":218152,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Synergy can hide behind redundancy in complex systems","keywords":[],"falsifier":"A system with a genuinely non-additive mechanism that is poorly captured by polynomial basis functions could yield Ss ≈ 0 (false negative), while a system with no non-additive mechanism but with a complex dependency structure not well captured by the additive polynomial model could yield Ss > 0 (false positive). The framework's ability to distinguish mechanism-driven from dependency-driven synergy therefore depends on the polynomial basis being a faithful representation of both the additive and non-additive components.","tokens_in":22745,"feed_emoji":"🧩","tokens_out":1722,"duration_ms":85830,"temperature":0.7,"pith_summary":"The paper introduces a quantity called structural synergy, defined as the gap between the best possible prediction of a target variable from two source variables and the best prediction achievable when those sources are constrained to contribute additively. The authors argue that this gap separates two distinct phenomena that are routinely conflated: high-order behaviours (HOBs), which are statistical patterns of synergy or redundancy visible in observed data, and high-order mechanisms (HOMs), which are genuine non-additive interaction rules in the data-generating process. The central claim is that these two levels can dissociate. A system can display redundancy-dominated HOBs — where the two sources appear to provide overlapping, redundant information about the target — while simultaneously containing a significant non-additive mechanism that is only detectable through structural synergy. The authors call this the masked structural synergy regime. They prove it analytically for a multiplicative model Y = cX₁X₂ + U, where structural synergy is strictly positive for all source correlations below one, even when the observed synergy-redundancy balance becomes negative (redundancy-dominated) as source correlation increases. They validate the framework in stochastic autoregressive simulations and demonstrate it in climate indices (ENSO) and source-reconstructed EEG motor data, both of which show redundancy-dominated HOBs coexisting with statistically significant structural synergy.","feed_headline":"Synergy can hide behind redundancy in complex systems","feed_subtitle":"A new predictability measure separates statistical patterns from genuine non-additive mechanisms, revealing hidden structure in climate and","key_machinery":"The framework operates on triplets of variables (two sources, one target). Four regression models are fit: single-source polynomial regressions for each source, a full bivariate polynomial model with interaction terms, and an additive bivariate model without interaction terms. From the residual variances of these models, three quantities are computed: Δ (interaction predictability, the observed synergy-redundancy balance), Δ_A (additive interaction predictability, the same balance under the additive constraint), and Ss = Δ − Δ_A (structural synergy). The classification of interaction regimes into five types — no high-order effect, dependency-driven synergy, dependency-driven redundancy,","core_discovery":"The paper's core result is the decomposition of observed high-order predictability into a dependency-driven component and a mechanism-driven component, operationalized through the comparison of two prediction errors: R* (the best achievable mean squared prediction error using both sources jointly and without restrictions) and R*_A (the best achievable error when the prediction is constrained to be additive in the two sources). Their difference, structural synergy Ss = R*_A − R*, is zero when the sources contribute additively to the target and positive when a non-additive combination of the sources is required. The analytical proof that Ss > 0 for all |r₁₂| < 1 in the multiplicative model Y =","pith_inferences":["The masked structural synergy regime suggests a general principle: whenever sources are correlated and the target depends on their interaction, increasing source correlation will eventually push the observed synergy-redundancy balance negative while leaving structural synergy intact. This means that in any real system with correlated inputs — which is most real systems — observed redundancy is not","The framework could be applied as a diagnostic for when linear or additive models are insufficient: if Ss is significant, one knows that additive models will fail to capture the system's predictive structure, which has direct consequences for model selection in fields ranging from econometrics to systems biology.","The polynomial regression implementation means structural synergy is sensitive to the choice of basis functions and polynomial order. A system with a non-additive mechanism that is poorly approximated by polynomials (e.g., discontinuous or highly oscillatory interactions) might show low Ss not because the mechanism is absent but because the model class cannot represent it."],"forward_implications":["If structural synergy is present even when observed redundancy dominates, then intervention strategies based solely on observed synergy-redundancy patterns may miss genuine non-additive mechanisms, leading to incorrect predictions about how a system responds to perturbations.","The framework could be extended beyond two sources using partial information decomposition, which would allow explicit separation of synergistic and redundant components rather than relying on the whole-minus-sum principle that conflates them.","In neuroscience, the finding that motor-network EEG dynamics show masked structural synergy suggests that redundancy-dominated information-theoretic analyses may have systematically underestimated the prevalence of non-additive neural interactions.","In climate science, the selective detection of structural synergy in ENSO-related index pairs (NINO34 with PDO or NTA) provides a data-driven criterion for identifying which variable combinations require mechanism-based models rather than additive descriptions."],"fun_headline_variants":["Redundant systems can hide non-additive mechanisms","Structural synergy separates statistical patterns from mechanisms","A predictability framework for detecting hidden structural synergy","Synergistic mechanisms can hide within redundant data","Distinguishing statistical dependencies from non-additive mechanisms"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The framework assumes that the additive model class — functions of the form h(X₁) + h(X₂) — adequately captures all effects arising from source dependencies, so that any excess predictability beyond the additive benchmark can be attributed to non-additive mechanism structure. In practice the additive benchmark is implemented via polynomial regression, where the choice of polynomial order and basis functions determines what counts as additive versus non-additive. If the basis","fun_headline_variants_meta":{"raw":{"variants":["Redundant systems can hide non-additive mechanisms","Structural synergy separates statistical patterns from mechanisms","A predictability framework for detecting hidden structural synergy","Synergistic mechanisms can hide within redundant data","Distinguishing statistical dependencies from non-additive mechanisms"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":1474,"prompt_tokens":612,"completion_tokens":862,"prompt_tokens_details":null},"tokens_in":612,"tokens_out":862,"duration_ms":29485,"temperature":1.0,"reasoning_tokens":830,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T15:06:07.737091+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A system with a genuinely non-additive mechanism that is poorly captured by polynomial basis functions could yield Ss ≈ 0 (false negative), while a system with no non-additive mechanism but with a complex dependency structure not well captured by the additive polynomial model could yield Ss > 0 (false positive). The framework's ability to distinguish mechanism-driven from dependency-driven synergy therefore depends on the polynomial basis being a faithful representation of both the additive and non-additive components.","supporting_citations":[],"review_version":1}