{"id":"76cb4144-2722-4e5e-b44b-7a64fd7ee41e","arxiv_id":"2602.04905","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A two-parameter random Markov ensemble shows heterogeneity, not asymmetry, controls most observables near a predicted critical locus; human fMRI and EEG both fall near that locus.","lead":"The paper extends a random Markov model with a new asymmetry parameter and finds that heterogeneity, not irreversibility, dominates most dynamical measures, with a predicted critical line. Applied to human fMRI and EEG, both rest-state datasets land near that line, suggesting a common regime for brain dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Empirical inference is not identifiable: Eq. 22 is a likelihood for latent Q, while HMM-MAR data provide column-normalized M; unobserved column scales S_b are not integrated out, so reported ε,γ may be gauge artifacts.","rationale":"The reader's weakest assumption is the RMT universality behind Eq. 15, which the authors explicitly flag and which can be checked by numerical spectral-edge measurements. I find a more decisive, unacknowledged issue in the inference pipeline. The likelihood used for MLE is written for Q (Eq. 22), but the observable data are transition probabilities M, related to Q by per-column normalization. Since M is invariant under independent column rescalings of Q, while the ensemble density is not, the MLE is gauge-dependent unless column scales are integrated out or raw counts are modeled. The paper does neither. For N ≈ 11 (fMRI), the LLN approximation that would make column scales self-averaging is quantitatively invalid: the coefficient of variation of a column sum is O(1), so the mapping from M to Q is highly non-unique. The reported proximity of both modalities to the critical locus could therefore be an artifact of the chosen gauge rather than a property of the data. A synthetic-data recovery test, using the actual generative model for M and the paper's MLE, would settle this directly. Because this concern targets the empirical super-universality claim but can in principle be resolved by reanalysis, the existing CONDITIONAL verdict is appropriate: the paper should not be accepted until the inference is validated on M, not Q.","tokens_in":13867,"tokens_out":21110,"duration_ms":199601,"concrete_test":"Synthetic recovery test: Draw Q from Eq. 7 for (ε, γ) at the reported fMRI/EEG values (e.g., ε = 0.2, γ = −0.4, N = 11 and N = 165), normalize columns to M_ab = Q_ab / Σ_c Q_cb, then apply the paper's MLE (Eqs. 23–25) to M treated as Q. Compare recovered (ε̂, γ̂) to true values. Also implement the correct marginal likelihood ∫ ∏_b dS_b S_b^{N−1} P(Q = M S; ε, γ) and compare. If the MLE bias exceeds roughly the observed fMRI–EEG separation (Δ(ε log(N+1)) ≈ 0.14), the empirical placement in Figs. 7 and 9 is unreliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is in Sec. II.A, not the RMT edge. The generative model is for the primitive matrix Q (Eq. 7), while the data pipeline (HMM-MAR) outputs transition probabilities M_ab = Q_ab / S_b with S_b = Σ_c Q_cb. The likelihood in Eq. 22 is the density of Q, not of the observed M: it contains no integral over the N unobserved column scales S_b, and it is not invariant under Q_ab → λ_b Q_ab (the log-quadratics in Eq. 4 shift under x → x + log λ_b). Thus the MLE equations (23)–(25) select a gauge for Q from M without justification. The paper's validation sentence — 'we confirmed ... reproduces imposed parameter values' — does not close this gap unless it was performed on M rather than Q. The issue is severe for the fMRI comparison because N ≈ 11: at ε ≈ 0.2 the column-sum coefficient of variation is O(1), so the large-N replacement S_b ≈ N⟨Q⟩ used implicitly to equate log Q with log M is badly violated. If the gauge choice is biased, the reported ε log(N+1) ≈ 0.53, γ ≈ −0.32 for fMRI and the analogous EEG values are not trustworthy, and the super-universality claim loses its empirical support.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a two-parameter ensemble of random discrete-time Markov models, extending an earlier heterogeneous ensemble by adding an asymmetry parameter γ that correlates forward and backward log-rates. The central theoretical object is the critical locus ε_c(γ,N) obtained in Eq. (15) by moment-matching the log-normal Q ensemble to the Gaussian elliptical law of Ref. [28] and using a shift lemma to transfer the spectral edge to the stochastic matrix M. The authors study relaxation times, entropy production, and predictive information across this ensemble, finding that most observables depend strongly on ε but only weakly on γ away from the symmetric limit. They then apply maximum-likelihood inference to human fMRI and EEG data, reporting that both datasets fall near the predicted critical locus with moderately negative γ (fMRI: mean ε log(N+1)≈0.53, γ≈−0.32; EEG: 0.39, −0.42), which they interpret as evidence for a \"super-universality\" of human resting-state brain dynamics. Numerical simulations in Figs. 2–5 validate several ensemble predictions, and the paper explicitly acknowledges that a rigorous distinguished limit for the critical locus remains to be shown.","tokens_in":14246,"tokens_out":4153,"duration_ms":49284,"significance":"If correct, the paper provides a useful null model that unifies critical slowing down and broken detailed balance in a single two-parameter ensemble, with falsifiable predictions about spectral reorganization, entropy production, and predictive information. The numerical validation of the critical locus (Figs. 2–3) and the observation that entropy production is controlled primarily by heterogeneity rather than by γ are valuable and extend the earlier γ=0 results in a natural way. The application to two independent neuroimaging datasets is ambitious and the claim of a common region in the ε–γ plane is provocative. However, the empirical support for this claim rests on a maximum-likelihood procedure whose identifiability is not established (Sec. II.A), and the theoretical locus itself relies on a moment-matching step whose quantitative accuracy for log-normal entries at finite N is not assessed beyond a few simulation points. The paper is transparent about the latter limitation, but the inference issue is not acknowledged and is load-bearing for the main data conclusion.","major_comments":[{"comment":"The log-likelihood is written for the primitive matrix Q, whereas the data pipeline (HMM-MAR) provides the column-normalized matrix M_ab = Q_ab / S_b with S_b = Σ_c Q_cb. The likelihood contains no integral over the N unobserved column scales S_b, and it is not invariant under Q_ab → λ_b Q_ab: the log-quadratic terms in Eq. (4) and the h, a, and d statistics in Eq. (22) all shift under x → x + log λ_b. Consequently the MLE equations (23)–(25) select a gauge for Q from M without justification. The validation sentence in Sec. II.A (\"we confirmed ... reproduces imposed parameter values\") does not close this gap unless the test was performed on M rather than Q. This is severe for the fMRI comparison because N≈11 and ε≈0.2 implies strong column-sum fluctuations, so the replacement S_b ≈ N⟨Q⟩ is badly violated. If the gauge choice is biased, the reported ε log(N+1)≈0.53, γ≈−0.32 (and the analo","section":"Sec. II.A, Eq. (22)"},{"comment":"The critical locus is derived by applying the Gaussian elliptical law of Ref. [28] to P_ij ∝ Q_ij/⟨Q⟩ − 1 after matching only the first two moments of the log-normal entries. Ref. [28] is a Gaussian result, and log-normal entries are heavy-tailed in the small-ε phase; the paper itself notes that the law-of-large-numbers approximation M_ab ≈ Q_ab/(N⟨Q⟩) \"will break down in the small-ε phase\" and that a rigorous distinguished limit \"remains to be shown.\" The numerical validation in Figs. 2–3 supports the locus at N=32 and N=64, but the empirical application uses this finite-N locus at N≈11, where heavy-tail and finite-N corrections may be substantial. The authors should quantify the sensitivity of ε_c(γ,N) to the Gaussian-moment-matching assumption, for example by comparing Eq. (15) with simulations using alternative heavy-tailed positive distributions or by a finite-N crossover analysis.","section":"Sec. I.B, Eqs. (14)–(15)"},{"comment":"The claim that human fMRI data exhibit significantly enhanced variability compared with ensemble expectations is based on width ratios at 10% of the histogram maximum, computed for a single representative γ=−0.3 and without error bars or bootstrap intervals. The data points in Fig. 7 show a broad distribution of γ values, so the appropriate null model should either integrate over the fitted (ε,γ) distribution or at least report the sampling uncertainty of the width ratios. As it stands, the conclusion that the spread \"does not reflect natural variability\" at any putative parameter set is not quantitatively supported; part of the apparent excess variability may simply reflect the spread of γ across subjects. This concern does not affect the central theoretical construction, but it weakens one of the paper's two main empirical conclusions.","section":"Sec. II.B, Fig. 8"}],"minor_comments":[{"comment":"The constant q is introduced as a scale but its role is not fully specified. Eq. (2) and Eq. (3) depend on q only through scale, whereas Eq. (22) treats q as a free parameter. Please clarify whether q is identifiable from M or whether it absorbs part of the column-scale gauge in the inference procedure.","section":"Sec. IA, Eq. (4)"},{"comment":"The abstract reports mean γ≈−0.32 for fMRI, while the text of Sec. II.B states \"the mean value for γ lies at γ≈−0.4.\" These numbers should be reconciled.","section":"Sec. II.B"},{"comment":"The caption refers to \"solid\" and \"dashed\" curves but does not specify which is which. The figure legend should be reproduced in the caption for clarity.","section":"Fig. 4 caption"},{"comment":"The large-ε estimate ⟨Σ̇⟩/k_B ≈ (1+γ)/(2ε) should state explicitly that it assumes π_i≈1/N and M_ij≈Q_ij/(N⟨Q⟩), and that the limit is ε≫1 at fixed γ. The figure comparison at ε≈ε_c includes substantial corrections, and the dotted lines might be misinterpreted as a general formula.","section":"Eq. (18) and Fig. 4"},{"comment":"Refs. [17] and [40] appear to be the same paper (Vidaurre et al., NeuroImage 2018). Please deduplicate. Also, the duplicate NeuroImage citation in [40] should be checked.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The inference identifiability problem in Sec. II.A is the main obstacle. If the authors can show that the MLE on column-normalized M (either by integrating out S_b or by deriving an exact likelihood for M) reproduces known parameters in simulations at N≈11, the empirical conclusions would be much more credible. The paper is otherwise a solid extension of the γ=0 ensemble, with honest acknowledgment of the RMT rigor gap; the data comparison is the part that needs the most work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core theory here is a clean extension of the γ=0 ensemble from the authors' earlier work. The new piece is the γ-dependent critical locus in Eq. 15, obtained by matching to the Gaussian elliptical law of Sommers et al. and then using a shift lemma. The authors are honest that the distinguished limit is not rigorous, and their own numerical validation in Figs. 2–3 gives good support for the locus at finite N. I also found the result that entropy production depends weakly on γ (except near γ→−1) genuinely informative: it warns against reading scalar irreversibility measures as clean proxies for microscopic asymmetry. That is the part of the paper that deserves to survive.\n\nThe soft spots are where the empirical claims start. The stress-test note is right: the likelihood in Eq. 22 is a density for the primitive matrix Q, but the data pipeline (HMM-MAR) produces column-normalized transition matrices M_ab = Q_ab/S_b. The unobserved column scales S_b are never integrated out, and the likelihood is not invariant under Q_ab → λ_b Q_ab. So the MLE equations (23)–(25) are maximizing the wrong objective. That makes the reported ε log(N+1) ≈ 0.53 and γ ≈ −0.32 for fMRI, and the analogous EEG values, unreliable. The paper's validation sentence — “reproduces imposed parameter values” — does not close the gap unless it was run on M rather than Q, and nothing in the text says it was. For N ≈ 11 the column-sum fluctuations are large, so this is not a small-N correction.\n\nIf that inference step is broken, the “super-universality” claim falls, because it rests entirely on both datasets landing near the same critical locus in the ε–γ plane. The variability comparison also lacks uncertainty quantification: the width ratios are reported without error bars or any statistical test, so “significantly more variability” is not demonstrated.\n\nSo my take: the theory is a legitimate contribution and the paper deserves referee time, but the empirical section needs to be substantially reworked or dropped. The authors should either integrate out the column scales and redo the inference, or clearly state that they are inferring Q-distribution parameters from Q under an assumption that the column sums are constant; the current presentation does not support the conclusions drawn from the data.","headline":"The γ-dependent critical locus is a solid theoretical extension, but the MLE on column-normalized data is mismatched to the Q-likelihood, so the reported brain-data ε and γ may be gauge artifacts.","tokens_in":738,"tokens_out":905,"would_cite":true,"duration_ms":27838,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that a two-parameter random Markov ensemble captures both critical slowing down and broken detailed balance, and that human resting-state fMRI and EEG both sit near the predicted critical locus.","keywords":["random Markov matrices","criticality","broken detailed balance","entropy production","predictive information","random matrix theory","brain dynamics","maximum likelihood inference"],"falsifier":"Compute the finite-N spectral edge of the log-normal ensemble for γ≠0 in the small-ε phase (e.g., N=64, ε=0.1 ε_c): if the largest real eigenvalue departs from λ* of Eq. 15 by more than O(1/N), the transferred elliptical-law bound that sets the critical locus fails. Alternatively, an independent resting-state fMRI or EEG dataset whose inferred (ε, γ) falls far from the predicted locus (e.g., γ>0 or ε log(N+1) not near 0.5) would falsify the super-universality claim.","tokens_in":13748,"feed_emoji":"🧠","tokens_out":6344,"duration_ms":61401,"temperature":0.7,"pith_summary":"The paper extends a previously studied random Markov ensemble to two parameters: heterogeneity ε, which controls how uneven the transition rates are, and asymmetry γ, which correlates forward and backward rates. It derives a critical locus ε_c(γ,N) at which relaxation times diverge and spectral universality breaks down. A central finding is that most observables—entropy production, predictive information, and relaxation time—depend strongly on ε but only weakly on γ, except near the symmetric limit γ→−1. Applying maximum-likelihood inference to human fMRI and EEG resting-state data, the paper finds that both modalities fall near the predicted critical locus with similar parameters, supporting a 'super-universality' of human brain dynamics. The data means are captured by the null model, but the data show substantially larger variability than the ensemble predicts, indicating subject-specific structure beyond random expectations.","feed_headline":"Brain data lands on predicted critical line of random Markov model","feed_subtitle":"A two-parameter random Markov ensemble ties criticality to irreversibility; fMRI and EEG match it.","key_machinery":"The central object is the two-parameter maximum-entropy ensemble on transition matrices, with Q_ab log-normally distributed and pairwise correlation γ between Q_ab and Q_ba; heterogeneity is measured by h(Q) = (1/N²)Σ log²(Q_ab/Q) and log-asymmetry by a(Q). The argument for criticality is carried by the spectral-edge formula λ* (Eq. 15), obtained by mapping the fluctuations of M to a Gaussian random matrix ensemble with an elliptical law, using a law-of-large-numbers approximation M_ab ≈ Q_ab/(N⟨Q⟩) and a shift lemma showing that the rank-one shift in M does not change eigenvalues. The empirical application uses maximum-likelihood equations for ε, γ, and q, solved by iteration, to infer para","core_discovery":"The paper introduces a two-parameter ensemble of random discrete-time Markov models in which log transition rates are drawn from a correlated log-normal distribution, with heterogeneity ε and pairwise asymmetry γ. Using the elliptical law for Gaussian random matrices and a shift lemma, it derives the spectral-edge formula λ* (Eq. 15) and identifies the critical locus λ*=1, where the largest real part of the eigenvalues reaches unity and relaxation times diverge as ε→ε_c(γ,N). Below this locus the spectrum reorganizes into 'bicycle spokes'. The paper shows numerically and analytically that entropy production, predictive information, and mean relaxation time are controlled primarily by ε, with","pith_inferences":["If heterogeneity, not asymmetry, drives most observables, then reports of heightened irreversibility in conscious versus unconscious states could be reinterpreted as changes in effective heterogeneity rather than stronger microscopic driving—a distinction testable by computing pairwise forward/backward statistics on inferred Markov models.","The same ensemble could serve as a null model for other biological time series (e.g., heart-rate dynamics, gene-regulatory networks) where criticality and nonequilibrium are both claimed; a similar collapse onto the critical locus would generalize the proposed super-universality beyond the brain.","The MLE framework could be extended to include subject-level covariates (age, disease, task) to test whether the residual variability correlates with phenotype; the current EEG dataset shows no systematic group differences, but the authors did not control for age or medication.","A rigorous distinguished limit (N→∞ with ε/ε_c fixed) would either confirm or move the predicted locus; until then, the data comparison should be read as a quantitative but approximate test."],"forward_implications":["If the predicted critical locus is correct, relaxation-time divergence and spectral universality breakdown occur together in any Markov model with log-normal heterogeneity and pairwise asymmetry, giving a concrete null model for criticality.","Entropy production, predictive information, and relaxation times are largely set by heterogeneity ε, so empirical changes in irreversibility need not imply changes in microscopic time-asymmetry.","Distance from criticality and irreversibility measures are strongly correlated in this ensemble, making scalar irreversibility metrics ill-conditioned for inferring asymmetry without pairwise or cycle statistics.","Human resting-state fMRI and EEG both sit just to the critical side of the predicted locus with similar (ε, γ), supporting a 'super-universality' of human brain dynamics across measurement modalities.","Data variability exceeding ensemble expectations provides a quantitative way to detect subject-specific structure beyond the null model."],"fun_headline_variants":["Random Markov model unifies brain criticality and irreversibility","fMRI and EEG match predicted critical locus of random model","Critical slowing and broken balance captured in brain data","Two-parameter random ensemble predicts brain dynamics near criticality"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The predicted critical line assumes that the heavy-tailed log-normal transition-rate distribution can be replaced, for locating the spectral edge, by a Gaussian ensemble matched only in mean and pairwise correlation—an approximation the authors note is not proven in the small-ε phase.","fun_headline_variants_meta":{"raw":{"variants":["Random Markov model unifies brain criticality and irreversibility","fMRI and EEG match predicted critical locus of random model","Critical slowing and broken balance captured in brain data","Two-parameter random ensemble predicts brain dynamics near criticality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000192,"raw_usage":{"total_tokens":1175,"prompt_tokens":729,"completion_tokens":446,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":380}},"tokens_in":473,"tokens_out":446,"duration_ms":5193,"temperature":1.0,"reasoning_tokens":380,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T04:46:03.346049+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the finite-N spectral edge of the log-normal ensemble for γ≠0 in the small-ε phase (e.g., N=64, ε=0.1 ε_c): if the largest real eigenvalue departs from λ* of Eq. 15 by more than O(1/N), the transferred elliptical-law bound that sets the critical locus fails. Alternatively, an independent resting-state fMRI or EEG dataset whose inferred (ε, γ) falls far from the predicted locus (e.g., γ>0 or ε log(N+1) not near 0.5) would falsify the super-universality claim.","supporting_citations":[],"review_version":1}