{"id":"efe24649-45e7-4ba7-85f8-2e647bb375fb","arxiv_id":"2501.02291","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Asymmetric Cauchy noise induces a nonlinear bias in the collective frequency of synchronized Kuramoto oscillators, an effect invisible to the Ott-Antonsen ansatz and captured via circular cumulants.","lead":"This paper shows that when the random noise acting on coupled oscillators has skewed, heavy-tailed statistics, the whole population's oscillation frequency shifts by an amount that depends on how synchronized it is. The result matters because it identifies a collective effect that standard models (Ott-Antonsen) completely miss, and provides equations to compute it.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the convergence caveat in Sec. III C does not affect the Kuramoto θ=0 regime where Eq. (46) is claimed.","rationale":"The paper's central claim is the nonlinear frequency bias in Eq. (46) for the Kuramoto ensemble with asymmetric Cauchy noise. The reader flagged the convergence of Eq. (29) and the approximate formula (23) as the weakest assumption. I examined the specific regime of the claim. For Kuramoto coupling with θ=0, the geometric multiplier in the κ2 series is bounded by 1/2, so the series converges uniformly for all ε>εcr; the paper's own caveat about divergence applies to a different coupling phase (mostly conservative), which is not used for the headline result. The approximation (23) appears only in the convergence analysis of Eq. (30), not in the derivation of S0 and S1, which use the exact C_{l+1} coefficients. The numerical validation is supported by an independent continued-fraction solver with 2000 modes, and the error plots in Fig. 5 show the rigorous expansion converging as more terms are kept. The only unproven numerical element is the self-similarity truncation at m∞=2000, but this is a standard and checkable numerical approximation; it does not invalidate the analytic derivation. Because the stated concern does not undermine the central result in the regime where it is claimed, the reader's CONDITIONAL verdict need not be changed. I agree with the reader that the paper is accept-shaped and well supported; I simply do not see a load-bearing residual risk that would require conditional acceptance rather than acceptance.","tokens_in":147,"tokens_out":19761,"duration_ms":186829,"concrete_test":"Recompute the continued-fraction solution for the Kuramoto ensemble with m∞=10^4 (or double m∞=2000 twice) at β=1 and ε/εcr=100, and compare the inferred collective frequency Ω with the m∞=2000 result; if the difference exceeds the plotted line width in Fig. 4b, the numerical ground truth used to validate Eq. (46) needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption concerns convergence of the perturbative series (29) and use of the approximate C_m formula (23). For the paper's central quantitative claim, this concern does not land. For the Kuramoto ensemble with θ=0 (h=εZ1/2), the multiplier in Eq. (29) satisfies |bZ1| = (ε−εcr)/(2ε−εcr) < 1/2 for all ε>εcr, so the series converges absolutely and rapidly; the admitted divergence risk is explicitly restricted to mostly-conservative coupling (θ≈π/2), which is outside the regime of Eqs. (45)-(46). Moreover, Eq. (23) is used only for the convergence heuristic in Eq. (30); the asymptotic constants S0 and S1 are defined via exact C_{l+1} from Eq. (22), and the reported numerical agreement in Fig. 5 confirms that the exact coefficients were used. The independent continued-fraction solver is a separate validation path and is not built on the CC truncation. The only residual uncertainty I see is the uncontrolled self-similarity truncation at m∞=2000 in the 'exact' solver, but this is a numerical check rather than a flaw in the central argument. I therefore find no load-bearing objection to the claim as stated for the Kuramoto θ=0 regime.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies populations of sin-coupled phase oscillators driven by independent asymmetric Cauchy noises, a case where the Ott–Antonsen (OA) ansatz is no longer exact. The authors derive a hierarchy of equations for circular cumulants (Sec. III A–B), a linear-in-βσ asymptotic expansion for time-independent states (Sec. III C), and low-dimensional truncations. For the Kuramoto ensemble, they predict and quantify a nonlinear bias of the collective oscillation frequency that is completely absent under the OA ansatz, with an explicit strong-coupling asymptotic law (Eq. 46). They also analyze the entrainment of individual oscillator frequencies (Sec. III D, IV C). The theoretical results are validated against a high-order continued-fraction solution (Sec. IV D) and reported in Figs. 4–6.","tokens_in":23945,"tokens_out":18709,"duration_ms":151166,"significance":"The reported frequency-bias effect is new and physically well motivated: the argument for generic asymmetry of non-Gaussian stable noises (Sec. II) is persuasive, and the quantitative prediction (Eq. 46) with the constants S_0, S_1 is a falsifiable result that goes beyond the OA ansatz. The paper has important strengths: the derivations are self-contained, the low-dimensional reductions are benchmarked against an independent numerical solver, and the paper honestly reports the regimes where the series in Eq. (29) may not converge. However, the manuscript contains at least two substantial typographical errors in key formulas (Eq. 23 and the definition of S_n in Eq. 44) that currently prevent a reader from reproducing the convergence argument and the asymptotic constants. These issues are fixable without changing the main conclusions.","major_comments":[{"comment":"The displayed approximation for C_m is misprinted. Eq. (23) reads C_m ≈ (−1)^m m! ln m, but the exact values (e.g., C_2 = ln 2, C_3 = (1/2) ln 3 − ln 2) and the approximate equality used in Eq. (30) show the intended formula is C_m ≈ (−1)^m / (m! ln m). With the printed formula, the terms of the series in Eq. (29) would be O(l! C_{l+1} (bZ1)^l) = O((l!)^2 ln l (bZ1)^l), which diverges for any nonzero bZ1; this would contradict the claim that the series converges for |bZ1| ≤ 1 and would undermine the asymptotic expansion (44)–(46). The authors should correct Eq. (23) and revise the convergence discussion in Sec. III C accordingly.","section":"Sec. III C, Eq. (23)"},{"comment":"The definition of the constants S_n in Eq. (44) is unclear. The displayed formula 'S_n = Σ_{l=1}^∞ (−1)^{l+1} ln l! / 2^l C_{l+1}' is ambiguous and does not follow from expanding Eq. (31) around bZ1 = 1/2. Since S_0 and S_1 enter the central asymptotic law Eq. (46) and are compared with numerics in Fig. 5, the authors must give the correct definition (presumably involving l! C_{l+1}/2^l multiplied by a polynomial in l) and explain how the numerical values 0.45158... and 0.60927... were obtained.","section":"Sec. IV B, Eq. (44)"}],"minor_comments":[{"comment":"There is a stray question mark at the end of the displayed formula for ν_ω in the β = 0 limit; this appears to be a typographical artifact and should be removed.","section":"Sec. III D, after Eq. (38)"},{"comment":"The 'exact' continued-fraction solution relies on the self-similarity closure B_{m∞+1} = B_{m∞} at m∞ = 2000. A brief convergence test (e.g., comparing m∞ = 1000 and 2000) would substantiate the claim of 'nearly machine accuracy' and would help the reader assess the reliability of the reference solution.","section":"Sec. IV D"},{"comment":"The statement that 'the case of a fast decay of the series in Eq. (29) is expected to be typical but not guaranteed' should be sharpened after the correction of Eq. (23): for |bZ1| < 1 the series converges absolutely, so the caveat applies only to the mostly-conservative coupling regime (θ ≈ π/2).","section":"Sec. III C"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely to be acceptable after the authors correct the misprints in Eqs. (23) and (44) and clarify the convergence discussion. The heavy self-citation is noticeable but the novel content is distinct from prior work. I recommend major revision because the misprints directly affect the reproducibility of the central quantitative claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper finds something real: for a Kuramoto ensemble driven by asymmetric Cauchy noise, the collective frequency is biased in a way that depends on the order parameter amplitude and vanishes at the synchronization threshold. This is genuinely new—the OA ansatz misses it entirely, and earlier circular-cumulant work did not identify the effect. The authors also give a tractable formalism: a circular-cumulant chain for asymmetric Cauchy noise, finite truncations, and a rigorous-looking linear-in-βσ expansion for stationary states. I checked the stress-test note on the reader's convergence worry, and I agree it does not land. For the Kuramoto case with dissipative coupling (θ=0), the multiplier in Eq. (29) is bounded by 1/2, so the series converges rapidly; the admitted divergence risk is explicitly restricted to mostly-conservative coupling, outside the regime where Eqs. (45)–(46) are claimed. The approximate C_m formula is only used for the convergence heuristic, not for the constants S0 and S1. So the central quantitative claim stands on solid ground. What the paper does well is validate. The continued-fraction solver with 2000 modes is an independent route from the CC truncation, and Fig. 5 shows the low-dimensional reductions track the exact solution across the parameter range, with honest error reporting. The asymptotic law (46) is checked against the full expansion and it works. The physical motivation—heavy-tailed asymmetric noise arising from synaptic pulse trains—is plausible, but it is qualitative; I would not want that sold as a theorem. The only real soft spot I see is that the ``rigorous'' language is a bit stronger than what is actually proven: the derivation of Eq. (27) and the recursive solution assume the series in (29) converges, and the paper only gives conditions for typical cases rather than a general proof. That is a legitimate caveat, but it does not weaken the Kuramoto θ=0 result, where the convergence bound is explicit and verified numerically. Self-citation is heavy but appropriate here; this is their own machinery and they use it correctly. This is a paper for researchers working on oscillator populations with non-Gaussian noise. I would take the results seriously and would cite the bias effect in my own work. It deserves a serious referee; send it out.","headline":"Asymmetric Cauchy noise produces a genuine nonlinear shift in Kuramoto collective frequency that the OA ansatz misses; the central claims are well supported and this deserves refereeing.","tokens_in":24393,"tokens_out":1774,"would_cite":true,"duration_ms":20829,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C31","37N25","60G51","34C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Asymmetric Cauchy noise biases the frequency of collective oscillations, an effect missed by the standard Ott-Antonsen ansatz; circular cumulants quantify it.","keywords":["asymmetric Cauchy noise","circular cumulants","Kuramoto model","collective oscillation frequency bias","Levy noise","Ott-Antonsen ansatz","synchronization","frequency entrainment"],"falsifier":"Direct numerical simulation of the Kuramoto ensemble with asymmetric Cauchy noise (or a high-order continued-fraction computation) for beta=1 and epsilon/epsilon_cr large should reproduce the asymptotic law (46); a persistent discrepancy larger than the stated truncation error, or a bias that does not saturate to a constant, would falsify the central claim.","tokens_in":2058,"feed_emoji":"🔄","tokens_out":2266,"duration_ms":83161,"temperature":0.7,"pith_summary":"This paper reports that in a population of sine-coupled phase oscillators driven by independent asymmetric Cauchy noises, the collective oscillation frequency is shifted away from the mean natural frequency. The shift is nonlinear: it depends on the amplitude of the collective mode and vanishes near the Kuramoto synchronization transition. The paper derives the strong-coupling asymptotic law $\\tilde{\\omega} = (2\\beta\\sigma/\\pi)(-S_0 + S_1(\\gamma+\\sigma)/\\varepsilon)$ with $S_0=0.45158...$ and $S_1=0.60927...$, valid for $\\varepsilon\\gg\\varepsilon_{\\mathrm{cr}}$. This effect is completely absent on the Ott-Antonsen manifold, which is applicable only for symmetric Cauchy noise. The paper develops a circular-cumulant formalism to capture the effect and validates it against high-accuracy continued-fraction solutions.","feed_headline":"Asymmetric noise biases collective oscillation frequency","feed_subtitle":"The shift vanishes at the synchronization threshold but saturates to a constant far from it.","key_machinery":"The central object is the infinite chain of circular-cumulant equations (Eq. 17), derived from the moment-generating function F(k,t)=langle $e^{{k e^{i phi}}$}rangle=sum_m Z_m k^m/m!. For asymmetric Cauchy noise the noise term is $\\sigma$ dot{Phi}^{(xi)}(m)=-|m|+i(2 $\\beta$ $\\sigma$/pi) m ln|m|, producing coefficients $G_m^{{(beta)}}$. The chain is truncated in two ways: finite 2- and 3-CC reductions (Eqs. 13-15), and a rigorous perturbative series for the second cumulant kappa_2 (Eq. 29), whose terms are controlled by the coefficient C_m approximately (-1)^m m! ln m. Substituting kappa_2 into the equation for Z_1 and solving the real and imaginary parts yields Eqs. (41)-(42) for (|Z_1|, tilde{omega}), and the strong-coupling limit gives the closed law (45)-(46). The continued-fraction solution of the stationary Fourier-mode equations (Eqs. 47-50) with 2000 modes serves as the exact benchmark.","core_discovery":"For a Kuramoto ensemble with intrinsic asymmetric Cauchy noise ($\\alpha$=1, skewness $\\beta$), the mean-field rotation frequency $\\Omega$ is biased relative to the mean natural frequency omega_0: the bias tilde{omega}=$\\Omega$-omega_0 is nonzero for $\\beta$ $\\sigma$ > 0 and epsilon > epsilon_cr, vanishes as the collective mode amplitude goes to zero near the transition, and saturates to -(2 $\\beta$ $\\sigma$/pi) S_0 plus a correction proportional to S_1(gamma+$\\sigma$)/epsilon in the strong-coupling limit. The bias arises from higher circular cumulants and is exactly zero on the Ott-Antonsen manifold. The paper shows that the 2- and 3-circular-cumulant reductions are accurate for the order parameter across the parameter range, but for the rotation bias they retain a finite non-decaying error in the strong-coupling limit, whereas the rigorous linear-in-$\\beta$-$\\sigma$ expansion of the cumulant chain makes the error tend to zero.","pith_inferences":["Because physical noise with Cauchy-type tails is generically asymmetric when microscopic fluctuations are asymmetric (e.g., different strengths of excitatory and inhibitory synaptic pulses), the reported bias should be observable in networks where the diffusion approximation breaks down.","The same circular-cumulant construction should carry over to alpha != 1 Levy noises, but the skewness term changes shape, so the saturation constants S_0 and S_1 will be alpha-dependent and require separate derivation.","The convergence failure of the series (29) for mostly conservative coupling suggests that in that regime a different resummation or a finite-CC closure may be needed; the continued-fraction benchmark remains available to test any alternative approximation."],"forward_implications":["The Kuramoto transition threshold stays at epsilon_cr=2(gamma+sigma), independent of noise skewness beta, while the collective frequency bias grows from zero at threshold to a nonzero constant in strong coupling.","Any Ott-Antonsen-based neural-mass model applied to networks with asymmetric heavy-tailed endogenous noise will systematically miss a rotation-frequency shift of order beta sigma, even when the order-parameter amplitude is accurately captured.","For individual oscillator frequency entrainment, the noise asymmetry breaks symmetry between subpopulations with fast and slow natural frequencies; the deviation of individual average frequencies from the mean-field rotation depends nonmonotonically on detuning.","The 2- and 3-CC reductions are accurate for the order parameter across the whole parameter range, but for the rotation bias one needs the full perturbative series (26)-(29) to make the error vanish in the strong-coupling limit."],"supporting_citations":[{"why":"Supplies the Ott-Antonsen ansatz, the low-dimensional manifold that the paper shows is inapplicable once the Cauchy noise is asymmetric.","marker":"[5]"},{"why":"Introduces the circular-cumulant framework that the paper uses to move beyond the Ott-Antonsen manifold.","marker":"[20]"},{"why":"Establishes the truncation and asymptotic analysis of circular-cumulant series that the paper adapts to the asymmetric-Cauchy case.","marker":"[21]"},{"why":"Derives circular-cumulant reductions for non-Gaussian (alpha-stable) noise, the direct starting point for the asymmetric-Cauchy equations.","marker":"[44]"},{"why":"Provides the symmetric-Cauchy (Lorentzian) low-dimensional description against which the asymmetric-noise results are contrasted.","marker":"[28]"},{"why":"Supplies the continued-fraction method used to compute the high-accuracy exact numerical solutions for validation.","marker":"[63]"}],"fun_headline_variants":["Asymmetric noise tilts collective oscillation frequency","Skewed Cauchy noise offsets collective frequency","Noise asymmetry induces nonlinear frequency shift","Collective frequency bias from asymmetric noise","Cauchy noise skewness warps sync frequency"],"cache_read_input_tokens":26496,"weakest_assumption_plain":"The quantitative results rely on the perturbative series for the second circular cumulant converging fast enough; the paper states that fast decay is typical but not guaranteed, and it can fail for mostly conservative coupling.","fun_headline_variants_meta":{"raw":{"variants":["Asymmetric noise tilts collective oscillation frequency","Skewed Cauchy noise offsets collective frequency","Noise asymmetry induces nonlinear frequency shift","Collective frequency bias from asymmetric noise","Cauchy noise skewness warps sync frequency"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000841,"raw_usage":{"total_tokens":3640,"prompt_tokens":899,"completion_tokens":2741,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":2676}},"tokens_in":515,"tokens_out":2741,"duration_ms":21320,"temperature":1.0,"reasoning_tokens":2676,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:13:48.453445+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Direct numerical simulation of the Kuramoto ensemble with asymmetric Cauchy noise (or a high-order continued-fraction computation) for beta=1 and epsilon/epsilon_cr large should reproduce the asymptotic law (46); a persistent discrepancy larger than the stated truncation error, or a bias that does not saturate to a constant, would falsify the central claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the circular-cumulant framework that the paper uses to move beyond the Ott-Antonsen manifold."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the truncation and asymptotic analysis of circular-cumulant series that the paper adapts to the asymmetric-Cauchy case."},{"cited_title":"V.\\ Dolmatova, I","cited_arxiv_id":null,"evidence_quote":"Derives circular-cumulant reductions for non-Gaussian (alpha-stable) noise, the direct starting point for the asymmetric-Cauchy equations."},{"cited_title":"Ya.\\ Khinchin, Continued Fractions (University of Chicago Press, Chicago and London, 1964)","cited_arxiv_id":null,"evidence_quote":"Supplies the continued-fraction method used to compute the high-accuracy exact numerical solutions for validation."}],"review_version":1}