{"id":"6a386911-45d1-4e8e-9db5-4b8952d08d55","arxiv_id":"2506.22160","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a finite Kuramoto population below the synchronization threshold, the order parameter fluctuations are described by the spectrum W(ω) = (2π/N)|(1+iω)/(1+iω-K/2)|² g(ω), with variance 1/[N(1-K/2)].","lead":"This paper derives a formula for the power spectrum of random collective oscillations that appear in finite groups of coupled oscillators even when the coupling is too weak to synchronize an infinite population. The result gives a quantitative handle on finite-size fluctuations in the Kuramoto model and matches numerical simulations away from the critical coupling.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The power spectrum (23) rests entirely on the self-consistency closure Eq. (22), which the paper asserts from a nested-configuration analogy but does not derive from the microscopic dynamics (Eq. 3).","rationale":"The paper's central claim is that in the subcritical finite-size Kuramoto model, the order-parameter power spectrum is given by Eq. (23), with the variance (25). The reader's weakest-assumption analysis correctly identifies Eq. (22) as the load-bearing closure. I re-derived the expected closure from the linearized microscopic dynamics and found that it indeed yields w = w0 + (K/2)Sw to leading order in 1/N, provided one replaces the sample frequency kernel by its average; this suggests the formula is correct. However, the paper as written does not supply this derivation, and instead invokes a nested-configuration heuristic that, taken literally, could imply a different two-loop feedback. Therefore the conditional verdict is appropriate: the gap is real, but it is pluggable. The variance (25) matching Daido's independent formula is strong supporting evidence. The numerical simulations are visually consistent but lack error bars, and the spectrum deviates near K=2 as expected. Overall, I would keep the CONDITIONAL verdict and require either the microscopic derivation of Eq. (22) or error-barred numerical verification of the spectrum shape.","tokens_in":10824,"tokens_out":28788,"duration_ms":299407,"concrete_test":"Derive Eq. (22) from Eq. (3) without the nested configuration: set θ_j(t)=ω_j t+θ_j0+φ_j(t), linearize the phase equation to dφ_j/dt = K Im[s(t)e^{-i(ω_j t+θ_j0)}], substitute into s(t)=(1/N)Σ e^{iθ_j}, and replace the sample average (1/N)Σ e^{iω_jτ} by its expectation e^{-|τ|} for the Lorentzian g(ω). If the resulting frequency-domain equation is w = w0 + (K/2)Sw with S=1/(1+iω), the closure is justified to leading order in 1/N; if it yields any extra factor or loop, Eq. (23) must be revised. Alternatively, measure the linear response kernel numerically by applying a weak external field to a finite-N population and checking that the kernel's Fourier transform equals (K/2)S(ω).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (22), w(ω) = (K/2)S(ω)w(ω) + w0(ω) with S(ω)=1/(1+iω), is the pivotal step: all subsequent results, including the power spectrum (23) and variance (25), are algebraic consequences. The paper justifies this equation by analogy with the 'nested' configuration from Refs. 51-54, but does not derive it from the finite-N dynamics of Eq. (3). The concern is not merely presentational: if the correct microscopic closure were, for instance, w = w0 + (K/2)^2 S^2 w (a two-loop feedback, which a literal reading of the nested set-up could suggest), the spectrum would acquire a different frequency dependence and the variance would no longer match Daido's formula at first order in K. Thus the central claim's validity depends on whether Eq. (22) is the correct linear self-consistency. The paper also linearizes around r=0, which fails near K=2, but that is acknowledged; the closure is the unacknowledged risk.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies finite-size fluctuations in the Kuramoto model below the synchronization threshold. It represents the order parameter of a finite population as the sum of the infinite-population order parameter and a 'shot noise' term, and proposes a self-consistency condition in Fourier space (Eq. (22)) that leads to an analytic power spectrum (Eq. (23)) and variance (Eq. (25)). Numerical simulations for N=10^5 agree with the predicted spectra for K=0.5, 1, and 1.5, and with the variance up to K=1.9; the variance matches Daido's earlier formula. The main claim is that finite-size populations exhibit coupling-dependent collective oscillations below K=2 even though the thermodynamic mean-field state is asynchronous.","tokens_in":10982,"tokens_out":12304,"duration_ms":125162,"significance":"If correct, the result provides a simple, parameter-free formula for finite-size collective spectra in a canonical model, extending the shot-noise approach from neural populations to the Kuramoto model and quantitatively connecting finite-size fluctuations to coupling strength. The agreement with independent numerics and with Daido's variance formula is strong evidence that the final expression captures the physics. The main weakness is that the central closure Eq. (22) is introduced heuristically; the paper's value would be substantially increased by deriving it from the microscopic equations.","major_comments":[{"comment":"In Eq. (19) the mean-field response of the infinite population to the finite-population order parameter s(t) has the wrong sign in the nonlinear term. Replacing r by s in the Ott-Antonsen equation (8) and evaluating at ω=-i yields dr/dt = -r + (K/2)(s - s* r^2), not -r + (K/2)(s r^2 - s*). The printed equation linearizes to dr/dt ≈ -r - (K/2)s*, which is not the response used in the feedback term of Eq. (22); the corrected equation linearizes to dr/dt ≈ -r + (K/2)s. Please correct Eq. (19) and make the linearization leading to Eq. (22) explicit.","section":"V, Eq. (19)"},{"comment":"Equation (22), w(ω) = (K/2)S(ω)w(ω) + w0(ω), is the pivotal self-consistency condition, and Eqs. (23) and (25) are algebraic consequences of it. The manuscript asserts this condition from the nested-population analogy but does not derive it from the microscopic dynamics (3). In particular, the condition assumes that the free shot-noise component w0 keeps its uncoupled spectrum after the feedback is switched on, and that the feedback enters linearly with coefficient K/2. Please provide a derivation from the linearized microscopic dynamics, or at least an explicit statement of the approximation and a justification of why w0 is unchanged. As written, the central spectral formula rests on an uncontrolled closure.","section":"V, Eq. (22)"}],"minor_comments":[{"comment":"The integration limits in Eq. (24) are printed as ∫_∞^∞; this should read ∫_{-∞}^∞.","section":"V, Eq. (24)"},{"comment":"The text says the variance is computed 'Using Eq. (21)', but Eq. (21) is the intermediate approximate spectrum that was superseded; the variance formula (25) follows from Eq. (23). Please correct this cross-reference.","section":"V, after Eq. (23)"},{"comment":"The caption contains grammatical errors such as 'a system Kuramoto system' and the colloquial 'Mind the difference'; please rewrite it.","section":"Fig. 1 caption"},{"comment":"The reference list is uneven: several entries lack DOIs or publisher information (for example Refs. 15, 16, 17, 46, and 51). Please standardize the format.","section":"References"},{"comment":"The theoretical spectrum is a sum of delta functions, while the numerics use FFT and moving-average smoothing; the comparison would be clearer if the smoothing window were defined in the text rather than only in the figure caption.","section":"V, Eq. (17)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope and the final formula is likely correct, as supported by numerical simulation and Daido's variance. However, the sign error in Eq. (19) and the underexplained closure in Eq. (22) are load-bearing and should be resolved before publication. I see no grounds for rejection, but the derivation needs a serious revision pass."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers what it promises: an explicit analytic power spectrum for finite-size order-parameter fluctuations in the subcritical Kuramoto model, Eq. (23), with no free parameters. That formula is new as a spectral shape — the prior finite-size scaling papers give exponents, and Daido's result (ref. 59) gives only the variance, which the paper reproduces exactly as D = 1/[N(1-K/2)]. Extending the authors' shot-noise approach from neural masses to Kuramoto is a meaningful transfer, and the numerics back them up: the simulated spectra line up for K=0.5, 1, 1.5 and the variance tracks the prediction up to K=1.9. This is a solid, useful contribution for the oscillator-network community.\n\nThe main soft spot is exactly where the reader put it: the self-consistency condition (22) is the load-bearing step, and it is stated as an ansatz. The paper says 'one should write' it, with no derivation from the microscopic dynamics. That is a genuine gap. The stress-test suggestion of a two-loop closure as an alternative doesn't land, because the paper's single-loop form with that particular sign matches both the simulated spectra and Daido's variance; if the two-loop form were right, the variance would not match at first order. So the closure is empirically validated even if it is not derived. A referee should ask the authors to derive it, perhaps from a linearized Vlasov equation, but should not treat it as a fatal flaw.\n\nMinor issues: no error bars in the spectral plots, and the linearization breaks down near K=2, which the authors acknowledge. The divergence of the spectrum at K=2 is real but unsurprising. The citation pattern is normal, with the authors' own prior work as the methodological basis, which is straightforward.\n\nWho it's for: people working on finite-size effects in Kuramoto-like systems, and anyone using Ott-Antonsen reductions or shot-noise approaches. It is a nice subfield result, not a paradigm shift. I would send it to a serious referee. The referee should request a derivation of (22) and some error quantification, then accept.","headline":"Useful subfield result: the first explicit spectrum for finite-size Kuramoto fluctuations below K=2, well-supported by numerics, but the central self-consistency condition is asserted rather than derived.","tokens_in":11554,"tokens_out":8444,"would_cite":true,"duration_ms":89259,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C15","34D06"],"pacs":["05.45.Xt","05.40.-a"],"model":"deepseek-v4-flash","headline":"Below the Kuramoto critical coupling, finite populations still show collective oscillations, and this paper derives their full power spectrum.","keywords":["Kuramoto model","finite-size effects","shot noise","power spectrum","collective oscillations","order parameter fluctuations","subcritical coupling","synchronization"],"falsifier":"At $K=1.8$ and $N=10^5$ with a Lorentzian frequency distribution of unit width, Eq. (23) predicts a zero-frequency spectrum $W(0)=(2\\pi/N)(1/(1-K/2))^2(1/\\pi)=8/[N(2-K)^2]\\approx 2.0\\times 10^{-4}$ and a total variance $D=1/[N(1-K/2)]=2.0\\times 10^{-4}$. If a direct simulation of the full Kuramoto dynamics yields a low-frequency plateau or variance that differs from these values by more than the statistical error, the feedback closure underlying the spectrum is wrong; the narrow non-thermal peaks the paper already observes at $K=1.5$ mark the natural place to look for the discrepancy.","tokens_in":10506,"feed_emoji":"🔄","tokens_out":8193,"duration_ms":81525,"temperature":0.7,"pith_summary":"The paper is trying to establish that a finite population of Kuramoto phase oscillators exhibits collective oscillations even when the coupling is below the critical value $K=2$, where the infinite-size mean-field theory predicts a perfectly asynchronous state with zero order parameter. Using a shot-noise picture, the authors derive an analytic power spectrum for these emergent oscillations and show it depends on the coupling strength, with a variance that grows as $D=1/[N(1-K/2)]$. The derivation treats the finite population as embedded in a much larger population whose response is described by the Ott–Antonsen reduction, and the finite-size fluctuations feed back on themselves through a self-consistency condition. Numerical simulations of populations of $10^5$ oscillators confirm the spectrum and variance for couplings up to about $K=1.9$, with deviations appearing only close to the critical point where linearization fails. If correct, this means finite-size fluctuations alone can sustain synchronization-like collective behavior below the thermodynamic threshold, which matters for real neural, grid, and coupled-oscillator systems that are finite.","feed_headline":"Finite-size Kuramoto populations oscillate below critical coupling","feed_subtitle":"Real oscillator networks are finite; this theory predicts their collective fluctuations below the mean-field threshold.","key_machinery":"The central object is the shot-noise decomposition $s(t)=r(t)+\\chi(t)$ together with the nested configuration, in which the finite population of size $N$ sits inside an infinite population driven by the finite population's order parameter. This makes the infinite population a linear filter with frequency response $S(\\omega)=1/(1+i\\omega)$, obtained from the Ott–Antonsen reduction around the asynchronous state. The argument is carried by the self-consistency condition $w(\\omega)=(K/2)S(\\omega)w(\\omega)+w_0(\\omega)$, which asserts that the total fluctuation spectrum is the free shot-noise spectrum $w_0$ plus the coupled population's linear response to that same fluctuation; solving it yields the closed form $W(\\omega)$.","core_discovery":"The central discovery is that the power spectrum of the order parameter fluctuations in a finite subcritical Kuramoto population is $W(\\omega)=(2\\pi/N)\\,|(1+i\\omega)/(1+i\\omega-K/2)|^2\\,g(\\omega)$, where $g(\\omega)$ is the frequency distribution and $N$ the population size. The variance obtained by integrating this spectrum, $D=1/[N(1-K/2)]$, matches a formula Daido derived earlier by a different method. The underlying mechanism is shot noise: the finite population's order parameter $s(t)$ is written as $r(t)+\\chi(t)$, where $r$ is the infinite-population mean field and $\\chi$ is the fluctuation. In the nested configuration, the finite population is part of an infinite population, so $\\chi$ drives the mean field through the linear response $S(\\omega)=1/(1+i\\omega)$, and the resulting response feeds back, producing the self-consistency condition $w=(K/2)Sw+w_0$. The paper shows this feedback amplifies the free shot noise, especially at low frequencies, and that the amplification diverges as $K\\to 2$, marking the approach to the synchronization transition.","pith_inferences":["The same feedback logic suggests a direct experimental test: artificially injecting an external fluctuation into a subcritical Kuramoto ensemble and checking whether the measured order-parameter spectrum obeys the same linear gain factor $|1/(1-K S(\\omega)/2)|^2$ would probe the closure without relying on the natural shot noise.","Because the linear response $S(\\omega)=1/(1+i\\omega)$ is derived for a Lorentzian frequency distribution, a natural extension is to recompute the spectrum for Gaussian or uniform frequency distributions; a failure of the formula there would mark the boundary of the shot-noise closure's validity.","The divergence of the zero-frequency spectrum as $K\\to 2$ implies that for finite $N$ the transition is smeared over a width set by a finite-size scaling law; connecting the shot-noise variance to known finite-size scaling exponents of the Kuramoto model is a natural next step."],"forward_implications":["For any finite population with $0