{"id":"82730fa7-9ab0-4712-b8ad-f2265c6ff4d0","arxiv_id":"2506.23181","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Finite-size fluctuations trigger a first-order synchronization transition in inertial Kuramoto oscillators with triadic coupling, with critical coupling scaling as N^gamma, where gamma grows with inertia.","lead":"This paper simulates inertial Kuramoto oscillators coupled through three-body (2-simplex) interactions and finds that finite network size allows a jump from desynchronized to synchronized behavior, even though the infinite-size theory predicts no such jump. The authors report a power-law relation between network size and the coupling where this jump occurs, with larger inertia shifting the jump to stronger coupling.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reported K2c-N power law rests on a fixed-time crossing rule, so the extracted exponents may be an artifact of the 200000-step horizon; vary T and trial count to test.","rationale":"The paper's central physical scenario is plausible: finite-size fluctuations can kick a finite-N system over the unstable branch and onto the synchronized branch in the bistable region, and the time series in Figs. 1 and 7 show real escapes. Eq. A5 provides a concrete deterministic mean-field picture for m=0, and the observed narrowing of P(r1) with N is consistent with O(1/sqrt N) fluctuations. The paper is not internally inconsistent or fraudulent. The soft spot is quantitative: Fig. 6 is the headline numerical result, and it is extracted from a finite-time, finite-sample crossing rule with no T-convergence check, no error bars, and a normalization note that conflicts with the text. Because the relevant dynamics are rare barrier crossings, a fixed time horizon can masquerade as a critical coupling: as N grows, the barrier grows, so for fixed T the apparent threshold moves and the slope of log K2c versus log N can reflect escape-rate scaling rather than a well-defined critical point. The insets in Figs. 3b and 7 make this concrete: varying K2 changes the number of transitions within T, so K2c is defined by a contour of N(T), not by a sharp bifurcation. This concern is the same as the reader's weakest assumption, and I agree with it. A concrete test with variable T and R would settle whether the exponents are robust. If they are not, the qualitative message (finite-size-induced forward transition, inertia shifting it) survives, but the power-law claim should be replaced by an escape-time characterization. The reader's CONDITIONAL verdict is therefore appropriate; no adjustment is needed.","tokens_in":10585,"tokens_out":6836,"duration_ms":72744,"concrete_test":"Take N=500 and m=1 (a midpoint of Fig. 6) and recompute K2c with the paper's own rule ('smallest K2 with at least one crossing in R runs within T') for T = 2e4, 2e5, 2e6 and R = 100, 1000, 5000. If K2c shifts by more than a few percent, or if the log-log slope estimated from N=100,...,5000 changes with T, then the reported gamma is controlled by the finite horizon and sample size, not by a converged critical coupling. As an analytic companion, fit the observed crossing probabilities to a barrier-escape / large-deviation form (mean first-passage time ~ exp(N Delta V(K2,m))) and check whether a fixed-T contour reproduces the reported power law.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The quantitative claim that K2c ~ N^gamma (Fig. 6, with gamma = 0.55, 0.58, 0.67, 0.73) is the headline numerical result, but K2c is not defined by any converged long-time criterion. For each K2, the simulation runs for a fixed horizon T = 200000, and K2c is the smallest coupling at which at least one of 500 (Fig. 7) or 1000 (Fig. 3b) realizations escapes from the incoherent branch. Since escape is a rare fluctuation over the unstable intermediate branch (Figs. 1b, 1d), the measured threshold is effectively the coupling at which the first-passage time becomes comparable to T times the number of trials. For barrier-crossing dynamics, such a threshold shifts logarithmically with T and with trial count, independent of any true critical coupling. The paper never varies T, reports error bars, or checks that the inferred K2c has converged, so the exponents are not demonstrated to be properties of the model rather than of the fixed observation window. A side ambiguity compounds this: the Fig. 6 caption says the results are 'normalized with respect to the network size', while the text interprets the plotted values directly as K2c ~ N^gamma; the reported exponents depend on which quantity is actually plotted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Kuramoto model with inertia and triadic (2-simplex) interactions on finite all-to-all networks. The authors report that, contrary to the thermodynamic-limit prediction of no forward synchronization transition, finite-size fluctuations cause an abrupt transition from the incoherent to the synchronized state at a finite coupling K2c. They further claim a power-law relation K2c ∝ N^γ with γ = 0.55, 0.58, 0.67, and 0.73 for m = 0, 1, 2, and 5, respectively, and that increasing inertia shifts K2c to larger values, counteracting finite-size effects. The explanation offered is that fluctuations in the order parameter, scaling as O(1/√N), allow the system to cross the unstable branch that separates incoherent and synchronized states in the bistable region.","tokens_in":10843,"tokens_out":3799,"duration_ms":40519,"significance":"If established, the result would be a useful cautionary example that thermodynamic-limit analyses of higher-order Kuramoto systems can miss finite-size-induced transitions, and the reported scaling of K2c with N and m would be a concrete quantitative prediction. The paper provides direct numerical evidence—time series and bimodal probability distributions in Figs. 1, 3, and 7—for finite-size-induced transitions in the bistable region, and the trend of K2c increasing with m is visually consistent across several system sizes. However, the headline power-law exponents are not yet demonstrated to be intrinsic model properties, and part of the analytical framework is imported from prior work. The manuscript does not ship machine-checked proofs or reproducible code, and the central quantitative claim currently rests on a single observational protocol with a fixed time horizon.","major_comments":[{"comment":"The quantitative claim K2c ∝ N^γ in Fig. 6 rests on defining K2c as the smallest coupling at which at least one of 500 or 1000 realizations crosses to the synchronized branch within a fixed integration horizon T = 200000. Because the crossing is a rare fluctuation over the unstable intermediate branch (Figs. 1b and 1d), this measured threshold is effectively the coupling at which the first-passage time becomes comparable to T times the number of trials; for barrier-crossing dynamics such a threshold typically shifts logarithmically with T and with trial count, independent of any true critical coupling. The paper does not vary T, vary the number of trials, report error bars, or show that the inferred K2c has converged. The exponents γ = 0.55–0.73 may therefore be properties of the fixed observation window rather than of the model. I request a T-convergence study (e.g., K2c for T = 10^4, 10^5, 10^6 at fixed N and m) or an explicit argument that the finite-time crossing measure is asymptotically equivalent to a long-time critical coupling.","section":"Fig. 6 and accompanying text"},{"comment":"The caption of Fig. 6 states that \"the results have been normalized with respect to the network size,\" while the text and the axis labels treat the plotted values directly as K2c and fit K2c ∝ N^γ. If the ordinate is actually K2c/N or another normalized quantity, then the fitted slopes do not correspond to the claimed K2c scaling; if the ordinate is K2c, the caption is incorrect. The manuscript must clarify which quantity is plotted and, if normalization was applied, refit the power law accordingly. This ambiguity directly affects the headline exponents and must be resolved before the scaling claim can be evaluated.","section":"Fig. 6"},{"comment":"The m > 0 analytical curves used as black lines in Figs. 4 and 5 are imported from reference [20] without derivation; the text states \"Following the further analytical derivation in [20]\" and later notes that the drifting-oscillator integral in Eq. (6) is difficult to solve, with only the strong-synchronization case presented. Nevertheless, the paper's central explanation of the finite-size transition—that fluctuations must cross the intermediate unstable state—uses these imported curves to locate that unstable state. The authors should either derive the self-consistent equations for m ≠ 0 within this manuscript or clearly attribute the unstable branch to ref. [20] and justify why a thermodynamic-limit unstable branch is a valid reference for finite-N dynamics. This is load-bearing because the origin claim rests on the position of that branch.","section":"Eqs. (5)–(6)"},{"comment":"The paper repeatedly attributes the finite-size transition to fluctuations that scale as O(1/√N) and illustrates that the FWHM of P(r1) decreases with N, but it never quantifies this scaling. Since the power-law dependence of K2c on N is the headline result, a quantitative demonstration that the variance or FWHM indeed scales as 1/N (or at least as a known function of N) would directly support the proposed mechanism. I recommend adding such a scaling plot or analysis.","section":"Figs. 1(c), 3(a)"}],"minor_comments":[{"comment":"Several figure captions contain rendering artifacts such as \"/uni27E8\" and \"/uni27E9\" (which should presumably be angle brackets), and the legend labels \"f\" and \"b\" are described as \"postscript\" rather than as subscripts. These should be corrected.","section":"Figs. 2, 4, 5 and captions"},{"comment":"The frequency distribution is called a \"Lorenzian distribution\" in the text; the correct spelling is \"Lorentzian.\"","section":"Analytical Derivation"},{"comment":"The paper states that the Runge-Kutta 4 method is used but does not give the time step dt, the total number of integration steps, or the initial phase/frequency sampling protocol beyond \"random phases.\" These details are needed for reproducibility, especially because the crossing statistics depend on the integration horizon.","section":"Model and Numerical Results"},{"comment":"The caption says the analytical prediction of r1 is obtained \"by solving Eq. A5 using bisection method,\" but Eq. A5 is a differential equation. It would be clearer to state that the fixed-point equation f(r1) = 0 is solved and that the dashed line is the unstable branch.","section":"Fig. 2 caption"},{"comment":"The text cites \"Tanaka et al. 9\" for the critical point and then \"Tanaka et al. 8\" for the perturbation method, which is chronologically confusing; reorder the citations or adjust the text so the numerical order matches the discussion.","section":"Introduction and references"}],"recommendation":"major_revision","confidential_remarks":"The core observation—finite-size-induced forward transitions in the bistable region—is credible and likely publishable, but the headline scaling is not yet supported because of the fixed-time-horizon protocol and the normalization ambiguity in Fig. 6. I would encourage the editor to ask for a T-convergence test and a clarification of the fitted quantity. The reliance on the authors' own earlier paper for the m>0 analytical branch is acceptable if properly credited, but the derivation gap should be stated openly in the revised text. The paper fits the journal's scope in nonlinear dynamics and complex systems, but the numerical evidence for the power-law claim needs to meet a higher standard before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: this is a legitimate extension of the finite-size story for higher-order Kuramoto to the m>0 case, and the basic observation—finite-size fluctuations can kick the system from the incoherent branch over the unstable branch into the synchronized state—is real and visible in the time series and probability distributions. The inertia dependence of that effect, with K2c shifting upward as m grows, is also convincing qualitatively. I part ways with the authors on the quantitative scaling law: K2c ~ N^gamma with gamma = 0.55–0.73 is not established by the evidence as reported.\n\nWhat is new: prior work by Suman and Jalan covered finite-size effects without inertia; Sabhahit et al. covered inertia in the thermodynamic limit. This paper fills the gap, and the result that inertia resists the finite-size-induced transition is worth knowing. The m=0 analytical appendix (Ott-Antonsen reduction) is sound and useful. The m>0 curves are imported from their own earlier paper [20]; that is fine, but they should reproduce the derivation or at least state the self-consistent equations clearly.\n\nSoft spots, in order of severity. First, the definition of K2c is a finite-time crossing threshold: they run T = 200,000 steps and take the smallest K2 where at least one of 500 or 1000 realizations escapes to the synchronized branch. For barrier-crossing dynamics, this threshold is expected to shift with T and with the number of trials, roughly logarithmically. They never vary T, report error bars, or check convergence. So the power-law exponents in Fig. 6 may be properties of the observation window rather than of the model. This needs to be addressed before I trust the quantitative claim. Second, the Fig. 6 caption says the results have been \"normalized with respect to the network size,\" which contradicts the text's K2c ∝ N^gamma interpretation. That ambiguity should be fixed. Third, minor: the exponents are fit to six points with no uncertainty; at minimum they should state the fit residuals.\n\nThe qualitative core of the paper holds up. It is a useful contribution for people working on finite-size and higher-order oscillator networks, and it deserves a serious referee—but the referee should push for a proper first-passage-time analysis or at least a T-dependence check. If the scaling law survives that test, it becomes a much stronger paper.\n\nRecommendation: send to peer review, with the expectation of major revision on the scaling-law section.","headline":"A legitimate but quantitatively fragile extension of finite-size synchronization to inertial higher-order Kuramoto; the observed forward transition is real, but the reported power-law exponents rest on a fixed-time crossing rule and need a T-dependence check.","tokens_in":11384,"tokens_out":3206,"would_cite":true,"duration_ms":33298,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C15","34D06","82C26"],"pacs":["05.45.Xt","89.75.-k"],"model":"deepseek-v4-flash","headline":"Finite-size fluctuations induce a first-order synchronization transition in Kuramoto oscillators with inertia on 2-simplex complexes, even though the thermodynamic-limit analysis allows no forward transition; the critical coupling grows…","keywords":["finite-size effects","Kuramoto model","inertia","higher-order interactions","simplicial complex","synchronization transition","order parameter fluctuations","power-law scaling"],"falsifier":"Doubling or decupling the integration time and realization count and re-extracting K2c(N): if the threshold moves systematically with T, or if the log-log slope of K2c versus N changes, the reported power law is a finite-time artifact rather than a true scaling of the transition. A complementary check computes the Kramers escape rate over the unstable branch from the one-dimensional order-parameter dynamics and compares the predicted mean crossing time with the observed transition-time distribution.","tokens_in":10408,"feed_emoji":"🔄","tokens_out":11755,"duration_ms":112304,"temperature":0.7,"pith_summary":"This paper asks what changes when a globally coupled Kuramoto model with inertia and triadic 2-simplex interactions is finite rather than infinite. The thermodynamic-limit analysis says that, starting from incoherence, there is no forward transition to synchronization at any coupling strength; only the backward path desynchronizes through a saddle-node bifurcation. The paper shows that finite size changes this picture: order-parameter fluctuations of size O(1/sqrt N) can carry the system over the unstable state that separates incoherence from synchronization, producing an abrupt first-order jump at a finite critical coupling K2c. Across network sizes from N=100 to N=5000, the paper finds K2c grows as a power law of N, with the exponent rising from 0.55 at m=0 to 0.73 at m=5, and larger inertia shifts K2c upward. If the claim is right, infinite-size predictions understate how easily triadic interactions synchronize real, finite oscillator systems, while inertia partially restores the infinite-size behavior.","feed_headline":"Finite size alone triggers abrupt synchrony in Kuramoto networks","feed_subtitle":"Even where the infinite-size theory predicts none, size-driven fluctuations make the jump; inertia delays it.","key_machinery":"The load-bearing object is the unstable branch of the mean-field order-parameter dynamics, the separatrix between the stable incoherent state r1=0 and the stable synchronized state r1 approximately 1. For m=0 the paper derives the one-dimensional equation dr1/dt = -r1 + (K2/2)($r1^{3}$ - $r1^{5}$), whose three fixed points give this bistable geometry; for m>0 the same geometry is obtained from locked and drifting oscillator contributions. The mechanism is finite-size noise: r1 fluctuates with a width that decreases as O(1/sqrt N), and once fluctuations are large enough to reach the unstable separatrix, the system escapes to the synchronized branch. Coupling strength K2 controls the distance between the incoherent state and the separatrix, which is why larger K2 compensates for smaller fluctuations in larger systems.","core_discovery":"The central discovery is that a forward synchronization transition exists for finite Kuramoto networks with inertia and 2-simplex (triadic, three-body) coupling, even though the same model in the thermodynamic limit has no such transition. In the continuum limit the stable incoherent state r1=0 persists for all K2 in the forward direction, with an intermediate unstable branch and a stable synchronized branch appearing only in the bistable region; finite-size fluctuations of r1, whose width scales as O(1/sqrt N), let the system cross the unstable branch and settle on the synchronized state. The paper defines the forward critical coupling K2c as the smallest K2 for which at least one of 500 or 1000 realizations crosses to the synchronized branch within a fixed integration time T=200000, and reports K2c proportional to N^gamma with gamma = 0.55, 0.58, 0.67, and 0.73 for m = 0, 1, 2, and 5. The backward desynchronization threshold is unchanged by N or m, so increasing size and inertia both widen the hysteresis loop.","pith_inferences":["If the crossing event is reinterpreted as Kramers escape over the unstable branch, the same numerical data could test whether the mean transition time follows an exponential law and whether that law reproduces the reported exponents gamma(m) from a barrier-height calculation; the paper does not attempt this.","The fixed-time, few-realization definition of K2c likely makes the reported exponents sensitive to the observation window; redefining K2c through the rate of crossings at fixed K2 would provide a cleaner scaling and might change the prefactor, possibly the exponent.","For sparse or non-globally coupled simplicial complexes, the O(1/sqrt N) fluctuation scaling will be modified by degree heterogeneity and local topology, so the same mechanism should produce a different N-scaling of K2c; this is a testable extension.","The inertia-dependent exponent could be interpreted as inertia lowering the effective fluctuation temperature of the order parameter, which would connect the numerical exponents to an effective-noise model of the mean-field dynamics."],"forward_implications":["For any finite globally coupled 2-simplex Kuramoto network starting from random phases, an abrupt forward jump to synchronization occurs at a finite coupling, contradicting the thermodynamic-limit prediction of no forward transition.","The critical coupling for this jump grows as a power law of network size, so larger systems need disproportionately stronger triadic coupling before fluctuations can trigger synchronization.","Adding inertia raises the critical coupling and increases the power-law exponent, meaning inertia acts as a stabilizing force that partly restores the infinite-size behavior in finite systems.","The backward transition point stays fixed with N and m, so hysteresis loops become wider as systems grow or as inertia increases.","The transition is an escape over an unstable state, so the waiting time to synchronization is distributed and shorter for larger K2, as the paper's transition-time statistics show."],"supporting_citations":[{"why":"Supplies the finite-size fluctuation analysis for the same higher-order Kuramoto model without inertia, the baseline the paper extends to m>0.","marker":"[24]"},{"why":"Provides the locked/drifting oscillator decomposition and steady-state integrals (Eqs. 5-6) used for the m>0 analytical comparison.","marker":"[20]"},{"why":"Establishes the bistable structure of 2-simplex Kuramoto in the thermodynamic limit: stable incoherent and synchronized states separated by an unstable branch.","marker":"[26]"},{"why":"Supplies the dimensional reduction that yields the closed m=0 order-parameter equation Eq. (A5).","marker":"[29]"},{"why":"Gives the scaling analysis of intrinsic finite-size fluctuations in large oscillator populations that underlies the O(1/sqrt N) fluctuation argument.","marker":"[21]"},{"why":"Introduces the inertial second-order Kuramoto model whose critical behavior the paper modifies with triadic interactions and finite size.","marker":"[9]"}],"fun_headline_variants":["Finite size flips Kuramoto to synchrony","Size-driven jump to synchrony in Kuramoto","Kuramoto with inertia: finite N triggers sync","Finite-size force kicks Kuramoto into sync"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central result rests on treating the first crossing within a fixed simulation window of length T=200000 among 500 or 1000 realizations as the critical coupling; if longer runs or more realizations shift that crossing value, the reported power-law exponents are finite-time artifacts.","fun_headline_variants_meta":{"raw":{"variants":["Finite size flips Kuramoto to synchrony","Size-driven jump to synchrony in Kuramoto","Kuramoto with inertia: finite N triggers sync","Finite-size force kicks Kuramoto into sync"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000124,"raw_usage":{"total_tokens":1080,"prompt_tokens":895,"completion_tokens":185,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":121}},"tokens_in":511,"tokens_out":185,"duration_ms":2680,"temperature":1.0,"reasoning_tokens":121,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:47:13.246294+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Doubling or decupling the integration time and realization count and re-extracting K2c(N): if the threshold moves systematically with T, or if the log-log slope of K2c versus N changes, the reported power law is a finite-time artifact rather than a true scaling of the transition. A complementary check computes the Kramers escape rate over the unstable branch from the one-dimensional order-parameter dynamics and compares the predicted mean crossing time with the observed transition-time distribution.","supporting_citations":[{"cited_title":"Suman \\ and\\ author S","cited_arxiv_id":null,"evidence_quote":"Supplies the finite-size fluctuation analysis for the same higher-order Kuramoto model without inertia, the baseline the paper extends to m>0."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the locked/drifting oscillator decomposition and steady-state integrals (Eqs. 5-6) used for the m>0 analytical comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the bistable structure of 2-simplex Kuramoto in the thermodynamic limit: stable incoherent and synchronized states separated by an unstable branch."},{"cited_title":"Ott \\ and\\ author T","cited_arxiv_id":null,"evidence_quote":"Supplies the dimensional reduction that yields the closed m=0 order-parameter equation Eq. (A5)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the scaling analysis of intrinsic finite-size fluctuations in large oscillator populations that underlies the O(1/sqrt N) fluctuation argument."},{"cited_title":"\\ Tanaka , author A","cited_arxiv_id":null,"evidence_quote":"Introduces the inertial second-order Kuramoto model whose critical behavior the paper modifies with triadic interactions and finite size."}],"review_version":1}