{"id":"048d3218-ac0a-42cd-81c4-5a5dcc5512b1","arxiv_id":"2502.07581","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A new 3D rough flow (WABC) is shown numerically to display spontaneous stochasticity: Lagrangian particle statistics converge to non-trivial random limits as regularisation vanishes.","lead":"This paper introduces a 3D model of a rough, turbulent-like velocity field made of superposed ABC flows and uses Monte-Carlo simulations to show that tracer particles in it become random even when the noise is removed. The model is a new testbed for 'spontaneous stochasticity', the idea that randomness in turbulence can appear in finite time rather than from butterfly-effect chaos.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed regularization-independence of the spontaneous stochastic limit is not established: the Langevin noise exponent p=2.4 was tuned to match the Cauchy regularisation at N=16, so the agreement in Fig. 9 is partly by construction.","rationale":"The reader's weakest assumption identifies the unproved ill-posedness of the limiting ODE (Eq. 12), and that is indeed a real gap: if the ODE were well-posed, no vanishing-noise limit could produce a non-Dirac distribution. I retain that concern. However, the more immediately load-bearing and testable weakness is methodological: the paper's evidence for regularization independence is calibrated by the same comparison used to demonstrate it. Appendix C explicitly states that p=2.4 was chosen by minimizing H_KL between the Cauchy and Langevin one-point distributions at N=16, and then Figure 9 shows that these two distributions agree at N=16. That does not establish that the Langevin limit is universal; it only shows that one parameter was tuned to make the two finite-N approximations coincide. The convergence-in-N figures for each regularization separately do not resolve this, because they do not compare the two regularizations to each other at increasing N. The proposed scan over p and N directly tests whether the limiting object is selected by the dynamics or by the fitted noise scaling. If the limits coincide for all p in the vanishing-noise regime, the concern is resolved and the claim is substantially strengthened. If not, the paper's central universality claim fails, although a non-universal spontaneous stochastic limit might still exist. Since the reader already returned CONDITIONAL, my read does not change that verdict; it sharpens the condition that should be met before acceptance.","tokens_in":11439,"tokens_out":6713,"duration_ms":64555,"concrete_test":"Perform a Langevin-WABC scan with the same a, N_p, and initial condition, for p ∈ {1.6, 2.0, 2.2, 2.4, 2.6, 3.0}, all of which give κ_N → 0 for h = 1/3, and compute the cross-regularization KL divergence H_KL(p_C^N, p_L^{p,N}) as a function of N = 12, 14, 16, 18 (and 20 if feasible). If spontaneous stochasticity is regularization-independent, these cross-KL curves should tend to zero for every p, and the tuned p = 2.4 curve should not be special. Also report the p = 2.4 cross-KL at N = 18 and N = 20, since the paper only shows the agreement at N = 16. If instead the cross-KL saturates at p-dependent positive values, the claimed universality fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim has two parts: (i) a nontrivial stochastic limit exists as the regularization vanishes, and (ii) this limit is independent of the stochastic regularisation. Part (ii) is the weaker link, and the paper's own procedure makes it circular. In Section 4 and Appendix C, the Langevin noise is set to κ_N = b^2 ω_N^2 / k_N^{2.4}, where p=2.4 was obtained by minimizing the Kullback-Leibler distance between the Langevin and Cauchy one-point distributions at N=16, t=0.875 (Figure 15). The agreement shown in Figure 9 is therefore a check at the fitting point, not an independent test of regularisation independence. Figures 7, 8, and 10 show convergence in N within each regularization separately; no cross-regularization KL divergence is reported as a function of N beyond that single comparison. If the limiting distribution actually depended on the regularization (or on the noise exponent), the optimization procedure would still select a p that makes the N=16 distributions look close, and the abstract's claim that the observed spontaneous stochasticity 'does not depend on the chosen stochastic regularisations' would be unsupported. This concern is distinct from the unproved ill-posedness of Eq. (12), but it is more directly checkable and it undermines the universality part of the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a three-dimensional incompressible velocity field, the 'WABC flow', built from a Weierstrass-type superposition of ABC flows with Hölder exponent h=1/3. The authors study Lagrangian particle trajectories in this rough flow under two stochastic regularizations: a Langevin white-noise forcing (Eq. 10) and a random-initial-condition (Cauchy) regularization (Eq. 11), both of which vanish as the number of modes N increases. They report that, as N→∞, one-point and two-point statistics converge to non-trivial, non-Dirac limit distributions, which they interpret as spontaneous stochasticity. They further claim that this limit is independent of the chosen stochastic regularization. The numerical evidence is based on Monte-Carlo simulations with Np=2^20 particles, KL-divergence convergence diagnostics, and an accuracy analysis in Appendix B.","tokens_in":11763,"tokens_out":6272,"duration_ms":54954,"significance":"If the conclusions are robust, the WABC model would provide a valuable tunable 3D testbed for spontaneous stochasticity, linking singular Eulerian flows to Lagrangian randomness and allowing future studies of the role of chaos. The paper has clear strengths: the construction of the WABC flow is novel and well-motivated; the numerical methodology is careful, with explicit treatment of time-step scaling, finite-particle-number effects, and a reproducibility-friendly fixed-parameter setup; and the convergence plots (Figs. 5–8, 10) give credible visual and quantitative evidence of non-Dirac limits within each regularization. The honest discussion of computational limits is also a positive feature. However, the claim of regularization independence is not fully supported by the presented experiments, because one of the two regularizations is calibrated against the other at the single comparison point used to demonstrate agreement.","major_comments":[{"comment":"The exponent p in the Langevin noise scaling κ_N = b^2 ω_N^2 / k_N^p is selected by minimizing the Kullback-Leibler divergence H_KL(p_C, p_L) between the Cauchy and Langevin one-point distributions at N=16 and t=0.875 (Fig. 15). Consequently, the agreement shown in Fig. 9 is a check at the fitting point, not an independent test of regularization independence. The paper does not report a cross-regularization KL divergence as a function of N (e.g., H_KL(P_C^N, P_L^N) for N=10, 12, 14, 16, 18), and it does not test how sensitive the limiting Langevin distribution is to the choice of p around the fitted value 2.4. As a result, the abstract's claim that the observed spontaneous stochasticity 'does not depend on the chosen stochastic regularisations' is stronger than the evidence supports. Please add a cross-regularization convergence study in N, and if the two regularized distributions do not become progressively closer as N increases, the universality claim should be softened accordingly.","section":"Section 4, Eq. (14); Appendix C, Eq. (15), Fig. 15"},{"comment":"The definition of spontaneous stochasticity requires that the limiting process x(t) solve the ideal problem dx/dt = u_W(x) with x(0)=x0, which in turn requires that this problem be ill-posed, i.e., that solutions be non-unique. The paper states that this problem is 'formally deterministic but generally ill-posed' but provides no proof and no reference establishing non-uniqueness for the specific WABC field with h=1/3. Without such a proof (or at least strong evidence, such as verification that the numerically observed limiting trajectories satisfy (12) in a weak sense), the non-Dirac limits could in principle arise from residual regularization effects rather than from a genuine selection of non-unique solutions. This point is load-bearing because the central claim—that the WABC model 'can build spontaneous stochasticity'—is explicitly tied to the ill-posedness of (12). The authors should either prove non-uniqueness (e.g., via the Osgood criterion for Hölder-continuous vector fields) or clearly state that the interpretation depends on this unproven assumption.","section":"Section 3.2, Eq. (12)"}],"minor_comments":[{"comment":"The abstract contains two typos: 'Richardon's regime' should be 'Richardson's regime', and 'fractal Brownian motion' should be 'fractional Brownian motion' (the standard term).","section":"Abstract"},{"comment":"The sentence 'Though the limiting equation (10) is deterministic' should refer to the limiting ideal problem (12), not to the regularized Langevin equation (10).","section":"Section 3.2, after Eq. (12)"},{"comment":"The phrase 'finite-dimentional marginals' should be 'finite-dimensional marginals'.","section":"Section 4, paragraph after Eq. (12)"},{"comment":"The sentence 'Computational burdens limits us to Np = 2^20' should be 'Computational burden limits us to Np = 2^20' (singular noun and verb agreement).","section":"Section 6.1"},{"comment":"The figure compares Cauchy and Langevin one-point distributions at N=16, but no error bars or shaded uncertainty regions are shown; adding them would help the reader judge whether the differences are statistically significant.","section":"Figure 9"},{"comment":"The second-order polynomial fit used to locate the minimum at p ≃ 2.4 is not described in the text; please specify the fit range, the data points included, and the uncertainty of the fitted minimum.","section":"Appendix C, Fig. 15"},{"comment":"The sentence 'In this paper we neither review the influence of the Hölder exponent nor the initial position' should use 'investigate' rather than 'review' to match the intended meaning.","section":"Section 6.2"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the circularity in the regularization-independence claim: the Langevin noise exponent p=2.4 is fitted to match the Cauchy regularization at N=16, so the agreement in Fig. 9 is not an independent test. This is fixable by adding a cross-regularization KL study as a function of N and by testing sensitivity to p. The second major issue, the unproven ill-posedness of the ideal problem, is also fixable by citing known results or by adding a numerical check. The paper's model and methodology are otherwise solid and would be of interest to the turbulence and dynamical-systems community, but the main claims as currently worded overstate the evidence. I recommend major revision rather than rejection, because the deficiencies are local and addressable within the scope of the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The WABC construction is a real step forward: a tunable 3D rough flow with a Weierstrass-like multi-scale superposition of ABC flows, where you can dial Lagrangian chaos. The paper gives a practical definition of spontaneous stochasticity for finite-dimensional marginals, and the numerical evidence for a nontrivial limit within each regularization is the strongest part. The KL convergence plots (Figs 5–8, 10) and the finite-size analysis in Appendix B are careful and convincing: the plateaus are explained by Np, and the approach to a non-Dirac limit for N ≥ 12–16 is clear. That part of the central claim holds up reasonably well.\n\nThe soft spot is the regularization-independence claim, and the stress-test note is right. The Langevin noise exponent p = 2.4 was chosen by minimizing the KL distance between Langevin and Cauchy one-point distributions at N = 16 and t = 0.875 (Appendix C). So Figure 9 is a check at the fitting point, not an independent test. The abstract's assertion that the observed spontaneous stochasticity 'does not depend on the chosen stochastic regularisations' is accordingly overstated. This does not kill the paper—the convergence within each regularization is still evidence for a stochastic limit—but the universality claim needs a better test, e.g. reporting the cross-regularization KL as a function of N or testing a different noise exponent and showing the limit is unchanged.\n\nA second, more conceptual gap is the unproved ill-posedness of the limiting ODE (12). The paper calls it 'generally ill-posed' but does not prove non-uniqueness for the WABC field specifically. If the ODE were well-posed, the zero-noise limit would be deterministic. This is a genuine caveat, but it is the kind of thing that can be addressed by citing or adapting existing non-uniqueness results for rough vector fields; it does not undermine the numerical observation itself.\n\nMinor: no code or data provided, and the paper is a bit over-optimistic in Section 6.2 when it claims compliance with earlier turbulence models without testing other regularizations. The fitting of p is also a free parameter that deserves more discussion.\n\nWho is this for? Anyone working on spontaneous stochasticity or Lagrangian chaos in rough flows. It is a useful new tool, and the numerical methodology section is worth reading. With the caveats addressed, it deserves a serious referee. I'd send it to review, and I'd cite it as a new model construction even if I remain skeptical of the universality claim until it is tested more honestly.","headline":"A genuinely new 3D toy model for spontaneous stochasticity with solid numerical evidence per regularization, but the cross-regularization universality claim is weakened by a fitted noise exponent.","tokens_in":12359,"tokens_out":899,"would_cite":true,"duration_ms":9707,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37N10","76F25","60H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that Lagrangian particles in a rough three-dimensional WABC flow remain stochastic in the vanishing-noise limit, with a limit law independent of the regularisation; this is spontaneous stochasticity in a tunable 3D model.","keywords":["spontaneous stochasticity","WABC flow","Lagrangian chaos","rough velocity fields","Hölder regularity","anomalous dissipation","stochastic regularisation","turbulent transport"],"falsifier":"Compute the limit of deterministic regularisations — spectral truncations of the WABC flow without noise and without initial spread — for the same initial point and h = 1/3: if the trajectories converge to a single deterministic solution of $dx/dt = u_W(x)$ as N → ∞, the central claim fails. Alternatively, prove uniqueness for the ideal WABC ODE at this initial data, which would rule out spontaneous stochasticity by definition.","tokens_in":11281,"feed_emoji":"🌀","tokens_out":7621,"duration_ms":70098,"temperature":0.7,"pith_summary":"The paper builds a three-dimensional velocity field, the WABC flow, by superposing infinitely many rescaled ABC flows with Hölder exponent h = 1/3. It asks whether fluid particles advected by this rough field remain randomly spread even when all regularisation noise is removed. Through Monte-Carlo simulations of about a million trajectories under two different stochastic regularisations, it finds that one-point and two-point statistics converge to nontrivial limits as the cutoff N grows. The paper concludes that the WABC model exhibits spontaneous stochasticity: the limiting ideal equation is deterministic, yet the vanishing-noise limit selects a genuinely stochastic distribution of trajectories. This matters because spontaneous stochasticity is a candidate mechanism for finite-time randomness in turbulence, and the WABC model is tunable enough to test whether chaos is needed for it.","feed_headline":"Rough 3D flow keeps particle randomness as noise vanishes","feed_subtitle":"A tunable Weierstrass-ABC flow shows Lagrangian trajectories stay stochastic in the vanishing-noise limit.","key_machinery":"The central object is the WABC velocity field, defined as $u_W(x) = \\sum_{i=1}^\\infty \\lambda^{-hi} U(\\lambda^i x)$ with $\\lambda = 2$, a superposition of ABC flows at scales $\\lambda^i$; it is incompressible and $h$-Hölder continuous. The exponent $h = 1/3$ places the flow at the critical value where the Duchon-Robert energy-transfer term tends to a finite nonzero limit as the mollification scale vanishes, mimicking the dissipation anomaly of turbulence. The two stochastic regularisations — white noise of amplitude $\\kappa_N$ and random initial data of radius $\\eta_N$, both vanishing with N — are the selection mechanism whose limit defines the spontaneously stochastic law.","core_discovery":"For the WABC flow with h = 1/3, the paper claims that regularised Lagrangian trajectories converge in distribution to a nontrivial stochastic process solving the ideal deterministic equation $dx/dt = u_W(x)$. The convergence is shown numerically through Kullback-Leibler divergences between one-point and two-point marginals for increasing numbers of modes N, reaching plateaus for N ≥ 14. The same limit is obtained whether the regularisation is additive white noise (Langevin-WABC) or random initial conditions (Cauchy-WABC), indicating that the limiting law is insensitive to the regularisation. The authors frame this as spontaneous stochasticity, possible because the ideal problem is formally deterministic but generally ill-posed.","pith_inferences":["Beyond the paper: a direct test it does not run is varying the ABC parameters between known chaotic and integrable regimes and measuring whether the limiting distribution collapses to a Dirac; that experiment would isolate the role of chaos.","Beyond the paper: the regularisation-independence shown for two noise types suggests the limiting law is an intrinsic property of the WABC field itself, but a convergence theorem would be needed to establish that rigorously.","Beyond the paper: the use of finite-dimensional marginals leaves open whether the limit is a genuine stochastic process on path space; checking two-time correlations or exit-time statistics would strengthen the identification."],"forward_implications":["For N ≥ 14 the one-point and two-point marginal distributions stop changing within statistical error, so the spontaneously stochastic regime is numerically accessible in this model.","The same limiting distributions appear under Langevin white-noise and random-initial-condition regularisations, so the result is not an artifact of one noise choice.","With ABC parameters that can be tuned to break Lagrangian chaos, the model gives a direct way to test whether positive Lyapunov exponents are necessary for spontaneous stochasticity.","The WABC construction provides a family of rough incompressible three-dimensional fields with tunable regularity h and tunable chaos, extending the study of spontaneous stochasticity beyond one-dimensional toy models."],"supporting_citations":[{"why":"Supplies the concept of spontaneous stochasticity and its role in turbulent transport.","marker":"[15]"},{"why":"Formalises intrinsic stochasticity through generalized flows, a conceptual precursor the paper builds on.","marker":"[36]"},{"why":"Gives a one-dimensional inviscid model where spontaneous stochasticity is established analytically, setting the pattern the WABC analysis adapts.","marker":"[23]"},{"why":"Provides the renormalization-group framework and the argument that irregularities are necessary for spontaneous stochasticity, motivating the rough WABC construction.","marker":"[13]"},{"why":"Argues that deterministic regularisations lead to ill-defined limits, justifying the use of stochastic regularisations here.","marker":"[7]"},{"why":"Reports regularisation-independent spontaneous stochasticity in singular shear flows, the comparison point for the numerical finding.","marker":"[33]"},{"why":"Defines the energy-transfer term used to identify anomalous dissipation at the critical exponent h = 1/3.","marker":"[9]"},{"why":"Describes ABC flows and their chaotic streamlines, the building blocks of the WABC model.","marker":"[6]"}],"fun_headline_variants":["Spontaneous stochasticity emerges in rough 3D flow","Rough flow yields randomness as noise fades","Noise-free limit still random in WABC flow","3D Weierstrass flow shows inherent randomness","Lagrangian randomness survives vanishing noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the ideal equation $dx/dt = u_W(x)$ is ill-posed for the chosen data — the paper states in Section 3.2 that it is “formally deterministic but generally ill-posed” but does not prove non-uniqueness — because if the ODE were well-posed, the zero-noise limit would be deterministic and the claimed spontaneous stochasticity could not occur.","fun_headline_variants_meta":{"raw":{"variants":["Spontaneous stochasticity emerges in rough 3D flow","Rough flow yields randomness as noise fades","Noise-free limit still random in WABC flow","3D Weierstrass flow shows inherent randomness","Lagrangian randomness survives vanishing noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000596,"raw_usage":{"total_tokens":2808,"prompt_tokens":983,"completion_tokens":1825,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":599,"completion_tokens_details":{"reasoning_tokens":1753}},"tokens_in":599,"tokens_out":1825,"duration_ms":11998,"temperature":1.0,"reasoning_tokens":1753,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T12:14:35.919859+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the limit of deterministic regularisations — spectral truncations of the WABC flow without noise and without initial spread — for the same initial point and h = 1/3: if the trajectories converge to a single deterministic solution of $dx/dt = u_W(x)$ as N → ∞, the central claim fails. Alternatively, prove uniqueness for the ideal WABC ODE at this initial data, which would rule out spontaneous stochasticity by definition.","supporting_citations":[{"cited_title":"Particles and fields in fluid turbulence","cited_arxiv_id":null,"evidence_quote":"Supplies the concept of spontaneous stochasticity and its role in turbulent transport."},{"cited_title":"Generalized flows, intrinsic stochasticity, and turbulent transport","cited_arxiv_id":null,"evidence_quote":"Formalises intrinsic stochasticity through generalized flows, a conceptual precursor the paper builds on."},{"cited_title":"Spontaneously stochastic solutions in one-dimensional inviscid systems","cited_arxiv_id":null,"evidence_quote":"Gives a one-dimensional inviscid model where spontaneous stochasticity is established analytically, setting the pattern the WABC analysis adapts."},{"cited_title":"Renormalization group approach to spontaneous stochasticity","cited_arxiv_id":null,"evidence_quote":"Provides the renormalization-group framework and the argument that irregularities are necessary for spontaneous stochasticity, motivating the rough WABC construction."},{"cited_title":"Statistical determinism in non-Lipschitz dynamical systems","cited_arxiv_id":null,"evidence_quote":"Argues that deterministic regularisations lead to ill-defined limits, justifying the use of stochastic regularisations here."},{"cited_title":"From the butterfly effect to spontaneous stochasticity in singular shear flows","cited_arxiv_id":null,"evidence_quote":"Reports regularisation-independent spontaneous stochasticity in singular shear flows, the comparison point for the numerical finding."},{"cited_title":"Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations","cited_arxiv_id":null,"evidence_quote":"Defines the energy-transfer term used to identify anomalous dissipation at the critical exponent h = 1/3."},{"cited_title":"Chaotic streamlines in the ABC flows","cited_arxiv_id":null,"evidence_quote":"Describes ABC flows and their chaotic streamlines, the building blocks of the WABC model."}],"review_version":1}