{"id":"c4bfe986-e163-43af-be34-90a61be2c085","arxiv_id":"2504.15795","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Eulerian spontaneous stochasticity is defined as liminf less than limsup of observables in the inviscid limit; all probability measures on the attainable set are shown selectable, and the Armstrong-Vicol passive scalar is shown, conditionally and numerically, to be spontaneously stochastic.","lead":"This paper builds a mathematical framework for 'Eulerian spontaneous stochasticity', the idea that deterministic flow equations can become genuinely random in the inviscid limit where viscosity or diffusion vanishes. It applies the framework to the Armstrong-Vicol passive scalar model and argues that its lack of a selection principle is a physical phenomenon with its own probability distribution, not a mathematical pathology.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 proof has two unproven steps: the interpolated regularisations need not be well-posed under (H0), and the conclusion M=M0 is drawn from density without a closedness/diagonal argument.","rationale":"The paper's most novel claim is Theorem 1, and it is also the load-bearing point for the abstract's 'measure selection principle' and for the claim that spontaneous stochasticity can generate any statistics of the inviscid system. The proof in Appendix 11.6 is incomplete in two concrete ways. First, H0 only assumes f0 continuous; the interpolated vector field is then not known to be well-posed. Without uniqueness, the regularisation function and the entire Bebutov/measure construction have no well-defined Phi_t. This matches the reader's first concern. Second, even under a Lipschitz or smoothing repair, the proof's final 'taking the closure' is not valid because M is not shown closed; the universal gamma construction in Appendix 11.7 only handles two-point measures, which are not dense. A diagonal argument with a dense sequence of finite combinations would fix it, but it is absent. Both gaps are repairable, so REJECT is not warranted; the Armstrong-Vicol SpSt conclusion itself is independently established in [4], and the numerics are clearly labelled as outside the theorem's parameter regime. The appropriate disposition is therefore unchanged: CONDITIONAL, with revision focused on making Theorem 1's proof rigorous.","tokens_in":57193,"tokens_out":13011,"duration_ms":134394,"concrete_test":"Analytically verify the two steps in Appendix 11.6 for a continuous non-Lipschitz f0 (e.g. x -> |x|^{1/2}) and a prescribed attainable point x: (1) prove the interpolated field is well-posed, i.e. satisfies a uniqueness criterion such as Lipschitz or Osgood along the selected trajectory; (2) either prove M in (28) is weak-* closed, or construct a single gamma such that M(gamma) contains a dense set of finite convex combinations and use compactness of each M(gamma) to conclude M = M0. If either step cannot be completed, Theorem 1 does not follow from the present proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Appendix 11.6 defines f_epsilon = theta_epsilon F_epsilon + (1 - theta_epsilon) f0 and asserts it is well-posed 'by construction'. Under Hypothesis (H0), f0 is only continuous, so the interpolation is generally only continuous; Peano existence holds but uniqueness does not. In the intended application f0 is non-Lipschitz (otherwise S0 would be a singleton), so global uniqueness of dx/dt = f_epsilon(x) cannot be inferred from continuity, and no Osgood/Lipschitz estimate is supplied. Thus the regularisation function A(kappa) = O(Phi_t[f(.,kappa)]x0) may not be well-defined. This is the main novel contribution: the proof of M = M0. Independently, even if each f_epsilon were made smooth, the proof only places finite convex combinations of Dirac masses in M and then says 'taking the closure'. But M is an infinite union of compact sets M(gamma); it is not shown to be closed, so density of co(E) in M0 does not imply equality. The universal gamma_un in Corollary 1 uses only two-point combinations, which are not weak-* dense in P(S0) when S0 has more than two points; a diagonal construction with a countable dense family of finitely supported measures would be needed. Theorem 1 is therefore unproven as written, although both gaps appear repairable.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a general framework for spontaneous stochasticity (SpSt) in finite-dimensional dynamical systems, defines SpSt through the liminf/limsup of a regularisation function, proves an equivalence with the lack of a selection principle, and develops a measure-theoretic description of the inviscid limit based on Bebutov flows and ergodic limits. Its central structural claim is Theorem 1: if M0 is the space of probability measures on the attainable set of the inviscid system, then M=M0, i.e. every probability measure on S0 can be attained as the inviscid-limit statistic of some well-posed regularisation. The paper also gives a necessary condition for nonuniqueness via Dini-type derivatives, applies the framework to the Armstrong-Vicol passive scalar, and claims in Theorem 3 that this model exhibits Eulerian spontaneous stochasticity in the sense of Definition 1. Part I contains a historical review, a description of the Armstrong-Vicol construction, and numerical experiments on anomalous diffusion and Lagrangian spontaneous stochasticity.","tokens_in":57407,"tokens_out":16213,"duration_ms":159208,"significance":"If the central claims hold, the paper would provide a rigorous reformulation of spontaneous stochasticity as a measure-selection phenomenon and would establish a clean dichotomy: non-uniqueness of the inviscid system is sufficient, through the M=M0 theorem, for the existence of regularisations exhibiting SpSt. The application to the Armstrong-Vicol model is also significant, since it identifies the absence of a selection principle with Eulerian spontaneous stochasticity. The paper is honest in crediting the Armstrong-Vicol result, and it provides detailed appendices, a reproducible numerical algorithm for the multiscale flow, and a large amount of historical context. However, the two pillars of the theoretical part — Theorem 1 and the paper's own proof of Theorem 3 — contain load-bearing gaps that need repair; the conclusions may be true, but they are not established as written.","major_comments":[{"comment":"The construction of the regularised vector field f_epsilon = theta_epsilon F_epsilon + (1 - theta_epsilon) f0 is not shown to satisfy Hypothesis (H0). Under (H0), f0 is only continuous, so the interpolation is generally only continuous; Peano's theorem gives existence but not uniqueness, and the assertion that g_epsilon is 'the unique solution' of dx/dt = f_epsilon(x) is unsupported. This matters because in the intended applications f0 is non-Lipschitz (otherwise S0 would be a singleton). The same problem affects the convex-combination step f_theta = a f_x + (1-a) f_y used to obtain arbitrary mixtures of Dirac measures: well-posedness is not an inherited property under convex combinations of well-posed vector fields. The proof needs either a modified construction that provably yields a unique solution for every kappa>0, or an additional hypothesis and an Osgood/Lipschitz estimate for f0.","section":"Section 5.3.2 / Appendix 11.6"},{"comment":"The statement M0 = co(E) is false as written because co(E) is defined as the set of finite convex combinations, while P(S0) is generally larger. The intended statement is that M0 is the closed convex hull of E. The proof only places finite convex combinations of Dirac masses in M and then says 'taking the closure'. But M is an infinite union of the compact sets M(gamma), and no closedness of M is established. Density of co(E) in M0 does not imply M = M0 unless M is closed. Corollary 1 has a related defect: the universal regularisation gamma_un is built from two-point combinations theta delta_x + (1-theta) delta_y only, and such measures are not weak-* dense in P(S0) when S0 has more than two points. A countable dense family of finitely supported measures and a diagonal argument would be required to obtain M(gamma_un) = M0.","section":"Section 5.3.2 / Theorem 1 and Corollary 1"},{"comment":"The proof of Theorem 3 depends crucially on the convention m* = 0 and on the lower bound (80), which is used to force A+/A- > 1. The Armstrong-Vicol estimates invoked in Lemma 2 are proved in [3,4] for m* >= 1. Appendix 11.13 simply asserts that 'all necessary estimates in [3] also hold' for the convention epsilon_0^{-1} = ceil(Lambda^{1/(q-1)}), without proof. This is a load-bearing assertion: Lemma 2, Lemma 3, and inequality (80) all rely on it. In addition, the statement of Theorem 3 does not explicitly include the analyticity condition (51) that is used in the cited estimates. The authors should either provide the missing verification for the m* = 0 extension, or present Theorem 3 explicitly as a corollary of Proposition 5.5 of [4] rather than as an independent proof.","section":"Section 7 / Appendix 11.13"},{"comment":"Definition 1 and the framework of Sections 5.3-5.5 are formulated for finite-dimensional phase space H = R^d, with compactness, Krein-Milman, and Prokhorov arguments used for probability measures on a compact subset of R^d. Section 5.6 explicitly states that the PDE extension is only formal and that the authors 'will not pursue this further'. Nevertheless, Theorem 3 states that the passive scalar equation (1) is spontaneously stochastic 'in the sense of Definition 1'. To make this a theorem, one needs an infinite-dimensional version of Definition 1 and of Hypothesis (H0), including a phase space on which the chosen observable is continuous. Without such a formulation, the claim that (1) exhibits SpSt is a corollary of the Armstrong-Vicol result [4], not a theorem of the framework developed in this paper.","section":"Sections 5.1-5.6 and Section 7"}],"minor_comments":[{"comment":"The reference 'Fig. .1' should read 'Fig. 1'.","section":"Section 3.1"},{"comment":"The norm in the final display is missing a closing parenthesis: it should be ||nabla theta^kappa||^2_{L^2((0,t) x T^2)}.","section":"Equation (14)"},{"comment":"The text describing Figure 4 refers to 'dotted black lines' and 'solid black line' in a way that is hard to reconcile with the figure caption; please harmonise the notation and the captions.","section":"Section 4.2"},{"comment":"Using epsilon both for the small parameter in (39) and for the scale parameter epsilon_m is confusing; consider renaming the small parameter to eta to avoid collision.","section":"Section 7"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely to interest the community, but the central theoretical machinery is not yet proved to the advertised level. Theorem 1 has two repairable but real gaps, and the independent proof of Theorem 3 depends on an unverified extension of the Armstrong-Vicol estimates. The authors' honest attribution of the Armstrong-Vicol result is a strength, but the manuscript should not claim an alternative proof until the m* = 0 extension is supplied. Given the scope of the gaps, major revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth your time. This paper tries to give a common language for the 'absence of selection principle' that mathematicians see and the 'spontaneous stochasticity' physicists mean. The finite-dimensional framework is new: Definition 1 via liminf/limsup of a regularisation function, the equivalence with non-uniqueness, the Bebutov-flow bridge to renormalisation, and the universality classes defined by ergodicity. It is a genuine synthesis, and the authors credit Armstrong-Vicol fairly.\n\nThe best part is the idea behind Theorem 1: every probability measure on the attainable set of the inviscid system can be selected by some well-posed regularisation. If it holds, it turns a perceived pathology into a measure-selection phenomenon. But the proof as written has two gaps. First, the interpolated regularisations f_epsilon = theta_epsilon F_epsilon + (1 - theta_epsilon) f0 are declared well-posed 'by construction,' yet under Hypothesis (H0) f0 is only continuous, so the interpolation is generally only continuous and uniqueness is not guaranteed. This needs a Lipschitz f0 or an explicit smoothing step. Second, the proof shows finite convex combinations of Diracs lie in M and says 'taking the closure,' but M is an infinite union of compact sets M(gamma) and is not shown to be closed; density of co(E) in M0 does not imply equality. Both gaps look repairable, but they are real.\n\nThe Armstrong-Vicol application is less of a problem. The paper's own proof of Theorem 3 depends on an unproven m* = 0 extension of the AV estimates, but the published Armstrong-Vicol result independently establishes the absence of selection principle, so the conclusion is safe via citation. What is not safe is the abstract's phrasing that there is a well-defined measure selection principle; Section 8 leaves the existence of the limit measure as a conjecture. The numerics are transparent about running outside the theorem's parameter regime and using arbitrary prefactors, which is honest.\n\nWho gets value: anyone working on inviscid limits of ODEs/PDEs, spontaneous stochasticity, or RG for turbulence. The historical review and the framework are worth reading even if Theorem 1 is not yet closed. Send it to a serious referee, not a desk reject, but the referee should insist the gaps be addressed. I would not cite it in its current form; if the proof is repaired, it becomes a natural citation.","headline":"A genuinely useful framework for Eulerian spontaneous stochasticity, but Theorem 1 has a real proof gap; the Armstrong-Vicol application rests on solid external work and is not the main risk.","tokens_in":58123,"tokens_out":2519,"would_cite":false,"duration_ms":23401,"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":"This paper proves spontaneous stochasticity is a measure-selection phenomenon, and that the Armstrong–Vicol passive scalar exhibits both Lagrangian and Eulerian forms of it.","keywords":["spontaneous stochasticity","passive scalar","anomalous dissipation","selection principle","measure selection principle","fractal homogenisation","Bebutov flow","renormalised diffusivity"],"falsifier":"Check well-posedness of the interpolated field $f_\\varepsilon=\\theta_\\varepsilon F_\\varepsilon+(1-\\theta_\\varepsilon)f_0$ for a continuous $f_0$ with nonunique inviscid solutions: if $\\dot x=f_\\varepsilon(x)$ has two solutions for arbitrarily small $\\varepsilon$, Theorem 1's construction is not a valid regularisation. For Theorem 3, compute the renormalised diffusivity sequence with $\\varepsilon_0^{-1}=\\lceil\\Lambda^{1/(q-1)}\\rceil$ and evaluate $A_+/A_-$ from the dissipation observable; if the lower bound (80) is violated so that $A_+/A_-\\le 1$ for all admissible observables, the claimed Eulerian spontaneous stochasticity does not follow.","tokens_in":56789,"feed_emoji":"🎲","tokens_out":9496,"duration_ms":80145,"temperature":0.7,"pith_summary":"This paper tries to establish two things. First, in a singular inviscid limit, the absence of a selection principle is not a pathology: it is the phenomenon of spontaneous stochasticity, defined as a continuous observable taking two distinct limit values along different vanishing-regularisation subsequences. Second, the Armstrong–Vicol passive scalar is spontaneously stochastic in both the Lagrangian and the Eulerian sense. The measure-theoretic core is a proof that every probability measure on the attainable set of the inviscid system is the inviscid-limit statistic of some admissible regularisation, so spontaneous stochasticity becomes a measure-selection problem whenever the inviscid system is non-unique. A sympathetic reader would care because it turns 'which solution is selected?' into 'which probability measure over solutions is selected?', with universality classes classified by the ergodic properties of regularisations.","feed_headline":"Spontaneous stochasticity is measure selection, not pathology","feed_subtitle":"Every inviscid-limit statistic is attainable; the Armstrong-Vicol scalar is Eulerian as well as Lagrangian.","key_machinery":"The machinery has two levels. In the general theory, the central object is the regularisation curve $\\gamma(\\tau)=\\Phi_t[f(\\cdot,1/\\tau)]x_0$ living in the space $\\mathrm{Reg}$, acted on by the Bebutov flow; Birkhoff averages along this flow yield the set $\\mathcal{M}(\\gamma)$ of subsequential statistics, and the paper proves $\\mathcal{M}=\\mathcal{M}_0=\\mathcal{P}(S_0)$, with $\\mathcal{M}_0$ the probability measures on the attainable set. This turns universality classes into genericity sets $\\mathcal{B}_\\mu$ of regularisations sharing the same limiting measure. For the Armstrong–Vicol model, the load-bearing object is the renormalised diffusivity sequence $\\kappa_{m-1}=\\kappa_m+c_0\\varepsilon_m^{2\\beta}/\\kappa_m$ with hypergeometric scale separation $\\varepsilon_m^{-1}=\\lceil\\Lambda^{q^m/(q-1)}\\rceil$, together with the homogenisation constraints that make the alternating shear-flow cascade produce anomalous diffusion; Lemma 2 ties the dissipation observable to the renormalised diffusivity at scale $m_\\star$, and Lemma 3 bounds $\\kappa_+/\\kappa_-$ from below, forcing the ratio $A_+/A_-$ above 1.","core_discovery":"The central claim is that spontaneous stochasticity — defined through the regularisation function $A(\\kappa)=O(\\Phi_t[f(\\cdot,\\kappa)]x_0)$ as $-\\infty<\\liminf_{\\kappa\\to 0}A(\\kappa)<\\limsup_{\\kappa\\to 0}A(\\kappa)<+\\infty$ — is exactly equivalent to finite-time trajectory splitting and to the lack of a selection principle, and that this equivalence is measure-theoretically complete. Theorem 1 states that the space $\\mathcal{M}$ of all probability measures obtainable from regularisations coincides with $\\mathcal{M}_0=\\mathcal{P}(S_0)$, the set of all probability measures supported on the attainable set of the inviscid system; hence any statistical behaviour compatible with the inviscid system is realisable by some well-posed regularisation. For the Armstrong–Vicol model, Theorem 3 proves that with $m_\\star=0$ and $\\Lambda$ sufficiently large the dissipation observable satisfies $A_+/A_- \\ge e^{-4\\varepsilon} A_+^{m_\\star}/A_-^{m_\\star} > 1$, so the passive scalar exhibits Eulerian spontaneous stochasticity, in addition to the Lagrangian spontaneous stochasticity that follows from anomalous diffusion.","pith_inferences":["If the measure-selection theorem extends to infinite-dimensional settings, then numerical and experimental searches for a selection principle in turbulence should target measure-valued fixed points of renormalisation rather than a single trajectory; shell models with isolated Hölder singularities are a minimal, computationally accessible testbed.","For the Armstrong–Vicol model, Eulerian spontaneous stochasticity predicts that under-resolved simulations should show the dissipation plateau jumping between two values as resolution or diffusivity is tuned; computing $A(\\kappa)$ along the renormalised sequence and measuring $A_+/A_-$ would test inequality (40) directly.","The conjecture that every non-renormalisable regularisation becomes renormalisable after a continuous reparametrisation suggests that the apparent ambiguity of the inviscid limit may be a gauge-like redundancy; if true, the physical statistics are the invariants under such reparametrisations, which parallels renormalisation-group fixed-point universality."],"forward_implications":["Absence of a selection principle is equivalent to spontaneous stochasticity and to finite-time trajectory splitting, so unbounded finite-time Lyapunov exponents in the inviscid limit are a direct signature of the phenomenon.","Because $\\mathcal{M}=\\mathcal{M}_0$, any probability measure on the set of attainable inviscid solutions is realisable as a limiting statistic; universality classes are exactly the genericity sets $\\mathcal{B}_\\mu$ and are distinguished by the ergodic properties of regularisations.","As soon as the inviscid system is non-unique, there exist regularisations for which the system is spontaneously stochastic; a necessary condition is that trajectories hit a non-Lipschitz singular set in finite time, detectable through Dini-type directional derivatives.","The Armstrong–Vicol passive scalar is Eulerian spontaneously stochastic: the dissipation rate has two distinct limit values as diffusivity vanishes, and it is also Lagrangian spontaneously stochastic through anomalous diffusion.","For the hypergeometric renormalised diffusivity sequence, a well-defined probability measure only emerges on the log-log scale; the limiting distribution of the renormalised diffusivity is non-Dirac and encodes the measure-selection information."],"supporting_citations":[{"why":"Supplies the Armstrong–Vicol model and the proof that the passive scalar has anomalous dissipation and lacks a selection principle; Theorem 3 builds on its Proposition 5.5.","marker":"[4]"},{"why":"Provides the fractal homogenisation construction and the key estimates (its Proposition 5.2 and Lemmas 3.3–3.4) used in Lemma 2 and the proof of Theorem 3.","marker":"[3]"},{"why":"Establishes the Lagrangian fluctuation-dissipation relation equating anomalous diffusion with Lagrangian spontaneous stochasticity.","marker":"[31]"},{"why":"Introduces Lagrangian spontaneous stochasticity in the model where only that form is possible, providing the contrast case for Eulerian spontaneous stochasticity.","marker":"[23]"},{"why":"Constructs a transport system whose inviscid limit has one energy-conserving and one dissipative subsequential solution, providing the liminf/limsup separation that motivates the Eulerian definition.","marker":"[24]"},{"why":"Gives the theory of non-Lipschitz singularities and statistical determinism used for the necessary condition and the finite-dimensional examples.","marker":"[34]"}],"fun_headline_variants":["Measure selection, not pathology: stochasticity explained","Passive scalar's randomness is a measure selection principle","Eulerian stochasticity emerges from measure selection","Every weak solution attainable: spontaneous stochasticity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central claim stands on two premises: that the interpolated regularisations built in Theorem 1 are well-posed even though only continuity of $f_0$ is assumed, and that the Armstrong–Vicol estimates proven for $m_\\star\\ge 1$ extend to the $m_\\star=0$ convention used in Theorem 3; if either fails, the corresponding proof collapses.","fun_headline_variants_meta":{"raw":{"variants":["Measure selection, not pathology: stochasticity explained","Passive scalar's randomness is a measure selection principle","Eulerian stochasticity emerges from measure selection","Every weak solution attainable: spontaneous stochasticity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000336,"raw_usage":{"total_tokens":1935,"prompt_tokens":1095,"completion_tokens":840,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":711,"completion_tokens_details":{"reasoning_tokens":781}},"tokens_in":711,"tokens_out":840,"duration_ms":7592,"temperature":1.0,"reasoning_tokens":781,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:20:18.329597+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check well-posedness of the interpolated field $f_\\varepsilon=\\theta_\\varepsilon F_\\varepsilon+(1-\\theta_\\varepsilon)f_0$ for a continuous $f_0$ with nonunique inviscid solutions: if $\\dot x=f_\\varepsilon(x)$ has two solutions for arbitrarily small $\\varepsilon$, Theorem 1's construction is not a valid regularisation. For Theorem 3, compute the renormalised diffusivity sequence with $\\varepsilon_0^{-1}=\\lceil\\Lambda^{1/(q-1)}\\rceil$ and evaluate $A_+/A_-$ from the dissipation observable; if the lower bound (80) is violated so that $A_+/A_-\\le 1$ for all admissible observables, the claimed Eulerian spontaneous stochasticity does not follow.","supporting_citations":[{"cited_title":"Drivas and G.L","cited_arxiv_id":null,"evidence_quote":"Establishes the Lagrangian fluctuation-dissipation relation equating anomalous diffusion with Lagrangian spontaneous stochasticity."},{"cited_title":"Chaves, K","cited_arxiv_id":null,"evidence_quote":"Introduces Lagrangian spontaneous stochasticity in the model where only that form is possible, providing the contrast case for Eulerian spontaneous stochasticity."},{"cited_title":"Colombo, G","cited_arxiv_id":null,"evidence_quote":"Constructs a transport system whose inviscid limit has one energy-conserving and one dissipative subsequential solution, providing the liminf/limsup separation that motivates the Eulerian definition."},{"cited_title":"Drivas, A.A","cited_arxiv_id":null,"evidence_quote":"Gives the theory of non-Lipschitz singularities and statistical determinism used for the necessary condition and the finite-dimensional examples."}],"review_version":1}