{"id":"a0908d53-485b-4383-a6f7-9c2b948fa315","arxiv_id":"2412.00708","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For a flat immobile interface in a particle system, the interface fluctuation converges, after suitable scaling, to Brownian motion in d=1 and to the stochastic heat equation in d>=2.","lead":"This paper introduces a new class of random equations (stochastic PDEs) for the fluctuating boundary between two phases, and proves that, after stretching space, the boundary wobbles like a random walk or a random surface. It matters because it connects microscopic particle models to the modern theory of singular random equations, giving concrete predictions that can be tested in simulations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4.2 and Theorem 4.4 assert joint-in-time convergence of Ψ^K, but Theorem 4.1 proves only single-time weak convergence; the proof does not bridge this gap, so the Gaussian interface limit as a process is not established.","rationale":"Good-faith reading: the paper proposes a class of singular SPDEs and studies a flat-interface scaling; the linear SPDE part is extensively proved via covariance computations, so I do not see a soundness failure there. The nonlinear and particle-system portions are explicitly heuristic (Section 1), so A.18 being unproved is acknowledged and does not by itself invalidate the SPDE-level theorem. The single most load-bearing problem is the unsupported inference from one-time convergence to finite-dimensional convergence. This is an internal logical gap inside the advertised rigorous part. It is fixable: the linear SPDE is Gaussian and the mild formula (4.8) gives enough structure to compute multipoint covariances. Thus the correct verdict remains CONDITIONAL, not REJECT. The reader's weakest assumption, A.18, is a different, real concern; I partially agree with it, but I weight the Corollary 4.2 gap more heavily because it affects the rigorous claim. A concrete two-time covariance check would settle whether the limit process is as stated.","tokens_in":44718,"tokens_out":6326,"duration_ms":66241,"concrete_test":"Using the mild representation (4.8), compute the two-time covariance E[⟨Ψ^K(t_1),H_1⟩⟨Ψ^K(t_2),H_2⟩] for 0<t_1<t_2 by Itô isometry, and evaluate its K→∞ limit, including the contribution of the K^{-1/2}∇_x·W_2 term and the semigroup cross terms. Compare with E[⟨ψ(t_1)e,H_1⟩⟨ψ(t_2)e,H_2⟩] for the additive SHE (4.5). If the limits agree for all H_1,H_2, Corollary 4.2 is true and requires only a missing proof; if they differ, the asserted multipoint Gaussian interface process is not the limit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive gap is in the passage from Theorem 4.1 to Corollary 4.2. Theorem 4.1 shows, for each fixed t, that ⟨Ψ^K(t),H⟩ → ⟨Ψ(t),H⟩ weakly in L2(Ω). Corollary 4.2 then claims that for every 0<t_1<...<t_n the joint law of {Ψ^K(t_k)} converges to that of {ψ(t_k)e(z)}, and Theorem 4.4 makes the same claim for d=1. This does not follow: single-time weak convergence says nothing about the joint distribution at two or more times unless additional information, such as convergence of the full covariance structure of the Gaussian field, is supplied. The proof of Corollary 4.2 contains only the sentence 'Theorem 4.1 implies ...,' and no two-time covariance computation is given. Since the central claim is that the interface fluctuation is the additive SHE solution ψ, the multipoint law is essential; without it, the Gaussian process statement is unproven. The higher-order Boltzmann-Gibbs principle (A.18) is also unproved and cited to [14], but the paper explicitly labels the particle-system derivation heuristic, so that is a stated limitation; the joint-convergence gap is an internal gap in the part of the argument advertised as rigorous.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a family of singular or regularized SPDEs, Eq. (2.3), intended as mesoscopic fluctuation equations for the density field of a Glauber–Kawasaki particle system near a flat interface. After stretching the coordinate normal to the interface by sqrt(K) and rescaling the fluctuation amplitude by K^{-3/4}, the linear member of the family is scaled to the SPDE (3.10)/(4.1). The central rigorous result, Theorem 4.1, asserts that for each fixed time the rescaled fluctuation field converges weakly, after testing against smooth functions, to a Gaussian field with spatial profile e(z) = U_0'(-z)/||U_0'||_{L^2}, whose amplitude solves an additive stochastic heat equation on the tangent torus (or a Brownian motion when d=1). Corollary 4.2 and Theorem 4.4 upgrade this to finite-dimensional convergence in time. Section 5 heuristically studies nonlinear corrections and finds that the quadratic correction vanishes and the cubic correction gives a dynamic P(phi)-type equation. Appendices provide the Carr–Pego spectral estimates and a heuristic derivation of the SPDE from the particle system via the higher-order Boltzmann–Gibbs principle.","tokens_in":45031,"tokens_out":15184,"duration_ms":151908,"significance":"If the process-level gap noted below is repaired, the paper gives a clean and testable route from a singular SPDE to Gaussian interface fluctuations: the interface shift N^{-d/2}K^{1/4}psi/||U_0'|| and the dominance of the neutral mode e(z) are explicit and falsifiable. The use of Carr–Pego spectral estimates to identify the projection onto the slow mode is a genuine technical strength, and the single-time convergence proof is carried out in considerable detail. The paper is also transparent about its heuristic parts: the derivation from the particle system is labelled as assuming the higher-order Boltzmann–Gibbs principle, and Section 5 is explicitly heuristic. The main value of the paper is as a proposal and partial rigorous validation of a new class of SPDEs; with the requested revisions it would be a solid contribution to the stochastic interface literature.","major_comments":[{"comment":"The proof of Corollary 4.2 does not establish the claimed joint-in-time convergence. Theorem 4.1 proves, for each fixed t, weak convergence in L^2(Omega) of the real variable <Psi^K(t),H> to <Psi(t),H>. The proof of Corollary 4.2 contains only the sentence 'Theorem 4.1 implies ...' and gives no computation of two-time covariances, no tightness argument, and no joint characteristic-function argument. Single-time marginal convergence says nothing about the joint law of {Psi^K(t_1),...,Psi^K(t_n)}. Since the central claim is that the interface fluctuation is the additive SPDE solution psi(t,x) (or, for d=1, the Brownian motion c_* B_t in Theorem 4.4), the multipoint law is essential and is currently unproven. The gap is fixable in principle, for example by proving convergence of E[<Psi^K(t_1),H_1><Psi^K(t_2),H_2>] to the corresponding covariance of the limiting Gaussian field, but that argument is absent. Until then Corollary 4.2 and Theorem 4.4 should be weakened to single-time statements.","section":"Corollary 4.2 and Theorem 4.4"},{"comment":"Equation (4.3) is not consistent with the weak-form limit used in the proof of Theorem 4.1. In the proof, the noise contribution I_1(t) is defined with the integrand g_1(check U_0(w)) partial_w H_{t-s}(w,y), and since H_{t-s}(w,y) is asymptotically e(w) times a function of (t-s,x), the limiting coefficient involves g_1(check U_0) e'(w), not e(w) partial_w g_1(check U_0). If Eq. (4.3) is read as an ordinary stochastic integral with integrand e(w) partial_w[g_1(check U_0(w))], it gives a different noise coefficient, and the formula for c_* in (4.7) becomes ambiguous. If the intended meaning is the distributional pairing <partial_w(g_1(check U_0) dot W_1), e> = - integral g_1(check U_0) e' W_1, then Eq. (4.3) should be rewritten in that dual form. As written, a reader cannot verify that the variance of the limiting SHE or Brownian motion is correct, and this coefficient is load-bearing for the main interface-fluctuation statement.","section":"Eq. (4.3) and the proof of Theorem 4.1"},{"comment":"The derivation of the SPDE (2.3) from the Glauber–Kawasaki dynamics relies on the higher-order Boltzmann–Gibbs principle, Eq. (A.18), which is assumed and cited to the in-preparation reference [14]; no proof is given. The paper explicitly labels this derivation heuristic, so this is not an internal inconsistency, but it is a genuine limitation: Theorem 4.1 and Corollary 4.2 are theorems about the model SPDE, not about the original particle system. The Introduction and Section 2.5 should state this distinction more prominently, and if the paper intends to claim interface fluctuation for the particle system, the Boltzmann–Gibbs principle must either be proved or formulated as a precise conjecture.","section":"Appendix A, Eq. (A.18)"}],"minor_comments":[{"comment":"There are typos: 'Bolt zmann-Gibbs' in the abstract should be 'Boltzmann–Gibbs', and 'due to the luck of regularity' in Section 2.3 should read 'due to the lack of regularity'.","section":"Abstract and Section 2.3"},{"comment":"The text refers to 'Lemma 4.5 (doubly called Lemma 4.4)'; this is confusing and should be corrected to a single unambiguous citation.","section":"Appendix B, proof of Lemma B.2"},{"comment":"In the formula for c_*, the expression 'partial_w e g_1(check U_0)' is ambiguous. Use parentheses, e.g. (partial_w e(w)) g_1(check U_0(w)) or partial_w(e(w)g_1(check U_0(w))), to match the intended definition of psi_1.","section":"Eq. (4.7)"},{"comment":"The higher-order Boltzmann–Gibbs principle is attributed to an in-preparation paper [14]. If the manuscript is revised before that paper is available, the dependence on [14] should be made explicit in the main text, and the reference should be updated if possible.","section":"Reference [14]"}],"recommendation":"major_revision","confidential_remarks":"The decisive issues are (i) the missing proof of joint-in-time convergence in Corollary 4.2/Theorem 4.4, and (ii) the ambiguity in the noise coefficient in Eq. (4.3) versus the proof. Both are fixable but require substantive revision. The Boltzmann–Gibbs principle is a stated heuristic assumption, so I do not view it as grounds for rejection, but the editor should be aware that the rigorous results concern the SPDE model, not the particle system. If the author can repair the joint-convergence gap and disambiguate the noise coefficient, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper has a genuinely new idea and a substantial rigorous core. Funaki proposes a class of SPDEs (2.3) for density fluctuations in Glauber-Kawasaki dynamics, with a scaling near the interface that produces, in the linear limit, a Gaussian height function: an additive SHE on T^{d-1} for d≥2 and a Brownian motion for d=1, with the explicit profile e(z)=U0'(-z)/||U0'|| along the stretched normal coordinate. The identities c2=0 and c3<0 are nice. The derivation of the linear limit in Section 4 is careful: the noise rescaling is done through explicit covariance computations, and the Carr-Pego spectral estimates are used properly. The paper is also honest about what is heuristic: the particle-system derivation relies on a higher-order Boltzmann-Gibbs principle (A.18) cited to [14], and the nonlinear section is explicitly formal.\n\nThe soft spot is an internal gap in the part presented as rigorous. Corollary 4.2 and Theorem 4.4 claim convergence of finite-dimensional distributions in time. Theorem 4.1 proves, for each fixed t and each test function H, weak convergence in L2(Ω) of the single-time marginal ⟨Ψ^K(t),H⟩. That does not imply joint convergence at several times: the proof of Corollary 4.2 just says 'Theorem 4.1 implies ,' with no computation of two-time covariances. Since the central claim is that the interface shift evolves as Brownian motion or SHE, the process-level statement is not optional. This is a missing proof, not a counterexample, and it looks fixable by a standard covariance estimate, but as written the conclusion outruns the argument.\n\nA minor related point: the proof of Theorem 4.1 uses weak convergence for the noise term I1, and Remark 4.2 notes that strong convergence would follow from the unproven (3.33). This does not undermine the single-time result, but it is another sign that the process-level step was not fully worked out.\n\nWho gets value: someone working on SPDE limits of particle systems or on singular SPDEs will find the scaling framework and the linear analysis useful, and the heuristic nonlinear section raises the right questions. The paper deserves a serious referee. I would send it to peer review with a request to close the gap in Corollary 4.2 and Theorem 4.4; the rest of the linear analysis is solid. I would cite the scaling framework and the single-time linear limit, but I would not cite the Brownian/SHE process statement until the proof is completed.\n\nBest,","headline":"New scaling framework for interface fluctuations with a solid single-time linear limit, but the claimed Brownian/SHE process limit needs a missing two-time covariance estimate.","tokens_in":45575,"tokens_out":5861,"would_cite":true,"duration_ms":58741,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60K35","82C21","82C24","82C26","74A50","35R60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that under the scaled SPDE (3.10) the fluctuation field converges to $\\psi(t,x)e(z)$, making microscopic interface fluctuations Gaussian at the stretched scale.","keywords":["stochastic PDE","fluctuating interface","Glauber-Kawasaki dynamics","Boltzmann-Gibbs principle","Allen-Cahn equation","interface fluctuation","Gaussian fluctuation","singular SPDE"],"falsifier":"Run the Glauber-Kawasaki dynamics with a bistable $f$, flat interface, and scaling $K\\ll N^{2d/3}$; if the rescaled fluctuation field $\\langle\\Psi^K(t),H\\rangle$ does not converge to $\\langle\\psi(t,x)e(z),H\\rangle$ with $\\psi$ solving the additive stochastic heat equation and noise constant $c_*$ given by (4.7), then either the Boltzmann-Gibbs replacement (A.18) or the SPDE limit fails. A more targeted check: in $d=1$, compare the variance of the interface location to $c_*^2t\\,N^{-1}K^{1/2}$; any other scaling would falsify.","tokens_in":44480,"feed_emoji":"🌊","tokens_out":10104,"duration_ms":134607,"temperature":0.7,"pith_summary":"The paper proposes a mesoscopic stochastic PDE for the density fluctuation of a Glauber-Kawasaki particle system whose density profile contains a flat interface between two stable phases. The author derives this SPDE from the particle dynamics through a higher-order Boltzmann-Gibbs principle, then stretches the coordinate normal to the interface by $\\sqrt{K}$ and rescales the fluctuation field by $K^{3/4}$. The central result is that the rescaled field collapses onto the derivative $U_0'$ of the standing-wave profile, and the interface height $\\psi(t,x)$ converges to a Gaussian process: an additive stochastic heat equation on the transverse torus for $d\\ge 2$, and a Brownian motion for $d=1$. If the derivation is correct, it is a rigorous linear bridge from microscopic phase separation to random interface motion, with the nonlinear versions proposed heuristically.","feed_headline":"Fluctuating interfaces become Gaussian shifts of a standing wave","feed_subtitle":"A particle-system SPDE limit gives additive stochastic heat equation in 2D and Brownian motion in 1D.","key_machinery":"The argument is carried by the Sturm-Liouville operator $A^K=-\\partial_z^2-f'(\\bar{v}_K(z))$, the linearization of the stationary Allen-Cahn equation around the stretched transition profile $\\bar{v}_K(z)=v_K(z/\\sqrt{K})$. Spectral estimates from the one-dimensional metastable Allen-Cahn literature show $A^K$ has two exponentially small eigenvalues, coming from the two interfaces, and a uniform spectral gap above them, so $e^{-tKA^K}$ projects any $L^2$ initial data onto the zero mode $e(z)=U_0'(-z)/\\|U_0'\\|_{L^2(\\mathbb{R})}$ after time $t>0$. The scaling $\\Psi=K^{-3/4}\\Phi(t,z/\\sqrt{K},x)$ is chosen so that exactly one noise term survives with $O(1)$ size, and the projection onto $e(z)$ turns the limiting SPDE into the additive stochastic heat equation, or into Brownian motion when $d=1$. The higher-order Boltzmann-Gibbs principle (A.18) is the derivation mechanism: it replaces the microscopic Glauber creation-minus-annihilation rate by $K$ times the Taylor expansion of $f$ around $u_K$ in powers of $N^{-d/2}\\Phi$, with a remainder that is ignored.","core_discovery":"Under the scaling $\\Psi^{K}(t,z,x)=K^{-3/4}\\Phi(t,z/\\sqrt{K},x)$, the linear SPDE (3.10) has a Gaussian limit $\\Psi(t,z,x)=\\psi(t,x)e(z)$ in which $e(z)=U_0'(-z)/\\|U_0'\\|_{L^2(\\mathbb{R})}$ is the normalized derivative of the standing-wave profile. The height function $\\psi$ solves the additive stochastic heat equation $\\partial_t\\psi=\\Delta_x\\psi+c_*\\dot{W}$ on $\\mathbb{T}^{d-1}$ when $d\\ge 2$, and equals $c_*B_t$ when $d=1$, with $c_*$ given by (4.7). This yields an interface shift of order $N^{-d/2}K^{1/4}\\psi/\\|U_0'\\|_{L^2(\\mathbb{R})}$ in the normal direction, meaning the interface moves as the graph $x_1=N^{-d/2}K^{1/4}\\phi(t,x)$ with $\\phi=\\psi/\\|U_0'\\|$ in the $d=2$ case. The author also shows that the quadratic nonlinearity in the nonlinear SPDE vanishes identically ($c_2=0$), so the first surviving interaction term is cubic; writing $K=N^{2d/5}$ gives the dynamic P($\\phi$)-type equation $\\partial_t\\psi=\\Delta_x\\psi+c_*\\dot{W}+c_3\\psi^3$ with $c_3<0$ under a convexity assumption on $f$.","pith_inferences":["If the higher-order Boltzmann-Gibbs principle is later proved from the particle dynamics, Theorem 4.1 would turn the interface-fluctuation problem into a spectral-gap plus martingale argument, so the same semigroup tools would produce a theorem about the particle system rather than about an auxiliary SPDE.","The vanishing of $c_2$ follows from $f(\\rho_\\pm)=0$ and needs no information about the shape of $f$ between the stable phases; this suggests the same cancellation, and hence the same cubic-then-linear ordering, will appear in other bistable particle systems and in anisotropic interface problems.","The condition $K^{7/4}\\ll N^{d/2}$ for the linear regime and the choices $K=N^{2d/7}$, $N^{2d/5}$ for nonlinear regimes imply a crossover: a simulation sweeping $(N,K)$ should see the interface-shift variance pass from Gaussian linear behavior to nonlinear behavior near those curves, which is a testable prediction the paper does not state."],"forward_implications":["In $d=1$ the two transition-layer positions converge to a Brownian motion with diffusivity $c_*^2/\\|U_0'\\|^2$, so microscopic phase separation produces random diffusive motion of the interface at scale $N^{-1/2}K^{1/4}$.","In $d=2$ the interface height is a function $\\psi(t,x)$ solving $\\partial_t\\psi=\\Delta_x\\psi+c_*\\dot{W}$ on $\\mathbb{T}$, giving a Gaussian random curve whose covariance is explicit from the heat kernel.","For $d\\ge 3$ the limit $\\psi(t,x)e(z)$ is distribution-valued, so the phrase 'fluctuating interface' loses a pointwise geometric meaning in high dimensions.","The quadratic coefficient $c_2$ in the nonlinear SPDE vanishes for any $f$ satisfying $f(\\rho_\\pm)=0$ and $\\int_{\\rho_-}^{\\rho_+}f=0$, so the first surviving nonlinearity is cubic and, under the convexity assumptions, has a negative coefficient $c_3<0$.","Away from the interface the density fluctuation collapses to independent centered Gaussian variables with variance given by (6.3), so fluctuations decouple in space-time at that scale."],"supporting_citations":[{"why":"Supplies the spectral estimates for the linearized Allen-Cahn operator, two small eigenvalues and a uniform gap, that drive the projection onto $e(z)$ in Proposition 3.3.","marker":"[4]"},{"why":"Provides the strong-solution theory for the dynamic P($\\phi$)-model used to identify the cubic SPDE (5.4) as the meaningful singular limit.","marker":"[7]"},{"why":"Earlier scaling limit for a stochastic phase-separation PDE; the Gaussian-layer-fluctuation argument is the model the present linear limit follows.","marker":"[9]"},{"why":"Gives the hydrodynamic limit and Dynkin decomposition for the Glauber-Kawasaki dynamics that the heuristic derivation of SPDE (2.3) starts from.","marker":"[12]"},{"why":"Establishes the macroscopic sharp-interface motion of Glauber-Kawasaki dynamics, so the flat stationary interface is the correct background for the fluctuation analysis.","marker":"[13]"},{"why":"The in-preparation companion paper is the stated source of the higher-order Boltzmann-Gibbs principle (A.18); the entire particle-system derivation of (2.3) leans on it.","marker":"[14]"},{"why":"Introduced the mesoscopic SPDE approach to fluctuating interfaces that this paper formalizes and makes rigorous in the linear case.","marker":"[20]"}],"fun_headline_variants":["Gaussian interface shifts from a standing wave derivative","SPDE limit gives additive stochastic heat equation for interfaces","Interface noises collapse to Brownian motion or heat equation","From particle density to Gaussian interface SPDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the higher-order Boltzmann-Gibbs principle, Eq. (A.18): that the microscopic Glauber term can be replaced, after local ensemble averaging, by $K$ times the third-order Taylor expansion of the bistable function $f$ around the macroscopic profile $u_K$ in powers of $N^{-d/2}\\Phi$, discarding all remainder terms; the paper cites this principle to an in-preparation companion work and gives no proof.","fun_headline_variants_meta":{"raw":{"variants":["Gaussian interface shifts from a standing wave derivative","SPDE limit gives additive stochastic heat equation for interfaces","Interface noises collapse to Brownian motion or heat equation","From particle density to Gaussian interface SPDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000911,"raw_usage":{"total_tokens":3960,"prompt_tokens":1038,"completion_tokens":2922,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":654,"completion_tokens_details":{"reasoning_tokens":2861}},"tokens_in":654,"tokens_out":2922,"duration_ms":22512,"temperature":1.0,"reasoning_tokens":2861,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:11:57.335289+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the Glauber-Kawasaki dynamics with a bistable $f$, flat interface, and scaling $K\\ll N^{2d/3}$; if the rescaled fluctuation field $\\langle\\Psi^K(t),H\\rangle$ does not converge to $\\langle\\psi(t,x)e(z),H\\rangle$ with $\\psi$ solving the additive stochastic heat equation and noise constant $c_*$ given by (4.7), then either the Boltzmann-Gibbs replacement (A.18) or the SPDE limit fails. A more targeted check: in $d=1$, compare the variance of the interface location to $c_*^2t\\,N^{-1}K^{1/2}$; any other scaling would falsify.","supporting_citations":[{"cited_title":"Carr and R.L","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral estimates for the linearized Allen-Cahn operator, two small eigenvalues and a uniform gap, that drive the projection onto $e(z)$ in Proposition 3.3."},{"cited_title":"Da Prato and A","cited_arxiv_id":null,"evidence_quote":"Provides the strong-solution theory for the dynamic P($\\phi$)-model used to identify the cubic SPDE (5.4) as the meaningful singular limit."},{"cited_title":"Funaki , The scaling limit for a stochastic PDE and the separation of ph ases, Probab","cited_arxiv_id":null,"evidence_quote":"Earlier scaling limit for a stochastic phase-separation PDE; the Gaussian-layer-fluctuation argument is the model the present linear limit follows."},{"cited_title":"Funaki , Hydrodynamic limit for exclusion processes , Comm","cited_arxiv_id":null,"evidence_quote":"Gives the hydrodynamic limit and Dynkin decomposition for the Glauber-Kawasaki dynamics that the heuristic derivation of SPDE (2.3) starts from."},{"cited_title":"Funaki, C","cited_arxiv_id":null,"evidence_quote":"The in-preparation companion paper is the stated source of the higher-order Boltzmann-Gibbs principle (A.18); the entire particle-system derivation of (2.3) leans on it."},{"cited_title":"Kawasaki and T","cited_arxiv_id":null,"evidence_quote":"Introduced the mesoscopic SPDE approach to fluctuating interfaces that this paper formalizes and makes rigorous in the linear case."}],"review_version":1}