{"id":"2f2adbb7-17d3-46d9-8975-4884a1e447dc","arxiv_id":"2502.02261","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"First theta-function asymptotic formulas for short-pulse soliton gases, covering two generalized reflection coefficients.","lead":"This paper derives explicit large-space and long-time asymptotic formulas for soliton gases of the short-pulse equation using Riemann-Hilbert analysis. The results give theta-function leading terms with 1/t and 1/|x| errors for two generalized reflection coefficients, but some core estimates are asserted rather than fully proved.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The phase theta in (1.8) is internally inconsistent with (3.87) and with the g-function normalization; as printed, g = theta - p behaves like 3*xi*lambda/4 at infinity instead of O(lambda^-1), so the central steepest-descent argument is not grounded as written.","rationale":"The reader's identified weakness, convergence of the discrete N-soliton Riemann-Hilbert problem to the continuum soliton gas problem, is real and appears in the unproved passage from (1.13) to (1.14) for beta_j in (-1,0). However, that gap is likely repairable by standard Riemann-sum estimates for improper integrals. The factor-of-4 phase inconsistency is a concrete internal contradiction in the central construction: the g-function is advertised as the key innovation, and every estimate in Propositions 4-6 uses its sign properties, which in turn depend on the phase definition. If theta as printed in (1.8) is used literally, the conjugation T = Y e^{t g sigma_3} does not normalize at infinity and the whole steepest-descent argument collapses; the text only becomes consistent if theta = (xi*lambda + lambda^-1)/4. Since Delta_alpha is written with xhat + t/(4*alpha*eta_2), the authors are implicitly using the corrected phase, so the final formula may be right, but the written problem statement is inconsistent with its proof. This reinforces the reader's CONDITIONAL verdict: the paper should be accepted only after the phase definition is corrected and the propagation of that correction through the constants is verified. Other completeness gaps, such as the omitted proof of Proposition 3 and the sketched matching of local parametrices, also remain, but they are secondary to this internal inconsistency.","tokens_in":50031,"tokens_out":18713,"duration_ms":180500,"concrete_test":"Recompute the g-function normalization directly: substitute (3.89)-(3.91) into g = theta - p and evaluate lim_{lambda -> +infinity} g(lambda)/lambda, first with theta = xi*lambda + 1/(4*lambda) as printed in (1.8), then with theta = (xi*lambda + 1/lambda)/4. The limit is 3*xi/4 in the first case and 0 in the second. Also evaluate the sign of Re(theta) at lambda = eta_2 under both definitions; only the second gives the claimed transition at xi = -eta_2^-2. If the second definition is confirmed, trace the corrected phase through (6.172)-(6.203) and verify that Delta_alpha and X_alpha in Theorem 2 are unchanged.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing problem is an internal factor-of-4 inconsistency in the phase. Equation (1.8) defines theta = xi*lambda + lambda^-1/4 with xi = 4*xhat/t, while (3.87) states Re(theta) = (Re(lambda)/4)*(xi + 1/|lambda|^2), which is the real part of theta = (xi*lambda + lambda^-1)/4, not of (1.8). The g-function construction settles which definition is intended: from (3.89)-(3.91), R ~ y^2 and Q ~ xi*y^4, so p(lambda) ~ xi*lambda/4 at infinity. If theta is as printed in (1.8), then g = theta - p ~ 3*xi*lambda/4, contradicting the required normalization g = O(lambda^-1) in RH Problem 5 and invalidating the conjugation T = Y e^{t g sigma_3} used in Section 6. If instead theta = (xi*lambda + lambda^-1)/4, then g ~ (4*lambda)^-1 and the g-function jumps, sign charts, and critical value -eta_2^-2 become consistent. Thus (1.8) is missing a factor 1/4 on the xi*lambda term. This is not cosmetic: Delta_alpha = Omega_alpha*(xhat + t/(4*alpha*eta_2) + phi_alpha) is built from the corrected phase, so the theorem statements and the Riemann-Hilbert problem from which they are derived are not the same problem. The central claim cannot be verified from the text until this phase normalization is corrected and all constants are re-checked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies asymptotics of soliton gases for the short-pulse equation. Starting from the N-soliton Riemann-Hilbert problem and taking N to infinity, it defines a continuum soliton-gas RH problem with two generalized reflection coefficients r0(λ)=(λ−η1)^{β1}(η2−λ)^{β2}|λ−η0|^{β0}γ(λ) and rc(λ)=(λ−η1)^{β1}(η2−λ)^{β2}χc(λ)γ(λ). It constructs a g-function and, via the transformations Y→T→S→E with Airy, modified Bessel and confluent hypergeometric parametrices, derives explicit theta-function formulas with O(1/t) error for u(x,t) and x=x̂+... in several ξ-regions, together with analogous large-x̂ formulas for u(x,0). The main results are Theorem 1 and Theorem 2. The paper is organized around the standard Deift-Zhou steepest descent machinery and states its constants explicitly, but several load-bearing steps are either internally inconsistent as printed or asserted without proof.","tokens_in":50422,"tokens_out":9238,"duration_ms":88890,"significance":"If correct, this would be a substantial extension of the rigorous soliton-gas asymptotics of Girotti, Grava, Jenkins and McLaughlin to the short-pulse equation, with explicit theta-function constants and no fitted parameters, and it would handle Jacobi-type endpoint singularities and an interior singularity or jump at η0. The paper deserves credit for assembling the full Deift-Zhou architecture and for giving explicit formulas for the outer parametrix. However, the central derivation is not yet certifiable as written: one phase normalization contradicts the g-function normalization, and the discrete-to-continuum limit, the local behavior estimates, and the small-norm Proposition 3 are asserted rather than proved. The result is plausible, but the manuscript needs substantial revision before the claims can be considered rigorous.","major_comments":[{"comment":"There is an internal factor-of-four inconsistency in the phase. Equation (1.8) defines θ=ξλ+λ−1/4, while Eq. (3.87) states Re(θ)=Re(λ)/4 (ξ+1/|λ|²), which is the real part of θ=(ξλ+λ−1)/4. From (3.89)–(3.91), Q∼ξy⁴ and R∼y², hence p∼ξλ/4 at infinity. With the printed definition (1.8), g=θ−p behaves like 3ξλ/4, contradicting the required normalization g=O(λ−1) in RH Problem 5 and invalidating the conjugation T=Y e^{t g σ3} used in Eq. (6.161). With the corrected phase θ=(ξλ+λ−1)/4, the g-function behaves like 1/(4λ), and the sign charts and critical value −η2^{-2} become consistent. This is not cosmetic: θ enters the residue conditions, the jump conditions, and the constants Δα and Δ1 in Eqs. (1.36) and (1.43). As printed, the theorem statements and the Riemann-Hilbert problem analyzed in Section 6 are not the same problem. The phase definition must be corrected and all subsequent constants involving θ rechecked.","section":"§1.2, Eq. (1.8); §3, Eqs. (3.87)–(3.91); RH Problem 5"},{"comment":"The passage from the discrete N-soliton RH problem to the continuum soliton-gas RH problem is foundational for the paper, but the convergence of the Riemann sums in (1.13) to the Cauchy integrals in (1.14) for β1,β2,β0∈(−1,0) is only asserted in one sentence. The text states that the proof relies on basic calculus, monotonicity and uniform continuity, but it does not provide the proof. This is load-bearing: if this limit fails, the matrix M∞ defined by RH Problem 3 and all subsequent asymptotic results are not grounded. A complete proof is needed, including control near the singular points η1, η0 and η2, where the equally spaced discrete points approach the singularities.","section":"§1.1, Eqs. (1.13)–(1.14)"},{"comment":"The local behavior of Y near ±η1, ±η2 and ±η0 is stated without derivation, even though it is used to make the RH problem well-posed and to match the parametrices. Section 6 explicitly says that the detailed examination of the local behaviors of T, S and E is omitted 'for the sake of clarity and simplicity.' These local estimates determine the admissible singularities and the matching conditions that produce the O(1/t) error terms in Propositions 4–6; they are not merely exposition. The estimates in (4.100)–(4.104) should be proved, or a precise reference with verified hypotheses should be supplied.","section":"§4, Eqs. (4.100)–(4.104); §6, first paragraph"},{"comment":"The small-norm estimate for the error matrix E is stated without proof: the proof is described as following 'standard procedures' and is omitted. This estimate is the mechanism behind the O(|x̂|^{-1}) term in Theorem 1, so it cannot be treated as routine in a paper whose title and abstract claim rigorous asymptotics. The authors should either provide a proof or state explicitly which standard theorem is being invoked and verify its hypotheses in this setting. The analogous estimates in Propositions 4–6 also depend on the omitted local behavior analysis.","section":"§5.3, Proposition 3"}],"minor_comments":[{"comment":"The sentence containing 'multi-solitons has found been been found' is garbled and should be corrected by proofreading.","section":"§1.1, first paragraph"},{"comment":"The caption of Figure 1 appears to swap the names of the Airy and modified Bessel parametrices: the left panel is described as 'Airy parametrix M_mB' and the right as 'first type of modified Bessel parametrix M_Ai'.","section":"§2, Figure 1"},{"comment":"In Theorem 2, the quantities Φ_1 and Φ̃_1 in (1.32)–(1.33) should presumably be indexed by α rather than 1, in order to match Ψα, Φα and the α-dependent modulus mα.","section":"§1.2, Eqs. (1.32)–(1.33)"},{"comment":"The error term in (6.203) is written as O(|x̂|^{-1}), while Theorem 2 states O(1/t). Since ξ is fixed away from zero in the regions considered, the two are equivalent, but the notation should be harmonized.","section":"§6.3, Eq. (6.203)"},{"comment":"In Eqs. (5.155) and (6.203), the statement that 'a straightforward calculation' yields the theta-function formulas in Theorems 1 and 2 hides a substantial amount of algebra. At least the main intermediate steps should be shown, since the paper's value depends on the correctness of these explicit constants.","section":"§5.4 and §6.3, final computations"}],"recommendation":"major_revision","confidential_remarks":"The factor-of-four phase inconsistency identified in the stress test is real and affects the theorem statements; however, it appears fixable by correcting (1.8) and rechecking the constants, rather than being an unfixable structural error. The paper would also be significantly strengthened by supplying the omitted discrete-to-continuum convergence proof and the local behavior estimates, since these are essential for the claimed rigor. I do not see a citation or attribution problem; the main issue is internal consistency and completeness."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious paper on a real open problem, and the stress-test note is right: there is a load-bearing phase inconsistency. Equation (1.8) defines θ = ξλ + λ^{-1}/4 with ξ = 4x̂/t, while (3.87) states Re(θ) = Re(λ)(ξ + |λ|^{-2})/4, which is the real part of θ = (ξλ + λ^{-1})/4. The g-function construction in §3 makes clear which is intended: from (3.89)–(3.91), Q(y) ~ ξ y^4 and R(y) ~ y^2, so p(λ) ~ ξλ/4 at infinity. If θ is as printed in (1.8), then g = θ − p ~ 3ξλ/4, violating the g = O(λ^{-1}) normalization in RH Problem 5 and breaking the conjugation T = Y e^{t g σ3} in §6. If instead θ carries the factor 1/4, then g ~ (4λ)^{-1} and the sign charts, jumps, and constants in Theorem 2 become consistent. So the paper as written does not verify its central claim; the theorem statements correspond to a different RH problem than the one actually written down.\n\nWhat is genuinely good: the piecewise g-function to handle the origin singularity, the second-type Bessel and confluent hypergeometric parametrices at the interior point η0, and the explicit theta-function formulas for two generalized reflection coefficients. This is a real extension of the KdV/mKdV soliton gas results, and the paper is carefully organized with explicit constants and legitimate reliance on prior machinery. The remaining soft spots are more standard for the genre: Proposition 3's proof is omitted, the long-time local behavior analysis is skipped, and the convergence of Riemann sums to Cauchy integrals for β_j ∈ (−1,0) is asserted rather than proved. Those are gaps a referee can ask to fill; the factor-of-4 is different because without it the steepest-descent argument is not grounded.\n\nMy sense: this is likely fixable, probably a typo in (1.8), but the authors need to correct the phase and re-check all constants and theorem statements. The paper deserves a serious referee, and I would not desk-reject it. As it stands, I would not cite it.","headline":"Serious paper on a real open problem, but a load-bearing phase inconsistency in (1.8) undercuts the central steepest-descent argument as written.","tokens_in":50895,"tokens_out":3786,"would_cite":false,"duration_ms":33176,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q15","35Q51","35P25","37K40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For generalized reflection coefficients with endpoint and interior singularities, the short-pulse soliton gas converges as t grows to an explicit theta-function profile with O(1/t) error.","keywords":["short-pulse equation","soliton gas","Riemann–Hilbert problem","steepest descent method","long-time asymptotics","theta functions","generalized reflection coefficients","g-function"],"falsifier":"Take $\\beta_0=-\\frac12$, $\\eta_1=1$, $\\eta_0=2$, $\\eta_2=3$, and $r(s)=(s-1)^{-1/2}(3-s)^{-1/2}|s-2|^{-1/2}$, then evaluate numerically the difference between the Riemann sums in (1.13) and the improper integral in (1.14) as $N_1,N_2\\to\\infty$; if the difference does not tend to zero, the continuum soliton-gas problem and Theorem 2 are not grounded.","tokens_in":49821,"feed_emoji":"🌊","tokens_out":10488,"duration_ms":86969,"temperature":0.7,"pith_summary":"This paper sets out to prove rigorous large-space and long-time asymptotics for soliton gases of the short-pulse equation $u_{xt}=u+\\tfrac16(u^3)_{xx}$. The gas is described by a continuum Riemann–Hilbert problem obtained as the $N\\to\\infty$ limit of $N$-soliton problems, with two generalized reflection coefficients that vanish or blow up at the endpoints $\\eta_1,\\eta_2$ and at an interior point $\\eta_0$. The central result is that, as $t\\to+\\infty$ in the region $\\xi\\in(\\xi_{\\mathrm{crit}},-\\eta_2^{-2})$ with $\\xi=4\\hat{x}/t$, the solution satisfies $u(x,t)=e^{\\Delta_\\alpha}(\\Psi_\\alpha\\tilde\\Phi_\\alpha+\\Phi_\\alpha\\tilde\\Psi_\\alpha)\\Theta_\\alpha+O(1/t)$, where every constant is an explicit function of the spectral data; a companion theorem gives the large-$\\hat{x}$ behavior of the initial value. A sympathetic reader should care because this turns a many-soliton statistical object into a deterministic oscillatory waveform governed by elliptic $\\theta$ functions, matching the level of description previously available only for other integrable soliton gases.","feed_headline":"Soliton gas asymptotics reduced to theta functions plus O(1/t)","feed_subtitle":"Long-time formulas for the short-pulse soliton gas give explicit theta-function waveforms with an O(1/t) error.","key_machinery":"The engine of the proof is a nonlinear steepest descent analysis of the continuum Riemann–Hilbert problem for $M_\\infty$. The central object is the $g$-function $g=\\theta-p$, defined piecewise so that the singularity of the phase $\\theta=\\xi\\lambda+\\lambda^{-1}/4$ at $\\lambda=0$ is controlled and the jump matrices become exponentially small away from the spectral interval; the parameter $\\alpha$ is fixed by the equation $\\xi=-\\eta_2^{-2}W(\\alpha/\\eta_2)$ involving complete elliptic integrals. Local parametrices then match the singular behavior near the endpoints and the interior point: the Airy parametrix and the first modified Bessel parametrix at $\\eta_1,\\eta_2$, the second modified Bessel parametrix at $\\eta_0$ for $r_0$, and a confluent hypergeometric parametrix for $r_c$. The outer parametrix is written as an explicit quotient of $\\theta$ functions $\\vartheta_3$, with phases and normalization constants built from the spectral data.","core_discovery":"On the paper's own terms, the discovery is that the short-pulse soliton gas has ordered long-time states of $\\theta$-function form. Precisely, Theorem 2 states that for the reflection coefficients $r_0(\\lambda)=(\\lambda-\\eta_1)^{\\beta_1}(\\eta_2-\\lambda)^{\\beta_2}|\\lambda-\\eta_0|^{\\beta_0}\\gamma(\\lambda)$ and $r_c(\\lambda)=(\\lambda-\\eta_1)^{\\beta_1}(\\eta_2-\\lambda)^{\\beta_2}\\chi_c(\\lambda)\\gamma(\\lambda)$, with $\\beta_j>-1$, $c>0$, $c\\neq1$, and appropriate sign of $\\xi$, the gas profile is $u(x,t)=e^{\\Delta_\\alpha}(\\Psi_\\alpha\\tilde\\Phi_\\alpha+\\Phi_\\alpha\\tilde\\Psi_\\alpha)\\Theta_\\alpha+O(1/t)$, with the $\\Psi,\\tilde\\Psi,\\Phi,\\tilde\\Phi,\\Theta$ built from the elliptic $\\theta$ function $\\vartheta_3$ and phases $\\Delta_\\alpha$ depending linearly on $\\hat{x}$ and $t$. The same analysis yields exponential decay for $\\xi>-\\eta_2^{-2}$ and a $\\theta$-formula for the initial value as $\\hat{x}\\to-\\infty$. In other words, the paper claims that the gas is not a featureless sea but a quasi-periodic coherent structure whose parameters are computable from the reflection coefficient.","pith_inferences":["Editorial: because every constant in the theta formula is explicit in the spectral data, the result can be checked numerically at finite $N$ by solving the $N$-soliton Riemann–Hilbert problem and comparing $u_N(x,t)$ with the right-hand side of (1.29); the $O(1/t)$ rate is the observable prediction.","Editorial: the piecewise $g$-function construction, designed to control the origin singularity of the phase, should transfer to other negative flows of the WKI hierarchy whose phases share the same $\\lambda^{-1}$ structure.","Editorial: a natural next step is the interaction of a short-pulse soliton gas with a single large soliton; the local parametrices built here are exactly the ingredients such a two-reflection-coefficient analysis would require."],"forward_implications":["For $\\xi>-\\eta_2^{-2}$, the soliton gas decays exponentially in $t$, so no oscillatory asymptotic state forms in that sector.","In the sectors $\\xi\\in(\\xi_{\\mathrm{crit}},\\xi_0)$, $\\xi\\in(\\xi_0,-\\eta_2^{-2})$, and $\\xi<\\xi_{\\mathrm{crit}}$, the leading waveform is an explicit quotient of theta functions with error $O(1/t)$.","The initial profile $u(x,0)$ has the same theta-function form as $\\hat{x}\\to-\\infty$ with error $O(1/|\\hat{x}|)$ and decays exponentially as $\\hat{x}\\to+\\infty$.","For $r_0$ with $\\beta_0=0$, the long-time formula holds on the full interval $(\\xi_{\\mathrm{crit}},-\\eta_2^{-2})$; for nonzero $\\beta_0$, the admissible ranges of the exponents differ from sector to sector.","The method extends to any finite number of interior singularities $\\eta_{0,j}$, with a modified Bessel parametrix for $r_0$ and a confluent hypergeometric parametrix for $r_c$ around each singularity, as stated in Remark 1."],"supporting_citations":[{"why":"Provides the Riemann–Hilbert formulation of the short-pulse equation and the recovery formulas for $u(x,t)$.","marker":"[16]"},{"why":"Supplies the interpolation trick that converts the $N$-soliton poles into jump conditions.","marker":"[26]"},{"why":"Supplies the theta-function outer parametrix for Riemann–Hilbert problems with exponential weights.","marker":"[27]"},{"why":"Supplies the strong-asymptotics steepest descent framework and the Airy parametrix.","marker":"[28]"},{"why":"Supplies the nonlinear steepest descent method for oscillatory Riemann–Hilbert problems.","marker":"[31]"},{"why":"Supplies the $g$-function technique for the zero-dispersion KdV limit that the paper adapts.","marker":"[30]"},{"why":"Gives the KdV soliton-gas Riemann–Hilbert model and the norming-constant discretization that the paper extends.","marker":"[50]"},{"why":"Gives the mKdV soliton-gas asymptotic analysis and the vanishing-lemma argument used for $M_\\infty$.","marker":"[51]"},{"why":"Gives the confluent-hypergeometric parametrix for jump-type singularities used at $\\eta_0$ for $r_c$.","marker":"[58]"},{"why":"Gives the modified-Bessel parametrix for endpoint singularities used at $\\eta_1,\\eta_2$.","marker":"[64]"}],"fun_headline_variants":["Theta functions tame soliton gas asymptotics","Soliton gas long-time limit: explicit theta-function waveforms","Rigorous asymptotics: SP soliton gas becomes theta-function quasi-periodic","Short-pulse soliton gas: theta waveforms emerge at large times","Soliton gas order emerges: theta functions govern long-time state"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, when any exponent $\\beta_j$ lies between $-1$ and $0$, the discrete Riemann sums in (1.13) converge to the improper Cauchy integrals in (1.14); the paper asserts this on the basis of calculus and uniform continuity but does not carry out the proof.","fun_headline_variants_meta":{"raw":{"variants":["Theta functions tame soliton gas asymptotics","Soliton gas long-time limit: explicit theta-function waveforms","Rigorous asymptotics: SP soliton gas becomes theta-function quasi-periodic","Short-pulse soliton gas: theta waveforms emerge at large times","Soliton gas order emerges: theta functions govern long-time state"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000846,"raw_usage":{"total_tokens":3873,"prompt_tokens":1329,"completion_tokens":2544,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":945,"completion_tokens_details":{"reasoning_tokens":2456}},"tokens_in":945,"tokens_out":2544,"duration_ms":15891,"temperature":1.0,"reasoning_tokens":2456,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T12:44:08.600884+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $\\beta_0=-\\frac12$, $\\eta_1=1$, $\\eta_0=2$, $\\eta_2=3$, and $r(s)=(s-1)^{-1/2}(3-s)^{-1/2}|s-2|^{-1/2}$, then evaluate numerically the difference between the Riemann sums in (1.13) and the improper integral in (1.14) as $N_1,N_2\\to\\infty$; if the difference does not tend to zero, the continuum soliton-gas problem and Theorem 2 are not grounded.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the interpolation trick that converts the $N$-soliton poles into jump conditions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theta-function outer parametrix for Riemann–Hilbert problems with exponential weights."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the strong-asymptotics steepest descent framework and the Airy parametrix."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the nonlinear steepest descent method for oscillatory Riemann–Hilbert problems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the $g$-function technique for the zero-dispersion KdV limit that the paper adapts."},{"cited_title":"Girotti, T","cited_arxiv_id":null,"evidence_quote":"Gives the KdV soliton-gas Riemann–Hilbert model and the norming-constant discretization that the paper extends."},{"cited_title":"Girotti, T","cited_arxiv_id":null,"evidence_quote":"Gives the mKdV soliton-gas asymptotic analysis and the vanishing-lemma argument used for $M_\\infty$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the confluent-hypergeometric parametrix for jump-type singularities used at $\\eta_0$ for $r_c$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the modified-Bessel parametrix for endpoint singularities used at $\\eta_1,\\eta_2$."}],"review_version":1}