{"id":"177b9630-85c5-4ce9-9fb8-ed52920dddd6","arxiv_id":"2412.09390","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every dilation set E in [1,2], the complete L^p to L^q mapping range of the spherical maximal operator on radial functions is computed: a Minkowski-dimension triangle for d at least 3, and a ν♯-spectrum region for d = 2.","lead":"This paper determines exactly when the spherical maximal operator, whose allowed dilation lengths lie in a fractal set E, maps radial functions from L^p to L^q. In three or more dimensions the answer is a simple triangle governed by the Minkowski dimension of E; in the plane it depends on a finer fractal spectrum of E.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 overclaims exact equality at the p=2 endpoint for β=1: for E=[1,2] the formula contains (1/2,1/2), while Lemma 3.4 and Theorem 1.4(v) exclude p=2; the theorem can at best describe the closure.","rationale":"Good-faith reading: the interior results and the necessary/sufficient matching for β<1 appear coherent; the ν♯ description is plausible for the closure of the type set, and my concern is not about the dyadic estimates being unverified. The decisive issue is a single endpoint where the stated equality fails by the paper's own necessary condition. Since the abstract says 'closure', the fix is local: re-state Theorem 1.2 as describing the closure of T^rad_E, amend Corollary 1.3 and the claims of complete endpoint determination for d=2, and check whether any endpoint statements in Theorem 1.4 are meant to fill the gap. This is a load-bearing correction to the advertised central claim as read by the reviewer, but it does not by itself invalidate the β<1 or d≥3 results. The reader's weakest assumption (Lemma 3.5 cap lower bound) is a different necessary condition; I do not see the sharpness of that cap estimate as the main risk. The risk is the missing Lemma 3.4 endpoint condition in Theorem 1.2. Verdict remains CONDITIONAL: acceptance should require the corrected statement and explicit discussion of the p=2, β=1 case.","tokens_in":27835,"tokens_out":15120,"duration_ms":140815,"concrete_test":"Evaluate the theorem at E=[1,2], (1/p,1/q)=(1/2,1/2). Lemma 3.4 gives the necessary condition sup_{0<δ<1/2}(log(1/δ))^{1/2}<∞, which is false, while the formula in Theorem 1.2 contains this point. If the authors intend closure, restate Theorem 1.2 with an overline and check whether the proof for β=1 can be extended to p=2; if not, the exact type-set statement must be withdrawn. A direct analytic check: apply M_{[1,2]} to the radial function f_0(s)=s^{-1}(log(1/s))^{-1}χ_{[0,1/2]}(s), which lies in L^2(rdr), and show the maximal function fails to be in L^2.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central d=2 equality overclaims endpoints. At β=1, Δβ contains the vertical side 1/p=1/2, 1/4≤1/q≤1/2. For E=[1,2] one has ν♯(α)=α, so every point on this side satisfies the displayed ν♯ condition in Theorem 1.2; in particular (1/2,1/2) is claimed to lie in T^rad_E. But Lemma 3.4, with d=2 and p_d=2, requires any L^2_rad→L^q bound to satisfy sup_{0<δ<1/2} N(E,δ)^{1/q}δ^{1/q}[log(1/δ)]^{1/2}<∞. For E=[1,2], N(E,δ)∼δ^{-1}, making this supremum infinite; hence no L^2_rad→L^q bound can hold. This is not a gap in a technical estimate: Theorem 1.4(v) of the same paper states that for β=1 with sup δ log(1/δ)N(E,δ)=∞, boundedness holds iff p>2 and p≤q≤2p. Thus Theorem 1.2 is internally inconsistent at the p=2 boundary. The proof in §5.4 for β=1 only establishes the interior p>2 and supplies no argument at p=2. The abstract's phrase 'closure' suggests the intended statement is the closure of T^rad_E, but the theorem and the reader's interpretation assert exact equality including endpoints.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the spherical maximal operator with dilations restricted to a set E⊂[1,2], acting on radial functions in R^d. For d≥3, Theorem 1.1 asserts that the radial type set T^rad_E is exactly the triangle Δ_β determined by the upper Minkowski dimension β of E, with a side removed when sup δ^β N(E,δ) is infinite and with a refined boundary description when β=1. For d=2, Theorem 1.2 claims a complete description of T^rad_E in terms of a new dimensional spectrum ν♯(α) defined from local covering numbers of E. Corollary 1.3 and Theorem 1.4 give consequences involving the quasi-Assouad and Assouad dimensions. The proof combines necessary conditions from test-function constructions (Lemma 3.4, Lemma 3.5) with sufficient estimates for the operators R^±_1, R^±_2, M^±_p in Propositions 5.2–5.5, following the reduction of Lemma 5.1.","tokens_in":28067,"tokens_out":3087,"duration_ms":29182,"significance":"If the stated theorems are correct, the paper would completely determine the radial L^p→L^q type set for every dilation set E in every dimension, including endpoints, and would introduce a new dimensional quantity ν♯ that appears naturally from the operator estimates. The logical architecture is careful: the necessary lower bound of Lemma 3.5 and the sufficient estimates of Propositions 5.4–5.5 are matched at the same boundary, and the quantity ν♯ is defined directly from covering numbers rather than fitted to operator bounds. The paper also gives a sharpness discussion for Corollary 1.3. However, the main theorem as stated overclaims the p=2 endpoint when β=1, a point that is internally inconsistent with Lemma 3.4 and Theorem 1.4(v). This issue is load-bearing and must be resolved before the claims can be accepted as stated.","major_comments":[{"comment":"Theorem 1.2 is false as stated at the p=2 endpoint for β=1. For E=[1,2], an elementary computation gives ν♯(α)=α, so every point on the vertical side 1/p=1/2, 1/4≤1/q≤1/2 satisfies the displayed inequality in Theorem 1.2; in particular (1/2,1/2) is claimed to lie in T^rad_E. But Lemma 3.4 with d=2 requires sup_{0<δ<1/2} N(E,δ)^{1/q}δ^{1/q}[log(1/δ)]^{1/2}<∞ for any L^2_rad→L^q bound; for E=[1,2] this supremum is infinite, so no such bound holds. This is not a technical gap: Theorem 1.4(v) of the same paper states that for β=1 with sup δ log(1/δ)N(E,δ)=∞, boundedness holds iff p>2 and p≤q≤2p, which excludes p=2. The theorem can at best describe the closure of T^rad_E, as the abstract itself says; the equality claim must be amended or the endpoint supplied with a genuine argument.","section":"§1, Theorem 1.2; §3, Lemma 3.4; §5.4"},{"comment":"The sufficiency part for β=1 treats only the interior p>2. For R^±_2 the argument uses the strict inequality 1/p−1/2 < (1/q)(1−ν♯(q/2−1)), and for M^±_p it invokes Proposition 5.5(i) which requires p>2. No argument covers the equality case 1/p=1/2. Since the necessary condition of Lemma 3.4 is not reflected in the statement of Theorem 1.2, the proof does not establish the claimed equality at that boundary. The statement should be changed to describe the closure, or an endpoint argument must be added that explains which q (if any) survive at p=2.","section":"§5.4, proof of Theorem 1.2, β=1 case"}],"minor_comments":[{"comment":"The abstract says the paper determines the closure of the L^p→L^q type set in two dimensions, while Theorem 1.2 states an exact equality. This mismatch should be resolved in the revision; if the closure is the correct statement, the theorem and its proof should be adjusted accordingly.","section":"Abstract and §1"},{"comment":"There is a typo in the proof of Proposition 5.5: \"we can assume without loss of generaltiy\" should read \"without loss of generality\".","section":"§5.3"},{"comment":"In the sentence following (1.2), the observation that ν♯(α)=β for α≤0 is stated without proof; while it is a quick exercise from the definition, a one-sentence justification would improve readability.","section":"§1, definition of ν♯"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper with a convincing overall architecture, and the endpoint issue is localized. The necessary lemma (Lemma 3.4) and the stated Theorem 1.4(v) make it clear that the p=2 endpoint is not a matter of a harmless typo: the theorem as printed claims points that are provably not in the type set. The authors should either change the statement to describe the closure (as the abstract already suggests) or find a correct endpoint formulation for β=1. With that fix, the paper would be publishable; in its current form the central two-dimensional theorem is not correct as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a serious piece of work: it resolves the radial spherical maximal type set problem for all dilation sets E in the sense of giving a closed formula (d=2) and a complete triangle (d≥3). The d≥3 part (Theorem 1.1) looks right to me—the necessary and sufficient estimates in Sections 3 and 4 match at the same boundaries, and the case distinctions depending on sup δ^β N(E,δ) are handled carefully. The new ν# quantity is a genuine idea, and the lower bound in Lemma 3.5 is a new necessary condition that correctly captures the Assouad-spectrum effects in two dimensions. The authors are also honest: they state the problem was open, cite prior work fairly, and flag the remaining gaps in Corollary 1.3. All of that is good, and the d≥3 half of the paper deserves to be published.\n\nThe soft spot is real and it is at the p=2 endpoint of Theorem 1.2 when β=1. Take E=[1,2], where ν#(α)=1 for 0≤α≤1. The theorem's right-hand side then contains the whole vertical side 1/p=1/2, 1/4≤1/q≤1/2, including (1/2,1/2). But the paper's own Theorem 1.4(v) says that for β=1 with sup δ log(1/δ)N(E,δ)=∞ (which holds for [1,2]), boundedness holds only for p>2. And Lemma 3.4 already rules out L^2_rad→L^q for E=[1,2]. So Theorem 1.2 is internally inconsistent at that boundary. The proof in §5.4 confirms the problem: for β=1 it only establishes the interior, p>2. The abstract says “closure,” and the theorem should say that too—or the boundary should be handled differently. This is not a minor typo; the stated equality is false for the most important example E=[1,2].\n\nThe other issues are minor: the abstract's “only depends on the upper Minkowski dimension” in higher dimensions oversimplifies the three cases of Theorem 1.1, and Corollary 1.3 is honestly limited to inclusions because part of the (β,γ)-only description remains open.\n\nWho is this for? Harmonic analysts working on maximal operators and anyone interested in how Assouad-type spectra govern operator bounds. The d≥3 theorem is a clean, citable result. The d=2 part needs a careful revision before it can be taken as stated.\n\nRecommendation: send it to peer review, but tell the editor openly that Theorem 1.2 appears to overclaim at p=2 when β=1. With that fixed—likely by stating the closure result and making the endpoint statements precise—this will be a valuable paper.","headline":"A strong paper with a real endpoint flaw: Theorem 1.2 overclaims exact equality at p=2 when β=1, at least for E=[1,2].","tokens_in":862,"tokens_out":986,"would_cite":true,"duration_ms":62271,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B25","28A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper determines, for every dilation set E⊂[1,2] and every dimension d≥2, the complete Lp→Lq mapping picture of the spherical maximal operator on radial functions.","keywords":["spherical maximal operators","radial functions","dilation sets","Minkowski dimension","Assouad spectrum","quasi-Assouad dimension","Lp-Lq estimates","type set"],"falsifier":"Take $d=2$ and an $E$ for which $\\nu^\\sharp$ is known exactly, such as the Assouad-regular examples used in the sharpness discussion, and choose $(1/p,1/q)$ on the boundary $(1/q)\\nu^\\sharp(q/2-1)+1/p-1/q=1/2$. Apply the lower-bound test of Lemma 3.5 to $g_\\delta(x)=\\mathbf{1}_{[t_L-\\delta,t_L+\\delta]}(|x|)$ with $t_L\\in E$ and $\\delta$ comparable to $|t-t_L|$: the theorem requires $|A_{t(x)}g_\\delta(x)|\\gtrsim(\\delta/|x|)^{1/2}$ for radii in the interval $I_t=[t-t_L-\\delta/10,t-t_L+\\delta/10]$. A direct computation showing any additional $\\delta^\\varepsilon$ decay there would push the necessary condition inward and falsify the two-dimensional formula; showing the lower bound is attained supports it.","tokens_in":27485,"feed_emoji":"📐","tokens_out":14965,"duration_ms":126206,"temperature":0.7,"pith_summary":"The paper solves the radial version of a long-standing question: for every set $E\\subset[1,2]$ of allowed dilation radii and every dimension $d\\ge2$, it gives the full list of $L^p\\to L^q$ bounds satisfied by the spherical maximal operator $M_E$ on radial functions. In dimensions $d\\ge3$ the answer is a closed triangle $\\Delta_\\beta$ whose vertices depend only on the upper Minkowski dimension $\\beta$ of $E$, with one side removed exactly when $E$ has too many $\\delta$-covers and with logarithmic endpoint conditions when $\\beta=1$. In two dimensions the answer is the same triangle cut by the curve $(1/q)\\nu^\\sharp(q/2-1)+1/p-1/q\\le1/2$, where $\\nu^\\sharp$ is a local counting exponent built from $N(E\\cap J,\\delta)$ over intervals $J$; this quantity is closely related to the upper Assouad spectrum. A reader should care because this determines the type set $T^{\\rm rad}_E$ completely, including all endpoints, for every dilation set in every dimension, leaving no dependence on finer structure beyond $\\beta$ in $d\\ge3$ and $\\nu^\\sharp$ in $d=2$.","feed_headline":"All radial spherical maximal bounds are now classified","feed_subtitle":"Higher dimensions need only Minkowski dimension; the plane needs a finer Assouad-type counting spectrum.","key_machinery":"The central object is the radial type set $T^{\\rm rad}_E$ together with the covering number $N(E,\\delta)$ and its local refinement $\\nu^\\sharp(\\alpha)=\\limsup_{\\delta\\to0}\\log\\sup_{\\delta\\le|J|\\le1}|J|^{-\\alpha}N(E\\cap J,\\delta)/\\log(1/\\delta)$, an Assouad-type spectrum of local dilation density. The proof route is the reduction of spherical averages on radial functions to one-dimensional weighted integral transforms with kernel $K_t(r,s)$; in $d\\ge3$ the operator is pointwise controlled by $M_p,R_1,R_2$, and in $d=2$ by $M^\\pm_p,R^\\pm_1,R^\\pm_2$. The matching of necessary and sufficient estimates is carried by Lemma 3.5's radial test function $g_\\delta$ supported in a $\\delta$-neighborhood of a dilation $t_L$, where the spherical average is bounded below by $(\\delta/|x|)^{(d-1)/2}$ on appropriate annuli, and by Proposition 5.4's decomposition of $R^\\pm_2$ using a dyadic resolution of the identity; the latter converts the local counting quantity $\\omega^{p,q}_m(E,k)$ into the $\\nu^\\sharp$ boundary.","core_discovery":"On its own terms, the paper's claim is that the radial type set $T^{\\rm rad}_E=\\{(1/p,1/q): M_E:L^p_{\\rm rad}\\to L^q\\text{ is bounded}\\}$ is now known for every $E\\subset[1,2]$ and every $d\\ge2$. For $d\\ge3$, $T^{\\rm rad}_E$ equals the triangle $\\Delta_\\beta$ with vertices $P_1=(0,0)$, $P_{2,\\beta}=((d-1)/(d-1+\\beta),(d-1)/(d-1+\\beta))$, and $P^{\\rm rad}_{3,\\beta}=(d(d-1)/(d^2-1+\\beta),(d-1)/(d^2-1+\\beta))$; when $\\sup_{0<\\delta<1}\\delta^\\beta N(E,\\delta)=\\infty$ the side $[P_{2,\\beta},P^{\\rm rad}_{3,\\beta}]$ is removed, and when $\\beta=1$ the endpoint at $p=d/(d-1)$ is governed by $\\sup_{\\delta<1/2}\\delta(\\log 1/\\delta)^{q/d}N(E,\\delta)<\\infty$. For $d=2$, $T^{\\rm rad}_E=\\Delta_\\beta\\cap\\{(1/p,1/q):(1/q)\\nu^\\sharp(q/2-1)+1/p-1/q\\le1/2\\}$. The boundary is made concrete by Corollary 1.3 and Theorem 1.4: if $2\\gamma-\\beta\\le1$ with $\\gamma$ the quasi-Assouad dimension of $E$, the whole triangle is the type set; if $2\\gamma-\\beta>1$, the quadrangle $Q^{\\rm rad}_{\\beta,\\gamma}$ is contained, and there are sets $E$ for which this inclusion is an equality.","pith_inferences":["A natural testable extension is to Lorentz spaces: the proof of Proposition 5.4 has room in its exponents, so the same $\\nu^\\sharp$ boundary likely controls $L^{p,r}\\to L^{q,s}$ refinements of the type set.","The two-dimensional theorem effectively identifies a Legendre-type transform of the Assouad spectrum as the operative invariant; similar formulas could be expected for other averaging operators, such as variable-coefficient circular means, with $N(E\\cap J,\\delta)$ replaced by an appropriate wave-packet count.","One could isolate the sharpness of Lemma 3.5 by computing $\\nu^\\sharp$ for a concrete Moran or self-similar set and checking the boundary pair on the test family $g_\\delta$; this would separate the lower-bound mechanism from the sufficiency arguments."],"forward_implications":["In dimensions $d\\ge3$ with $\\beta<1$, the side $[P_{2,\\beta},P^{\\rm rad}_{3,\\beta}]$ is bounded exactly when $\\sup_\\delta\\delta^\\beta N(E,\\delta)<\\infty$; when this supremum is infinite, that whole side is removed from the type set.","For $\\beta=1$ in $d\\ge3$, the endpoint $p=d/(d-1)$ is bounded into $L^q$ precisely for those $q$ satisfying $\\sup_{\\delta<1/2}\\delta(\\log 1/\\delta)^{q/d}N(E,\\delta)<\\infty$, so the endpoint condition depends on $q$ logarithmically.","In two dimensions, if the quasi-Assouad dimension $\\gamma$ satisfies $2\\gamma-\\beta\\le1$, the radial type set is exactly $\\Delta_\\beta$; if $2\\gamma-\\beta>1$, the quadrangle $Q^{\\rm rad}_{\\beta,\\gamma}$ is a sharp lower bound for some $E$.","For self-similar dilation sets, where $\\beta=\\gamma<1$, the theorem gives $T^{\\rm rad}_E=\\Delta_\\beta$, so the Minkowski dimension alone fixes all radial $L^p\\to L^q$ bounds.","The necessary condition of Lemma 3.5 shows no $L^p_{\\rm rad}\\to L^q$ bound can hold outside the $\\nu^\\sharp$-cut triangle in $d=2$, so the formula in Theorem 1.2 is the largest possible type set."],"supporting_citations":[{"why":"Supplies the integral representation (4.1)-(4.2) of spherical averages on radial functions, the starting point for the pointwise reductions.","marker":"[9]"},{"why":"Supplies the pointwise reduction Lemma 4.1 for d≥3 and Lemma 5.1 for d=2, as well as Lemma 2.7 used in the endpoint necessary condition.","marker":"[20]"},{"why":"Introduces the upper Assouad spectrum, the dimensional quantity behind the two-dimensional statement and Corollary 1.3.","marker":"[5]"},{"why":"Provides the (β,γ)-Assouad regular sets and the outer quadrangle Q(β,β) used to show sharpness of the inclusions and to frame the radial result against known full type sets.","marker":"[13]"},{"why":"Contains the standard necessary conditions of Lemma 3.2 (scaling and covering lower bounds) that force T^rad_E⊆Δ_β.","marker":"[1]"},{"why":"Supplies the L^{d/(d-1)} endpoint example with logarithmic divergence used in Lemma 3.4 for the β=1 endpoint.","marker":"[21]"},{"why":"Gives the single-sphere L^p→L^q bounds used in Theorem 1.4(ii) when β=0.","marker":"[10]"}],"fun_headline_variants":["Radial spherical maximal bounds fully classified","Minkowski and Assouad spectra settle radial maximal maps","Spherical maximal operators: radial type set known for all E","Higher-dim radial bounds from Minkowski, plane from Assouad","Radial maximal type set determined for every dilation set"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the cap computation in Lemma 3.5 gives the exact order of magnitude of the spherical average on the radial test function, namely $|A_{t(x)}g_\\delta(x)|\\gtrsim(\\delta/|x|)^{(d-1)/2}$ on the annuli where $|x|$ lies in $I_t$; if this lower bound were weaker by any power of $\\delta$, or if the pointwise reduction Lemma 5.1 failed to capture $M_E$ on radial functions, the boundary curve in Theorem 1.2 would shift and the equality would fail.","fun_headline_variants_meta":{"raw":{"variants":["Radial spherical maximal bounds fully classified","Minkowski and Assouad spectra settle radial maximal maps","Spherical maximal operators: radial type set known for all E","Higher-dim radial bounds from Minkowski, plane from Assouad","Radial maximal type set determined for every dilation set"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1435,"prompt_tokens":1030,"completion_tokens":405,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":646,"completion_tokens_details":{"reasoning_tokens":324}},"tokens_in":646,"tokens_out":405,"duration_ms":4070,"temperature":1.0,"reasoning_tokens":324,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T17:08:13.912791+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $d=2$ and an $E$ for which $\\nu^\\sharp$ is known exactly, such as the Assouad-regular examples used in the sharpness discussion, and choose $(1/p,1/q)$ on the boundary $(1/q)\\nu^\\sharp(q/2-1)+1/p-1/q=1/2$. Apply the lower-bound test of Lemma 3.5 to $g_\\delta(x)=\\mathbf{1}_{[t_L-\\delta,t_L+\\delta]}(|x|)$ with $t_L\\in E$ and $\\delta$ comparable to $|t-t_L|$: the theorem requires $|A_{t(x)}g_\\delta(x)|\\gtrsim(\\delta/|x|)^{1/2}$ for radii in the interval $I_t=[t-t_L-\\delta/10,t-t_L+\\delta/10]$. A direct computation showing any additional $\\delta^\\varepsilon$ decay there would push the necessary condition inward and falsify the two-dimensional formula; showing the lower bound is attained supports it.","supporting_citations":[{"cited_title":"Fraser, Kathryn E","cited_arxiv_id":null,"evidence_quote":"Introduces the upper Assouad spectrum, the dimensional quantity behind the two-dimensional statement and Corollary 1.3."},{"cited_title":"Leckband","cited_arxiv_id":null,"evidence_quote":"Supplies the integral representation (4.1)-(4.2) of spherical averages on radial functions, the starting point for the pointwise reductions."},{"cited_title":"Sph erical maximal operators on radial functions","cited_arxiv_id":null,"evidence_quote":"Supplies the pointwise reduction Lemma 4.1 for d≥3 and Lemma 5.1 for d=2, as well as Lemma 2.7 used in the endpoint necessary condition."},{"cited_title":"Spherical maximal funct ions and fractal dimensions of dilation sets","cited_arxiv_id":null,"evidence_quote":"Provides the (β,γ)-Assouad regular sets and the outer quadrangle Q(β,β) used to show sharpness of the inclusions and to frame the radial result against known full type sets."},{"cited_title":"Lp → Lq bounds for spherical maximal operators","cited_arxiv_id":null,"evidence_quote":"Contains the standard necessary conditions of Lemma 3.2 (scaling and covering lower bounds) that force T^rad_E⊆Δ_β."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the L^{d/(d-1)} endpoint example with logarithmic divergence used in Lemma 3.4 for the β=1 endpoint."},{"cited_title":"Lp − Lq-estimates for singular integral operators arising from hy perbolic equations","cited_arxiv_id":null,"evidence_quote":"Gives the single-sphere L^p→L^q bounds used in Theorem 1.4(ii) when β=0."}],"review_version":1}