{"id":"f16ea6d5-7b66-4183-bea1-90d28a390264","arxiv_id":"2412.03868","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"If two active scalar equations with fractional dissipation produce the same observations on a small open set, their nonlocal drift operators must agree on the whole exterior, according to the paper's main theorem.","lead":"An inverse problem for fluid-like active scalar equations with fractional dissipation is studied: knowing the solution in a small observation window is claimed to determine the nonlocal drift operator everywhere outside that window. The proof combines second-order linearization with unique continuation for the fractional Laplacian, but a key algebraic step in the linearization does not close.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.3 is not proven as written: the four-term decomposition in Eq. (4.10) is algebraically false, and the asserted convergence of each S_epsilon,j to zero fails, so the second-order linearization underlying Theorem 1.1 is unsupported.","rationale":"The reader's weakest-assumption analysis and my independent reading converge on the same load-bearing defect: Proposition 4.3, which is essential to Theorem 1.1, rests on the Eq. (4.10) decomposition of S_epsilon, and that decomposition does not hold algebraically. I checked the expansion with A, B, C as the three relevant solutions; the four terms do not sum to S_epsilon, and the individual terms do not vanish in the ε→0 limit. The leftover leading-order expression is a genuine obstruction to the proof as written, not merely a missing regularity argument or a consensus disagreement. The surrounding framework — the fractional unique continuation, the Runge approximation, and the exterior recovery argument in Section 6 — appears structurally coherent, and the theorem may be true; however, the submitted manuscript does not contain a valid proof of the central linearization result. Because the reader already reached REJECT and my analysis supports that outcome, I recommend no change to the verdict.","tokens_in":10086,"tokens_out":8642,"duration_ms":80553,"concrete_test":"Recompute Proposition 4.3 by substituting A = θ_{ε(f1+f2)}, B = θ_{εf1}, C = θ_{εf2} into Eq. (4.10) and expanding to order ε². Concretely, test the leading limit of the four-term sum using A = ε(w1+w2) + O(ε²), B = εw1 + O(ε²), C = εw2 + O(ε²); if the limit is N(w1,w1) + N(w2,w2) - N(w1,w2) - N(w2,w1) rather than 0, the displayed decomposition is false and the Proposition 4.3 proof is invalid. If a corrected polarization identity does make the limit zero, then check whether the remainder of Section 6 goes through unchanged.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.3 is the only bridge from the second-order linearization to the identity (R1-R2)w1·∇w2 + (R1-R2)w2·∇w1 = 0 used in Section 6 to recover R on W^e. Its proof asserts S_epsilon = S_epsilon,1 + S_epsilon,2 + S_epsilon,3 + S_epsilon,4 with the displayed definitions, and then asserts each S_epsilon,j converges to 0 in L2(0,T;H^{-α}). Direct expansion with A = θ_{ε(f1+f2)}, B = θ_{εf1}, C = θ_{εf2} and N(U,V) = R(U)·∇V gives the numerator of the four-term sum as [N(A,A) - N(A,B) + N(A,C) + N(B,B) - N(C,C) - 2N(B,C)]/ε², whereas S_epsilon has numerator [N(A,A) - N(B,B) - N(C,C)]/ε². The difference is nonzero. At leading order B ≈ εw1, C ≈ εw2, A ≈ ε(w1+w2), so S_epsilon,1 → N(w2,w2) - N(w2,w1) and S_epsilon,2 → N(w1,w1) - N(w1,w2), while S_epsilon,3 and S_epsilon,4 tend to 0; hence the claim that each S_epsilon,j goes to 0 is false. The residual N(w1,w1) + N(w2,w2) - N(w1,w2) - N(w2,w1) does not vanish for generic w1,w2. Since Proposition 4.3 is the step that produces v(1) = v(2) and thus the determining identity, Theorem 1.1 is not established by the submitted proof. This is an internal algebraic inconsistency, not a disagreement with the surrounding consensus; the proposition may be repairable by a correct polarization identity, but that repair is not present in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies an inverse problem for the dissipative active scalar equation ∂tθ + Rθ·∇θ + (−∆)αθ = f on the torus T^2, with α ∈ (1/2,1) and with drift velocity u = Rθ given by a divergence-free Fourier multiplier of order −1 (for example, the SQG Riesz-vector case). The main result, Theorem 1.1, asserts that if the source-to-solution maps LR1 and LR2, which record both θ and u in an arbitrarily small observation window W × (0,T), agree for all sources supported in W, then R1g = R2g on the exterior We = T^2 \\ W̄ for all g ∈ Cc∞(We). The proof combines forward well-posedness, first- and second-order linearizations, a unique continuation property of the fractional Laplacian, a Runge approximation property for the linear fractional heat equation, and a final convolution-kernel argument. The paper is clearly organized and the overall strategy is standard for the fractional Calderón program, but the key second-order linearization proposition is not proven correctly as written.","tokens_in":10441,"tokens_out":32641,"duration_ms":309178,"significance":"If the main theorem is correct, it is a meaningful extension of the fractional Calderón methodology to nonlinear active scalar equations, and it is unusual in that the recovered object is the nonlocal drift operator R rather than a local coefficient. The paper is well written, the well-posedness result is quoted from the SQG literature with a proof sketch, and the UCP/Runge ingredients are standard and used naturally. There are no fitted parameters and no circular assumptions. The obstruction is the proof of Proposition 4.3, which is the sole bridge from the second-order linearization to the determining identity used in Section 6; as submitted, that proof contains a false algebraic decomposition and an unjustified convergence claim. The central idea is defensible, but the manuscript needs a corrected proof of Proposition 4.3 before the main theorem can be accepted.","major_comments":[{"comment":"The identity Sε = Sε,1 + Sε,2 + Sε,3 + Sε,4 is algebraically false. Write A = θε(f1+f2), B = θεf1, C = θεf2, and N(U,V) = R(U)·∇V. The sum of the four numerators is [N(A,A) − N(A,B) + N(B,B) + N(A,C) − N(C,C) − 2N(B,C)]/ε², whereas the numerator of Sε is [N(A,A) − N(B,B) − N(C,C)]/ε². The difference does not vanish. Using the first-order asymptotics A ≈ ε(w1+w2), B ≈ εw1, C ≈ εw2, the residual is N(w1,w1) + N(w2,w2) − N(w1,w2) − N(w2,w1), which is nonzero for generic w1,w2. In particular, Sε,1 tends to N(w2,w2) − N(w2,w1) and Sε,2 tends to N(w1,w1) − N(w1,w2), so the assertion that each Sε,j converges to 0 in L2(0,T;H−α) is false. Since Proposition 4.3 is the only step that produces v(1) = v(2) and hence the identity (R1−R2)w1·∇w2 + (R1−R2)w2·∇w1 = 0 used in Section 6, Theorem 1.1 is not established by the submitted proof. The proposition is likely repairable by a standard two-term expansion θεf = εw + ε²q + o(ε²), but a correct proof must be supplied.","section":"§4, Proposition 4.3, Eq. (4.9) and the displayed decomposition after it"},{"comment":"Even apart from the algebraic error, the asserted convergence Sε,j → 0 in L2(0,T;H−α) is not justified by the displayed embedding argument. For example, Sε,1 is the product of R(θε(f1+f2)−θεf1)/ε and ∇(θε(f1+f2)−θεf1)/ε; by Proposition 4.2 both factors converge in H2α−1, not one in Hα. The embedding chain H2α−1 ↪ L1/(1−α) ↪ L1/α places each factor in L1/α, but the multiplier statement for L1/α does not control the product of two such factors. A different product estimate is needed even for a corrected decomposition.","section":"§4, proof of Proposition 4.3"}],"minor_comments":[{"comment":"After deriving v = 0 in W × (0,T), the text says 'so (−∆)αv = 0 in W × (0,T)'; this uses the fact that ∂tv = 0 in W as well, which follows from v = 0 but is not stated explicitly.","section":"§5, proof of Proposition 5.2"},{"comment":"The passage from the time-integrated identity to the pointwise identity 'in particular' would be clearer if it explicitly used arbitrary time cutoffs χ(t) to remove the time integral.","section":"§6, proof of Theorem 1.1"},{"comment":"The notation W e = T2 \\ W̄ is introduced in the introduction, but later the text sometimes writes T2 \\ W; standardizing this notation would avoid ambiguity about whether the boundary of W is included.","section":"§4 and §6, notation"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The stress-test concern about Proposition 4.3 is valid and is the main issue. The four-term decomposition in the proof is arithmetically incorrect, and the asserted convergence of the remainder terms is not supported. I recommend major revision rather than rejection because the statement of Proposition 4.3 is a standard second-order linearization and the overall strategy (UCP + Runge approximation + convolution argument) appears sound; the authors should be asked to provide a complete, correct proof of Proposition 4.3 and to re-verify the estimates in Section 6 that depend on it. If a corrected proof cannot be supplied, the manuscript should not be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is the first Calderón-style inverse problem I know of for active scalar equations with fractional dissipation, and the unknown is the nonlocal drift operator R rather than a local coefficient — a real novelty worth attention. Second, the central proof step, Proposition 4.3, is wrong as written; the algebra does not close, so Theorem 1.1 is not established in the submitted version.\n\nThe paper has real merits. The problem is well chosen (it includes SQG), the well-posedness and Runge/unique continuation machinery are quoted from the literature and look standard, and the final argument recovering R on W^e from the bilinear identity is clean. The authors are also honest about the limitation to the exterior.\n\nThe soft spot is load-bearing. In the proof of Prop 4.3, Eq. (4.10) claims S_epsilon is the sum of four terms. Direct expansion with A = θ_{ε(f1+f2)}, B = θ_{εf1}, C = θ_{εf2}, and N(U,V) = R(U)·∇V shows the sum contains extra terms −N(A,B) + N(B,B) + N(A,C) − 2N(B,C) compared with S_epsilon. In the ε→0 limit the asserted convergence of each S_epsilon,j to zero fails; the leftover is N(w1,w1) + N(w2,w2) − N(w1,w2) − N(w2,w1), which is nonzero for generic w1,w2. Since Prop 4.3 is the only bridge from the second-order linearization to v(1) = v(2) and the identity (R1−R2)w1·∇w2 + (R1−R2)w2·∇w1 = 0, the main theorem is unsupported.\n\nMy guess is this is a repairable slip rather than a dead end: a correct polarization identity should produce the desired bilinear expression, and the rest of the paper (Runge approximation, the recovery of R from the identity) seems fine. But as submitted, the proof does not go through.\n\nWho is this for? People working on inverse problems for nonlocal or nonlinear PDEs will find the setup and the strategy worth reading, even if the central proof needs repair. It deserves a serious referee — send it to peer review with a clear request to fix the second-order linearization. If the repair works, it is a solid contribution.","headline":"Genuinely novel Calderón-type inverse problem for active scalar equations with fractional dissipation, but Proposition 4.3 is algebraically wrong, so Theorem 1.1 is not proven as written.","tokens_in":11007,"tokens_out":7141,"would_cite":false,"duration_ms":83094,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35R11","35Q35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Observing $\\theta$ and $u$ in an arbitrarily small open set $W$ uniquely determines the nonlocal drift operator $R$ on the exterior $W^e$ for dissipative active scalar equations with fractional dissipation.","keywords":["active scalar equations","fractional dissipation","Calderón inverse problem","second-order linearization","fractional Laplacian","unique continuation","Runge approximation","nonlocal drift operator"],"falsifier":"A decisive check is to compute the difference between $S_\\epsilon$ as defined in Eq. (4.9) and its four-term decomposition in Eq. (4.10) for an explicit choice of $R$ (for instance the periodic Riesz transform) and two sources supported in $W$. If the remainder contains self-interaction terms of the form $R(w_1-w_2)\\cdot\\nabla(w_1-w_2)$ whose $L^2(0,T;H^{-\\alpha})$ norm does not tend to zero as $\\epsilon \\to 0$, then Proposition 4.3 and the theorem's linearization step are not established.","tokens_in":9820,"feed_emoji":"🌊","tokens_out":16953,"duration_ms":146284,"temperature":0.7,"pith_summary":"This paper studies an inverse problem for dissipative active scalar equations on the two-dimensional torus, where a scalar field $\\theta$ is carried by a divergence-free velocity $u = R\\theta$ generated from $\\theta$ through a nonlocal operator $R$ of order $-1$. The authors claim that if two such operators $R_1$ and $R_2$ produce the same observations of both $\\theta$ and $u$ in a fixed, arbitrarily small open set $W$, then $R_1$ and $R_2$ must agree on all smooth functions supported in the exterior $W^e$. In other words, a tiny interior window of data determines the nonlocal drift mechanism outside the window. The reason such a statement can be true is that the fractional dissipation $(-\\Delta)^\\alpha$ with $\\alpha > 1/2$ gives the equation nonlocal unique continuation, so information from the observation region reaches the whole torus, while the quadratic nonlinearity lets the inverse problem be reduced to a linear fractional diffusion equation. If correct, this is the first Calderón-type uniqueness result for this class of fractional active scalar equations.","feed_headline":"One open patch of data fixes the nonlocal drift outside the patch","feed_subtitle":"For this fractional fluid model, a tiny interior observation determines nonlocal drift R outside the window.","key_machinery":"The argument is carried by three linked mechanisms. First, second-order linearization: the mixed second difference quotient of the solution map, built from two small sources supported in $W$, converges to a linear equation whose source is $-Rw_1 \\cdot \\nabla w_2 - Rw_2 \\cdot \\nabla w_1$, and this is the step that turns recovery of a nonlinear operator into recovery of a term in a linear equation. Second, the unique continuation property of the fractional Laplacian on the torus, which states that a sufficiently regular function with $(-\\Delta)^r u = u = 0$ in any nonempty open set vanishes everywhere; this is what forces agreement of the two nonlinear solutions to propagate from $W$ to the whole torus. Third, a Runge approximation property for the linear fractional diffusion equation, which says that solutions sourced in $W$ are dense in $L^2(0,T;L^2(T^2 \\setminus W))$ when restricted to the exterior; this is what converts the final integral identities into statements about arbitrary test functions and produces the kernel-difference condition. The divergence-free structure $\\mathrm{div}\\,R\\theta = 0$ is used throughout, in the uniqueness argument and in the integration by parts.","core_discovery":"The central claim is Theorem 1.1: for $T>0$ arbitrarily small, if the source-to-solution maps $L_{R_1}$ and $L_{R_2}$ agree for all sources $f$ supported in $W \\times (0,T)$, where each map records the restriction of $\\theta$ and $u$ to $W \\times (0,T)$, then $R_1g|_{W^e} = R_2g|_{W^e}$ for every smooth $g$ supported in $W^e$. The discovery is that the nonlocal drift operator $R$, a Fourier-multiplier object rather than a local coefficient, can be recovered outside the observation region from interior data. The proof uses a second-order linearization: with two small sources $f_1$ and $f_2$ supported in $W$, the mixed second difference quotient of the solution map converges to the solution $v$ of $\\partial_t v + R w_1 \\cdot \\nabla w_2 + R w_2 \\cdot \\nabla w_1 + (-\\Delta)^\\alpha v = 0$, where $w_1, w_2$ solve the linear fractional diffusion equation. Equality of the two maps forces this mixed term to agree for $R_1$ and $R_2$; the divergence-free property of $R$ permits an integration by parts, unique continuation of the fractional Laplacian globalizes equality from $W$ to the torus, and a Runge approximation property lets the resulting identities be tested against arbitrary functions in $W^e$, yielding the exterior equality of $R_1$ and $R_2$.","pith_inferences":["A natural extension the authors do not pursue is higher-order linearization: third- or fourth-order mixed terms encode iterated compositions of $R$ and might allow recovery of more of the operator's structure, possibly on the whole torus.","The proof is qualitative, but a quantitative version of the Runge approximation could turn the uniqueness statement into a stability estimate, likely logarithmic or Hölder, for the difference $R_1-R_2$ in terms of the discrepancy between the two source-to-solution maps.","A numerical check of the second-order linearization on a known kernel would isolate whether the convergence of the mixed difference quotient is the only delicate step; this is a testable extension of the paper's analytic argument.","If the divergence-free condition were dropped, the integration-by-parts step that removes velocity terms would fail, so some form of velocity information appears essential; identifying the minimal measurements is a separate question."],"forward_implications":["If the theorem is correct, measurements taken in any arbitrarily small open set, with no boundary access, determine the drift operator $R$ on the complement of that set.","For models such as the surface quasi-geostrophic equation, where $R$ is a Riesz-type operator, the result gives exterior uniqueness for the velocity law from local observations of the scalar and velocity.","Because $W$ can be chosen arbitrarily small, the uniqueness statement is insensitive to the size of the observation window.","The proof gives a template for nonlinear fractional inverse problems: second-order linearization combined with unique continuation can recover nonlocal terms rather than only local variable coefficients.","The exterior-only conclusion is a genuine limitation of the method, since the Runge approximation is available only outside $W$; determination on the whole torus is left open."],"supporting_citations":[{"why":"Establishes the fractional Calderón problem and supplies the nonlocal unique-continuation and exterior-density ideas that this paper adapts.","marker":"[9]"},{"why":"Provides the unique continuation property of the fractional Laplacian on closed manifolds used in Proposition 5.1.","marker":"[5]"},{"why":"Gives the analogous Runge approximation/density result for the fractional heat equation that Proposition 5.2 extends.","marker":"[20]"},{"why":"Supplies the well-posedness and estimate machinery for the SQG equation that Proposition 3.1 adapts to the more general active scalar model.","marker":"[18]"},{"why":"Introduces multiple-fold linearization for nonlinear inverse problems, the technique underlying the second-order step.","marker":"[12]"},{"why":"Provides the linear parabolic estimates behind the well-posedness of the linear fractional diffusion equation in Proposition 4.1.","marker":"[4]"},{"why":"Gives the Lp boundedness of Riesz transforms on the torus that justifies the standing assumptions on $R$.","marker":"[22]"},{"why":"Provides an analogous Runge approximation for a fractionally damped wave equation, supporting the density step of Proposition 5.2.","marker":"[17]"}],"fun_headline_variants":["Local patch reveals nonlocal drift beyond the patch","A tiny observation patch determines the whole nonlocal flow","Fractional fluid: hidden drift pinned by local measurements","Interior data recovers nonlocal drift outside the window"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is that the nonlinear response to two small sources can be split cleanly into first-order and second-order pieces with no leftover cross-terms; if those cross-terms do not vanish, the equation that determines $R$ is not the one the proof analyzes.","fun_headline_variants_meta":{"raw":{"variants":["Local patch reveals nonlocal drift beyond the patch","A tiny observation patch determines the whole nonlocal flow","Fractional fluid: hidden drift pinned by local measurements","Interior data recovers nonlocal drift outside the window"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000738,"raw_usage":{"total_tokens":3306,"prompt_tokens":961,"completion_tokens":2345,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":2282}},"tokens_in":577,"tokens_out":2345,"duration_ms":15925,"temperature":1.0,"reasoning_tokens":2282,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:01:35.863970+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive check is to compute the difference between $S_\\epsilon$ as defined in Eq. (4.9) and its four-term decomposition in Eq. (4.10) for an explicit choice of $R$ (for instance the periodic Riesz transform) and two sources supported in $W$. If the remainder contains self-interaction terms of the form $R(w_1-w_2)\\cdot\\nabla(w_1-w_2)$ whose $L^2(0,T;H^{-\\alpha})$ norm does not tend to zero as $\\epsilon \\to 0$, then Proposition 4.3 and the theorem's linearization step are not established.","supporting_citations":[{"cited_title":"The Calder´ on problem for the fractional Schr¨ odinger equation","cited_arxiv_id":null,"evidence_quote":"Establishes the fractional Calderón problem and supplies the nonlocal unique-continuation and exterior-density ideas that this paper adapts."},{"cited_title":"Quantitative approximation properties for the fractional heat equation.Math- ematical Control & Related Fields , 10(1):1–26, 2020","cited_arxiv_id":null,"evidence_quote":"Gives the analogous Runge approximation/density result for the fractional heat equation that Proposition 5.2 extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the well-posedness and estimate machinery for the SQG equation that Proposition 3.1 adapts to the more general active scalar model."},{"cited_title":"Inverse problems for lorentzian manifolds and non-linear hyperbolic equations","cited_arxiv_id":null,"evidence_quote":"Introduces multiple-fold linearization for nonlinear inverse problems, the technique underlying the second-order step."},{"cited_title":"Partial differential equations , volume 19","cited_arxiv_id":null,"evidence_quote":"Provides the linear parabolic estimates behind the well-posedness of the linear fractional diffusion equation in Proposition 4.1."},{"cited_title":"Stein and Guido Weiss","cited_arxiv_id":null,"evidence_quote":"Gives the Lp boundedness of Riesz transforms on the torus that justifies the standing assumptions on $R$."}],"review_version":1}