{"id":"7520a1ff-4236-45fa-9b45-27a8834bb19d","arxiv_id":"2506.12535","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims an anisotropic Calderon uniqueness theorem for a logarithmic Laplacian of order 2+, but the central Paley-Wiener argument is invalid.","lead":"This paper claims that Cauchy data for a logarithmic Laplacian on a closed manifold determine the manifold up to isometry. The proof contains an invalid analytic step, so the claimed theorem is not established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's bridge from vanishing power moments to φ=0 uses a Taylor expansion of the Paley-Wiener transform at the boundary z=0, where analyticity is not available; this implication is false in general, so Theorem 1.1 is unsupported.","rationale":"The reader's weakest_assumption pinpoints the exact load-bearing step. Equation (3.8) gives only vanishing of all power moments of ψ; the proof then leaps to ψ=0 via a Taylor expansion of the Paley-Wiener transform at the boundary point z=0. This is not a theorem: analyticity in the upper half-plane does not imply analyticity up to the boundary, and even if it did, the Taylor coefficients at an interior point are not the power moments. The concrete counterexample ψ_0 shows the general implication is false, so the paper's bridge from the integral identities to (3.16) collapses. The earlier parts of the proof (well-definedness, commutator computation, Hardy estimate) are reasonable, and the final use of [18, Thm 1.5] would be fine if heat-kernel equality were available. I also note a secondary, likely repairable slip in (3.20) where applying (2.15) to e^{-t0A}u requires an extra factor A in the integrand; it is not the main obstruction. The central theorem is therefore unsupported by the proof as written, and the reader's REJECT verdict should stand.","tokens_in":13676,"tokens_out":18025,"duration_ms":219090,"concrete_test":"Compute the moments of ψ_0(τ)=e^{-τ^{1/4}} sin(τ^{1/4}): for each j≥0, ∫_0^∞ τ^j ψ_0(τ)dτ = 4∫_0^∞ u^{4j+3} e^{-u} sin u du = Γ(4j+4) 2^{-2j-2} sin((j+1)π)=0, while ψ_0≠0. This directly refutes the moment-vanishing-to-zero inference used in Section 3. Alternatively, re-derive the step from (3.8) to (3.16) without appealing to analyticity at ζ=0 and check whether any hypothesis beyond the vanishing moments is used; the current proof supplies none.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Between (3.17) and (3.18) the argument requires: if ψ(τ)=φ(1/τ)∈L2(0,∞) and ∫_0^∞ τ^j ψ(τ)dτ=0 for all j≥0, then ψ=0. The paper tries to prove this from the Paley-Wiener theorem by writing the Taylor expansion of \\hatψ(ζ)=∫_0^∞ e^{iτζ}ψ(τ)dτ as Σ_j (∫τ^jψ) ζ^j/j!=0. This is not justified. Paley-Wiener gives holomorphy only in H+; \\hatψ need not be holomorphic at ζ=0, and expanding e^{iτζ} as a power series near ζ=0 is not valid on the whole integration range. The Taylor coefficients at an interior point ζ0∈H+ are ∫(iτ)^j e^{iτζ0}ψ(τ)dτ, not the power moments, so the displayed series cannot be the expansion used. Moreover the implication is simply false: ψ_0(τ)=e^{-τ^{1/4}}sin(τ^{1/4}) is nonzero, belongs to L2(0,∞), and has ∫_0^∞ τ^jψ_0(τ)dτ=0 for every j≥0 (substitute u=τ^{1/4}; the integral is Γ(4j+4)2^{-2j-2} sin((j+1)π)=0). Thus the conclusion (3.16), the heat-kernel equality (3.18), and Theorem 1.1 are not established by the given proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates the anisotropic Calderón problem for the logarithmic Laplacian L_g = (-Δ_g + mI) log(-Δ_g + mI) on closed Riemannian manifolds of dimension greater than two. The main theorem (Theorem 1.1) claims that equality of the Cauchy data sets on a nonempty open set where the metrics agree forces the two manifolds to be isometric. The proof strategy is to show that the equality of Cauchy data implies certain integral identities for the difference of heat semigroups, to conclude the heat semigroups agree on the open set, then to deduce equality of the heat kernels, and finally to invoke the rigidity theorem [18, Theorem 1.5]. A central analytic step in this chain is not justified.","tokens_in":13945,"tokens_out":10126,"duration_ms":109625,"significance":"If correct, the result would be a valuable contribution to the program of extending nonlocal Calderón-type inverse problems to operators of order 2+, thereby connecting the fractional and local anisotropic cases. The paper contains a careful spectral and semigroup setup for the logarithmic Laplacian, and the overall strategy is natural and worth pursuing. However, the decisive step in the proof is invalid, and the main theorem is not established by the arguments presented.","major_comments":[{"comment":"The Paley-Wiener theorem guarantees that the Fourier transform \\hat{\\psi}(\\xi+i\\eta) = \\int_0^\\infty e^{i\\tau(\\xi+i\\eta)}\\psi(\\tau)\\,d\\tau is holomorphic only in the upper half-plane H_+. The Taylor expansion of this function at an interior point \\zeta_0 \\in H_+ has coefficients of the form \\int_0^\\infty (i\\tau)^j e^{i\\tau\\zeta_0}\\psi(\\tau)\\,d\\tau, not the power moments \\int_0^\\infty \\tau^j \\psi(\\tau)\\,d\\tau. Writing the Taylor series with coefficients \\int_0^\\infty \\tau^j \\psi(\\tau)\\,d\\tau would require analyticity of \\hat{\\psi} at \\zeta=0, which Paley-Wiener does not provide. Consequently, the assertion that the identity (3.8) forces \\hat{\\psi} \\equiv 0 is not justified.","section":"Section 3, between (3.17) and (3.18)"},{"comment":"The implication 'if \\int_0^\\infty \\tau^j \\psi(\\tau)\\,d\\tau = 0 for every j \\geq 0, then \\psi \\equiv 0' is false in general. For example, \\psi_0(\\tau)=e^{-\\tau^{1/4}}\\sin(\\tau^{1/4}) belongs to L^2(0,\\infty) and satisfies \\int_0^\\infty \\tau^j \\psi_0(\\tau)\\,d\\tau = 4\\Gamma(4j+4)2^{-2j-2}\\sin((j+1)\\pi)=0 for every j\\geq 0, yet \\psi_0 is not identically zero. Therefore the vanishing of the moment integrals (3.8) cannot, by itself, yield (3.16). The equality of heat semigroups (3.18) and the heat kernels (3.19) do not follow from the given proof.","section":"Section 3, implication from (3.8) to (3.16)"},{"comment":"The deduction of the heat kernel equality and the final appeal to [18, Theorem 1.5] rests entirely on (3.18), which in turn depends on the unjustified (3.16). Since the step from (3.8) to (3.16) is invalid, the proof does not supply any alternative route to the heat kernel equality. Thus Theorem 1.1 is unsupported by the present argument.","section":"Section 3, derivation of heat kernel equality (3.19)-(3.24)"}],"minor_comments":[{"comment":"The abstract contains typographical errors ('Logarithemic', 'close' for 'closed') and should be proofread.","section":"Abstract"},{"comment":"In the Taylor formula (3.13), the denominator '(m-1)!' appears to be a misprint for '(l-1)!'.","section":"Section 3, equation (3.13)"},{"comment":"The notation H(M) is overloaded: it is defined as D(L_g) in (1.4) but later used to denote Sobolev spaces H^l(M_i). Using distinct notation would improve clarity.","section":"Section 3, regularity discussion"}],"recommendation":"reject","confidential_remarks":"The manuscript requires a fundamentally new argument to bridge the integral identities (3.8) and the vanishing of the heat semigroup difference (3.16); the current gap is not a local fix. The bibliographic context and the statement of the theorem are reasonable, but the proof is not valid in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the paper announces a real new operator for the anisotropic Calderón program, but the proof has a load-bearing gap in the bridge from integral identities to heat semigroup equality. I don't think the theorem is established.\n\nWhat's good: the operator L_g = (-Δ_g + mI) ln(-Δ_g + mI) is a natural order-2+ candidate, and the main statement fits squarely into the fractional Calderón framework. The direct problem setup, the regularity bootstrapping, and the reduction to heat kernel equality are all competently handled. Invoking [18, Thm 1.5] is legitimate; I see no circularity and no fitted parameters. The paper is clearly written and the author knows the surrounding literature.\n\nThe trouble is Section 3, between (3.17) and (3.18). The proof claims that the moment identities ∫ τ^j ψ(τ)dτ = 0 for all j≥0 force ψ = 0. The argument uses a Taylor expansion of the Paley–Wiener transform of ψ, but expands at a boundary point where analyticity is not available. The coefficients it writes are the power moments, not the actual Taylor coefficients at an interior point. More importantly, the implication is false in general: ψ0(τ)=exp(-τ^{1/4}) sin(τ^{1/4}) is nonzero, lies in L2(0,∞), and has all power moments zero. The paper's φ does have extra structure (it solves a heat equation), but that structure is not used in this step, and the proof relies on the general statement. So the heat semigroup equality (3.18), the heat kernel equality (3.23), and Theorem 1.1 all rest on an invalid step.\n\nMinor issue: the 'order 2+' description is informal; L_g belongs to the intersection of Ψ^{2+ε} for every ε>0, which is not a standard order class. That's cosmetic next to the main gap.\n\nThis paper is for specialists in nonlocal inverse problems. It deserves a serious referee because the question is meaningful and the framework is standard, but the current proof should not be accepted. I'd recommend sending it out—the gap is concrete and a referee can pinpoint it—but the result needs a repaired argument, probably a genuinely new idea at the moment step.","headline":"Genuine new result and a clean setup, but the proof's decisive step from vanishing moments to φ=0 is invalid, so Theorem 1.1 is not established as written.","tokens_in":773,"tokens_out":2142,"would_cite":false,"duration_ms":56813,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","58J35","35S05","58J40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Observation data for a log-Laplacian recover the whole manifold up to isometry.","keywords":["anisotropic Calderón problem","logarithmic Laplacian","near Laplace–Beltrami operator","Cauchy data set","heat kernel","pseudodifferential operator","closed Riemannian manifold","inverse problems"],"falsifier":"One concrete check is to look for a nonzero $L^2(0,\\infty)$ function $\\psi$ with $\\int_0^\\infty \\tau^j \\psi(\\tau)\\, d\\tau = 0$ for every $j\\ge 0$ (moment-indeterminate functions of this kind are known to exist) and to determine whether $\\psi$ can have the specific form of the rescaled heat-semigroup difference between equations (3.17) and (3.18); if such a difference exists, the proof's central vanishing inference fails. Short of that, checking whether the Paley–Wiener transform of the relevant $\\psi$ is analytic in a full neighborhood of $z=0$ would settle the disputed step.","tokens_in":13404,"feed_emoji":"🔍","tokens_out":8329,"duration_ms":94099,"temperature":0.7,"pith_summary":"This paper claims a positive answer to the anisotropic Calderón problem for an operator that sits at order 2+, just past the Laplace–Beltrami operator: the logarithmic Laplacian $L_g = (-\\Delta_g + mI)\\ln(-\\Delta_g + mI)$. The claim is that on any closed connected Riemannian manifold of dimension greater than two, knowing the Cauchy data of $L_g$ on a small open set, together with the metric on that set, pins down the entire manifold up to diffeomorphism. A sympathetic reader would care because this moves nonlocal inverse problems from fixed fractional orders below 2 to operators arbitrarily close to the classical Laplacian, where the anisotropic Calderón problem remains open in smooth geometry. The proof works by converting the Cauchy data into equality of heat semigroups on the observation set and then invoking a known result that heat-kernel equality implies isometry.","feed_headline":"Log-Laplacian data fix the whole manifold","feed_subtitle":"Cauchy data for an order-2+ operator determine a closed Riemannian manifold up to isometry.","key_machinery":"The central object is the logarithmic Laplacian operator $L_g = (-\\Delta_g + mI)\\ln(-\\Delta_g + mI)$, whose spectral action multiplies the $k$-th Fourier mode by $(\\lambda_k+m)\\ln(\\lambda_k+m)$; its principal symbol is $(|\\xi|^2_g + m)\\ln(|\\xi|^2_g + m)$, giving an operator of order $2+\\varepsilon$ for every $\\varepsilon>0$. The load-bearing mechanism is a chain of reductions: equality of Cauchy data implies vanishing integral identities involving the heat semigroups of the two manifolds; after integration by parts, Hardy's inequality, and a change of variables, the Paley–Wiener theorem is invoked to turn those vanishing moment integrals into vanishing Taylor coefficients of a holomorphic function, forcing the heat semigroup difference to vanish on $O$; from there the equality of heat kernels follows and a known theorem converts heat-kernel equality into an isometry.","core_discovery":"The paper's central claim, Theorem 1.1, states that if two closed connected Riemannian manifolds $(M_1, g_1)$ and $(M_2, g_2)$ of dimension greater than two agree on a non-empty open set $O$, and if their Cauchy data sets for the operator $L_g = (-\\Delta_g + mI)\\ln(-\\Delta_g + mI)$ agree on $O$ for a fixed $m>1$, then there is a diffeomorphism $\\Phi: M_1 \\to M_2$ with $\\Phi^* g_2 = g_1$. The operator is defined spectrally through the eigenvalues $(\\lambda_k+m)\\ln(\\lambda_k+m)$, and its symbol $(|\\xi|^2_g + m)\\ln(|\\xi|^2_g + m)$ places it in the pseudodifferential class of order $2+\\varepsilon$ for every $\\varepsilon>0$, which is what makes it a nearly Laplace–Beltrami operator of order $2+$. The theorem is established by deriving from the Cauchy data a family of integral identities, using Hardy's inequality and the Paley–Wiener theorem to conclude that the heat semigroups coincide on $O$, and then invoking the heat-kernel uniqueness theorem quoted as Theorem 3.1 of the paper's reference [18].","pith_inferences":["A natural companion question, not addressed in the paper, is whether the same scheme works for the operator obtained by replacing $\\ln(-\\Delta_g + mI)$ with any slowly varying complete Bernstein function of the Laplacian, since only the order-$2+$ symbol growth appears to be used.","If the moment-vanishing step at the boundary can be made rigorous, the proof actually yields a reconstruction route: recover the heat semigroup on $O$ from the Cauchy data, then apply boundary-control or heat-kernel methods to recover the metric.","The example of the logarithmic Laplacian may serve as a testbed for whether local anisotropic Calderón uniqueness can be obtained as a limit of order-$2+$ nonlocal problems as the logarithmic correction is scaled away.","A direct way to stress-test the paper's central inference is to study the flat-torus or round-sphere case, where heat kernels are explicit, and check whether two different metrics can produce heat-semigroup differences whose moment integrals all vanish."],"forward_implications":["If the theorem is correct, the Cauchy data of the logarithmic Laplacian on any open set determine the isometry class of a closed Riemannian manifold of dimension greater than two.","The result extends the nonlocal anisotropic Calderón program from fractional Laplace–Beltrami operators of order $2\\alpha<2$ to an operator of order $2+$, bringing it closer to the unresolved smooth local problem.","Because the proof reaches equality of heat kernels, all heat-kernel invariants of the manifold become observable from the Cauchy data, not just the conformal class.","The same reduction, if valid, could be applied to other functions of the shifted Laplacian whose symbols grow like order $2+$.","The method suggests that knowledge of the Cauchy data for one such nearly local operator may be enough to recover the full metric, without boundary measurements on the whole manifold."],"supporting_citations":[{"why":"Supplies the final theorem: equality of heat kernels on an open set implies the manifolds are isometric, which closes the proof of Theorem 1.1.","marker":"[18]"},{"why":"Provides the Gaussian upper bound for heat kernels that is used throughout to prove the integral identities and the heat-semigroup estimates are well defined.","marker":"[28]"},{"why":"The classical Paley–Wiener theorem is invoked to pass from vanishing moment integrals to vanishing of the holomorphic transform, the critical step forcing heat-semigroup equality.","marker":"[42]"},{"why":"Hardy's inequality gives the weighted $L^2$ control that places the rescaled heat-semigroup difference in $L^2(0,\\infty)$, making the Paley–Wiener application possible.","marker":"[20]"},{"why":"Supplies the symbol-class theorem for fractional powers of the Laplace–Beltrami operator that fixes the order and principal symbol used to identify $L_g$ as order $2+$.","marker":"[17]"},{"why":"Records spectral properties of the logarithmic Laplacian that motivate treating $\\ln(-\\Delta_g + mI)$ as a legitimate operator and give the derivative-at-zero representation of the logarithm.","marker":"[35]"},{"why":"Provides an extension-problem formulation of the logarithmic Laplacian that supports the functional-calculus definition of the operator.","marker":"[7]"}],"fun_headline_variants":["Cauchy data for log-Laplacian determine the manifold","Log-Laplacian boundary measurements recover the metric","Anisotropic log-Laplacian: boundary data fix geometry","From Cauchy data to isometry: log-Laplacian theorem"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the step where vanishing of all the integrals $\\int_0^\\infty \\varphi(t)/t^{1+k}\\, dt$ for $k=0,1,2,\\dots$ is taken to force $\\varphi$ to be identically zero; that inference requires a Paley–Wiener transform to be analytic at a boundary point, and the cited Paley–Wiener theorem does not provide that analyticity.","fun_headline_variants_meta":{"raw":{"variants":["Cauchy data for log-Laplacian determine the manifold","Log-Laplacian boundary measurements recover the metric","Anisotropic log-Laplacian: boundary data fix geometry","From Cauchy data to isometry: log-Laplacian theorem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000288,"raw_usage":{"total_tokens":1642,"prompt_tokens":854,"completion_tokens":788,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":721}},"tokens_in":470,"tokens_out":788,"duration_ms":8608,"temperature":1.0,"reasoning_tokens":721,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:49:17.967105+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check is to look for a nonzero $L^2(0,\\infty)$ function $\\psi$ with $\\int_0^\\infty \\tau^j \\psi(\\tau)\\, d\\tau = 0$ for every $j\\ge 0$ (moment-indeterminate functions of this kind are known to exist) and to determine whether $\\psi$ can have the specific form of the rescaled heat-semigroup difference between equations (3.17) and (3.18); if such a difference exists, the proof's central vanishing inference fails. Short of that, checking whether the Paley–Wiener transform of the relevant $\\psi$ is analytic in a full neighborhood of $z=0$ would settle the disputed step.","supporting_citations":[{"cited_title":"Gaussian upper bounds for the heat kernel on arbitrary manifolds","cited_arxiv_id":null,"evidence_quote":"Provides the Gaussian upper bound for heat kernels that is used throughout to prove the integral identities and the heat-semigroup estimates are well defined."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classical Paley–Wiener theorem is invoked to pass from vanishing moment integrals to vanishing of the holomorphic transform, the critical step forcing heat-semigroup equality."},{"cited_title":"L., Laptev, A., and Weidl, T","cited_arxiv_id":null,"evidence_quote":"Hardy's inequality gives the weighted $L^2$ control that places the rescaled heat-semigroup difference in $L^2(0,\\infty)$, making the Paley–Wiener application possible."},{"cited_title":"Spectral properties of the logarithmic Laplacian","cited_arxiv_id":null,"evidence_quote":"Records spectral properties of the logarithmic Laplacian that motivate treating $\\ln(-\\Delta_g + mI)$ as a legitimate operator and give the derivative-at-zero representation of the logarithm."}],"review_version":1}