{"id":"fc1f7045-29c7-4470-aa0c-504b2ee9c34f","arxiv_id":"2505.12255","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For closed Riemannian manifolds, the Cauchy data of the operator ((−Δ)^2 + m^2)^(1/2) determines the manifold up to diffeomorphism.","lead":"This paper proves that measurements of a nonlocal second-order operator in a small region of a curved space determine the entire shape of the space up to isometry. The result extends a known uniqueness theorem from fractional powers of the Laplacian to the square root of a bi-Laplacian.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.5 rests on a Kannai formula that is false as printed; the constant mode gives a t^{7/6} mismatch, and the inverse Laplace step needs an unstated uniqueness theorem.","rationale":"The reader identified the Kannai transmutation/inverse-Laplace passage in Proposition 3.5 as the weakest assumption, and I agree that this is the load-bearing step. My stress-test sharpens the concern: the transmutation formula (27) is not just under-justified; as printed it is quantitatively wrong on the constant eigenfunction, because the prefactor t^{1/3} should be t^{-3/2}. This is a concrete, checkable error in the written proof. It does not, by itself, sink the mathematical claim, because the standard Kannai identity with the correct normalization is a valid spectral-theoretic identity and the inverse Laplace step can likely be supplied by a standard uniqueness theorem for Laplace transforms of polynomially growing functions. The situation therefore calls for a conditional verdict: the result is plausible and the strategy is sound, but the manuscript must correct the formula and fill the missing justification before the proof is complete. I do not see an independent reason to reject the central theorem, nor do I see a reason to accept the proof as currently written.","tokens_in":13304,"tokens_out":19104,"duration_ms":198270,"concrete_test":"Evaluate identity (27) on a closed manifold, e.g. S^1, with f≡1. Compute the left-hand side e^{-tΔ^2}1=1 for all t; compute the right-hand side with the printed constant 1/(4√π t^{1/3}), which equals C t^{7/6}. If the two differ, formula (27) as stated is false. Then re-run the same test with t^{-3/2} in place of t^{1/3}; if equality holds, the intended Kannai formula is the corrected one, and the author should state and prove it, and additionally justify the inverse Laplace step by proving that ∫_0∞ e^{-uτ}F(τ)dτ vanishes for all u>0 implies F≡0 for F of polynomial growth.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim depends on Proposition 3.5 converting equality of bi-Laplace heat semigroups into equality of Laplace heat kernels. That conversion uses Kannai's transmutation formulas (27) and (35). As printed, (27) is not correct on a closed manifold. On the zero eigenspace of Δ, e^{-tΔ^2}=I, while Δ^{-1} sin(√τ Δ) acts as √τ I; hence the right-hand side of (27) equals [1/(4√π t^{1/3})] ∫_0∞ e^{-τ/(4t)} √τ dτ · I = C t^{7/6} I, which is not I. The correct normalization is t^{-3/2}, for which the identity is a standard sine-transform representation. Since the proof uses (27) and (35) to obtain (28) and then (34), the printed argument is invalid at this step. Moreover, the passage (28)→(29) invokes an inverse Laplace transform without specifying the class of F(τ)=[sin(√τ Δ1)/Δ1 − sin(√τ Δ2)/Δ2]f(x); this function has at best polynomial (√τ) growth on the constant component, so the usual L^2/Paley–Wiener uniqueness does not apply directly, and a Laplace-uniqueness theorem for polynomially growing functions/distributions is needed. These issues are repairable, but as written they are genuine gaps in the central chain.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the anisotropic Calderón problem for the nonlocal second-order operator L_g^{1/2}=((−Δ_g)^2+m^2 I)^{1/2} on closed Riemannian manifolds. Theorem 1.1 claims that if two such manifolds share an open set O on which the metrics agree, and if the interior data L_{g_1}^{−1/2}f|_O = L_{g_2}^{−1/2}f|_O for all f∈C_0^∞(O), then the manifolds are isometric. The proof derives vanishing moment identities for the difference of heat semigroups of L_g, uses Paley–Wiener to conclude the semigroup difference vanishes, then invokes Kannai transmutation formulas to pass from equality of e^{−tΔ^2} to equality of e^{tΔ}, and finally applies a heat-kernel rigidity theorem from [20] to recover the diffeomorphism. A second, closely related theorem for A_g=L_g^{1/2}−mI is stated without proof.","tokens_in":13610,"tokens_out":10170,"duration_ms":100483,"significance":"The main result is a natural extension of the nonlocal anisotropic Calderón program to an operator of order 2, and if the proof can be repaired it would be a worthwhile contribution to the area. The paper's strategy is clear and follows the now-standard template of [20], and the final rigidity step is honestly imported from an independent source rather than reproved. The central new step, however, depends on an incorrect transmutation formula and an unsubstantiated inverse-Laplace uniqueness argument; these must be corrected before the result is reliable. The result's significance is therefore conditional on those repairs.","major_comments":[{"comment":"The Kannai formulas are not correct as stated on a closed manifold. For the constant function v≡1, Δ1=0 and sin(√τΔ)/Δ acts as √τ, so the right-hand side of (27) equals (1/(4√π t^{1/3}))∫_0^∞ e^{−τ/(4t)}√τ dτ = t^{7/6}, whereas e^{−tΔ^2}1=1. The same calculation applies to (35) with e^{tΔ}. Since (27) is used to derive (28) and (35) is used to obtain equality of the Laplace heat semigroups, this is a load-bearing error. The author must supply a correct transmutation identity with the proper normalization and state the hypotheses under which it holds on compact manifolds.","section":"Section 3, Eqs. (27) and (35)"},{"comment":"The step from equality of the t-integrals in (28) to pointwise equality in σ invokes an inverse Laplace transform, but no uniqueness theorem is stated. After setting s=1/(4t), one has ∫_0^∞ e^{−sτ}F(τ)dτ=0 for all s>0 with F(τ)=[sin(√τΔ1)/Δ1−sin(√τΔ2)/Δ2]f(x). The function F is not known to lie in L^2 or to satisfy Paley–Wiener growth; on the constant component it grows like √τ. A Laplace-uniqueness theorem for polynomially growing functions or distributions, together with sufficient regularity to justify pointwise evaluation in x, must be supplied.","section":"Section 3, passage from (28) to (29)"},{"comment":"Theorem 1.4 is a separate theorem but is followed only by 'The proof is quite similar to Theorem 1.1' and no argument is given. If this theorem is to remain in the paper, the author must either prove it or give an explicit reduction of the Cauchy data for A_g to the data used in Theorem 1.1; otherwise it should be demoted to a remark.","section":"Section 1, Theorem 1.4"}],"minor_comments":[{"comment":"Equation (14) identifies ∂_t^l((e^{−tL_{g1}}−e^{−tL_{g2}})f) with a term involving L^k_g f; this should involve L^l_g f up to sign. Equation (17) mixes L^{2l} and L^k, and the bounds in (18) are not immediate from (17) without an additional argument. These inconsistencies appear repairable, but the displayed estimates should be corrected.","section":"Section 3, Eqs. (14), (17), and (18)"},{"comment":"Using (18) and the definition φ(s)=diff/√s gives |φ(s)|≤Cs^{(n−1)/2} for s∈(0,1), not Cs^{n/2} as printed. The conclusion φ∈L^2(0,∞) is unaffected, but the exponent should be corrected.","section":"Section 3, Proposition 3.2"},{"comment":"There are several typographical issues: 'psedodifferential' in Section 1; 'From (26) and (26)' before Eq. (28); a garbled displayed formula for ψ in (34); and Eq. (12) omits the harmless constant 1/Γ(1/2). These should be fixed in revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The most serious issue is the incorrect Kannai formula in Eqs. (27) and (35); the verification on the constant eigenspace shows the printed formulas cannot hold, and the inverse-Laplace uniqueness step also needs a precise theorem. Because the overall strategy is sound and the gaps appear repairable, I recommend major revision rather than rejection. The novelty is incremental relative to [20], and the paper should make clear exactly which parts are new."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWhat you should know: this paper proves that the Cauchy data for the nonlocal operator L_g^{1/2} = ((−Δ_g)^2 + m^2 I)^{1/2} on a closed manifold determines the isometry class. That is a genuinely new boundary case: the fractional Calderón results so far stop at order <2. The proof follows the now-standard template from Feizmohammadi–Ghosh–Krupchyk–Uhlmann [20]: show that the exterior data produce vanishing moments of the heat semigroup difference, turn that into equality of the bi-Laplace heat semigroups, then convert to equality of Laplace heat kernels, then invoke the heat kernel rigidity theorem from [20].\n\nThe good parts. The operator is new in this context and the main theorem is not in the literature. The paper is honest that the strategy is a rerun of [20] rather than a new idea; that's fine. The semigroup arguments in Section 2 are standard and mostly correct. The reliance on the heat kernel rigidity theorem from [20] is legitimate, since that's an independent established result.\n\nThe soft spots. There is a real error in the central step. Equation (27) is Kannai's transmutation formula, but as written it is false on a closed manifold. On the constant eigenfunction, the left side is e^{-tΔ^2} 1 = 1, while the right side evaluates to C t^{7/6} (the integral of sqrt(τ) e^{-τ/(4t)} gives t^{3/2}, and the prefactor contains t^{-1/3}). The correct normalization is t^{-3/2}, not t^{-1/3}. So the equality (28) does not follow from (26) as written. This is likely a typo, but it is load-bearing: everything downstream uses (28) and (34). Also, the passage from (28) to (29) applies an inverse Laplace transform without justifying the uniqueness for functions that grow like sqrt(τ) on the zero mode; that needs a short argument, but it isn't there. These are repairable, but the proof as printed is not complete.\n\nMinor issues: bound (17) mixes L^l and L^k, the exponent in Proposition 3.2 is slightly off relative to (16)/(18), and Theorem 1.4 is stated without proof (the paper says it's similar, which is acceptable but should be detailed or flagged). These are easy fixes.\n\nWho this is for: researchers working on nonlocal Calderón problems and anisotropic inverse problems. For that audience, the extension to order 2 is a meaningful step. The paper deserves a serious referee. I'd send it to review, asking the author to fix the Kannai normalization and justify the Laplace inversion. If those are fixed, the main theorem should hold.","headline":"A useful extension of the fractional anisotropic Calderón theorem to order 2, but the central Kannai formula has a normalization error that needs fixing.","tokens_in":14148,"tokens_out":5302,"would_cite":false,"duration_ms":43436,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","58J35","35S05","47G30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Internal data of a nonlocal order-2 operator on a shared open patch force the two manifolds to be isometric.","keywords":["anisotropic Calderón problem","nonlocal elliptic operator","fractional power of bi-Laplace operator","heat kernel rigidity","Riemannian manifolds","isometry recovery","pseudodifferential operator","internal Cauchy data"],"falsifier":"Evaluate equation (27) on a single eigenfunction $\\varphi_k$ of $\\Delta_g$ with $\\Delta_g\\varphi_k=\\lambda_k\\varphi_k$; the identity as printed gives a prefactor $t^{7/6}e^{-\\lambda_k^2 t}$ rather than $e^{-\\lambda_k^2 t}$, so verifying the correct prefactor and weight against Kannai's statement settles whether the bridge from bi-Laplace to Laplace heat kernels is valid.","tokens_in":13118,"feed_emoji":"📐","tokens_out":7201,"duration_ms":66597,"temperature":0.7,"pith_summary":"This paper claims an anisotropic Calderón theorem for the nonlocal second-order operator $L_g^{1/2} = ((-\\Delta_g)^2 + m^2 I)^{1/2}$ on a smooth closed connected Riemannian manifold: if the internal Cauchy data of two such operators agree on a common open set, the manifolds are isometric. The point of the result is that a genuinely nonlocal, order-2 elliptic operator still carries complete metric information, even though the classical local anisotropic Calderón problem for the Laplacian remains open in smooth geometry. The proof works by converting the assumed data equality into equality of the associated heat kernels and then invoking a known rigidity theorem for heat kernels on Riemannian manifolds.","feed_headline":"A nonlocal operator's data reveals a manifold's whole shape","feed_subtitle":"Measuring the square root of a perturbed bi-Laplacian on one open patch pins down the entire metric.","key_machinery":"The load-bearing chain is: (i) the fractional inverse $L_g^{-1/2}$ is written through the heat semigroup $e^{-tL_g}$ by a Gamma-integral formula, so assumed equality of data becomes vanishing of weighted $t$-integrals of the semigroup difference; (ii) integration by parts and the estimates of the bi-Laplace heat kernel (Theorem 2.1) turn those integrals into vanishing moments $\\int_0^\\infty \\phi(s)s^k\\,ds=0$; (iii) the Paley-Wiener theorem forces $\\phi\\equiv0$, giving equality of the bi-Laplace heat semigroups on a subdomain; (iv) Kannai's transmutation formulas formally convert this into equality of the Laplace heat kernels; and (v) a known heat-kernel rigidity theorem (Theorem 3.6) upgrades local kernel equality to a global isometry. The central object carrying the argument is the transmutation identity expressing $e^{t\\Delta_g}$ (and $e^{-t\\Delta_g^2}$) as an integral of $\\sin(\\sqrt{\\tau}\\,\\Delta_g)/\\Delta_g$ against a Gaussian weight.","core_discovery":"The paper's central claim is Theorem 1.1: let $(M_1,g_1)$ and $(M_2,g_2)$ be smooth closed connected Riemannian manifolds sharing an open set $(O,g)$, and suppose $L_{g_1}^{-1/2}f|_O = L_{g_2}^{-1/2}f|_O$ for every source $f\\in C_0^\\infty(O)$, where $L_g=(-\\Delta_g)^2+m^2I$. Then a diffeomorphism $\\Phi:M_1\\to M_2$ exists with $\\Phi^*g_2=g_1$. In other words, the full isometry class of a closed manifold is encoded in the restriction of the operator's inverse to an arbitrarily small open patch. A companion theorem (Theorem 1.4) states the same conclusion when the Cauchy data set of $A_g=L_g^{1/2}-mI$ agree.","pith_inferences":["If the transmutation step can be repaired with verified hypotheses, the same moment-to-heat-kernel strategy should apply to other fractional powers or functions of $\\Delta_g$ that admit a sine/cosine transmutation, such as the relativistic Schrödinger operator $(-\\Delta_g+m^2)^{\\alpha}$.","The proof's structure suggests a quantitative cousin: replacing 'all moments vanish' with 'finitely many moments are small' might yield a stability estimate or a partial-data reconstruction statement; the paper itself does not pursue this.","Letting the mass parameter $m$ tend to zero would connect the result to the unperturbed bi-Laplacian inverse problem; the author keeps $m\\neq0$ fixed and does not discuss the limit."],"forward_implications":["If Theorem 1.1 is correct, the assignment $(M,g)\\mapsto L_g^{-1/2}$ restricted to any open patch is injective up to isometry: no two non-isometric closed manifolds can have matching internal data.","Theorem 1.4 extends the same rigidity to the operator $A_g=L_g^{1/2}-mI$ and its Cauchy data set of solutions to $A_gu=0$.","The result gives an order-2 example in the nonlocal Calderón program, whose earlier operators had orders in $(0,2)$.","Together with the heat-kernel rigidity theorem, the argument shows that local information about a nonlocal inverse problem determines global topology and geometry, not just the metric on the observed patch."],"supporting_citations":[{"why":"Supplies Theorem 3.6, the heat-kernel rigidity result that turns equality of Laplace heat kernels on a common open set into a global isometry.","marker":"[20]"},{"why":"Supplies the Kannai transmutation formulas (27) and (35), the bridge from the bi-Laplace to the Laplace heat semigroup.","marker":"[34]"},{"why":"Supplies the pointwise estimate for the biharmonic heat kernel used to get the decay bounds in the moment argument.","marker":"[32]"},{"why":"Supplies the Paley-Wiener theorem used to conclude that the function φ vanishes from its vanishing Fourier-Laplace transform.","marker":"[43]"},{"why":"Supplies the long-time semigroup decay estimate used in the integration-by-parts step.","marker":"[52]"},{"why":"Supplies the unique continuation property for the heat equation used to pass from a subdomain to all of the shared open set.","marker":"[41]"},{"why":"Establishes that fractional powers of the Laplacian are classical elliptic pseudodifferential operators, locating $L_g^{1/2}$ in $\\Psi^2_{1,0}$.","marker":"[19]"}],"fun_headline_variants":["A tiny patch of operator data recovers the whole manifold's metric","One open patch of operator data determines the manifold's isometry class","Small patch of operator data fixes the full Riemannian metric","Operator inverse on a tiny set reveals the whole manifold's shape","Recover the whole metric from operator data on one patch"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on Kannai's transmutation formulas and an inverse Laplace transform step being valid on closed manifolds under the needed regularity and support conditions; the paper imports these formulas without checking those hypotheses.","fun_headline_variants_meta":{"raw":{"variants":["A tiny patch of operator data recovers the whole manifold's metric","One open patch of operator data determines the manifold's isometry class","Small patch of operator data fixes the full Riemannian metric","Operator inverse on a tiny set reveals the whole manifold's shape","Recover the whole metric from operator data on one patch"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000606,"raw_usage":{"total_tokens":2736,"prompt_tokens":768,"completion_tokens":1968,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":384,"completion_tokens_details":{"reasoning_tokens":1882}},"tokens_in":384,"tokens_out":1968,"duration_ms":12545,"temperature":1.0,"reasoning_tokens":1882,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:38:37.869310+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate equation (27) on a single eigenfunction $\\varphi_k$ of $\\Delta_g$ with $\\Delta_g\\varphi_k=\\lambda_k\\varphi_k$; the identity as printed gives a prefactor $t^{7/6}e^{-\\lambda_k^2 t}$ rather than $e^{-\\lambda_k^2 t}$, so verifying the correct prefactor and weight against Kannai's statement settles whether the bridge from bi-Laplace to Laplace heat kernels is valid.","supporting_citations":[],"review_version":1}