{"id":"2d8217f9-e727-4f41-864e-d022c77f09d9","arxiv_id":"2502.00710","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exterior Dirichlet-to-Neumann data for fractional Laplace-Beltrami operators determine a Euclidean-asymptotic Riemannian metric up to a diffeomorphism fixing the exterior.","lead":"Mathematicians prove that nonlocal boundary measurements taken entirely outside an unknown object determine the object's internal geometry up to a natural coordinate change. The result resolves an open problem from the fractional Calderón program and is proved by two independent methods.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorem relies on an unproved exterior-fixing diffeomorphism claim (Remark 3.6); without it, recovery is only up to a gauge that need not fix the exterior.","rationale":"The reader's conditional verdict is well targeted: the central proof is long and largely self-contained, but the final recovery of the metric up to a diffeomorphism fixing the exterior depends on a theorem that is not proved in the manuscript. I read the paper in good faith and found two candidate weak points: (i) the exterior-fixing property imported via Remark 3.6, and (ii) the compressed derivations in Lemma 3.4 and Proposition 4.25. The first is the most load-bearing because it is exactly the difference between the advertised conclusion and a weaker recovery up to an arbitrary diffeomorphism. The second is a genuine issue of exposition but can likely be repaired by expanding the cited argument; the first cannot be repaired without a new mathematical proof of the modification of [59, Theorem 2]. The presence of two independent proof strategies does not remove this dependence, since both strategies conclude via the same Theorem 3.5. The issue is not that the theorem is likely false; rather, the paper's own Remark 3.6 admits that a key step is outsourced to a private communication. For a result of this significance, that step must appear in the text or in a publicly available reference. A conditional verdict is therefore appropriate: the manuscript should be accepted only after the exterior-fixing claim is fully proved. Since the reader already chose CONDITIONAL with the same underlying concern, my recommendation is to leave that verdict unchanged.","tokens_in":78637,"tokens_out":6093,"duration_ms":68648,"concrete_test":"Replace the one-sentence Remark 3.6 with a complete proof of the exterior-fixing claim, or provide a public reference containing that proof. Concretely, formulate the modified version of [59, Theorem 2] that is claimed, and verify it in the minimal Euclidean trial case (N = R^n, g = δ, O = R^n \\ B(0,1)) by checking that the reconstructed isometry is forced to be the identity on O and cannot be, say, a translation. If the reconstruction procedure from [59] does not force identity on O, then Theorem 3.5 as stated is false and Theorem 1.1 needs a different final step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The final step of both proofs (Section 3.5 and the conclusion of Section 4.2.3) invokes Theorem 3.5, imported from [41, Theorem 1.5], to pass from equality of heat kernels on O = Ωe to the conclusion g1 = Φ*g2 with Φ(x) = x for x ∈ Ωe. However, [41, Theorem 1.5] does not assert the exterior-fixing property. The paper's Remark 3.6 states only that this property follows from a 'small modification' of [59, Theorem 2], communicated privately by T. Saksala, with no proof or even a precise statement of the modified theorem. This is load-bearing because the diffeomorphism gauge obstruction in the problem is exactly Φ|Ωe = Id: if the constructed Φ is only an isometry mapping Ωe to itself but not fixing it pointwise, Theorem 1.1 is not established. The issue cannot be dismissed from the assumption g1 = g2 on Ωe, since the boundary-control reconstruction must still rule out hidden isometries of (Ωe, g2) that are not the identity, e.g., translations when g2 is Euclidean on an unbounded piece of the exterior. A smaller but related gap is that the key transitions in Lemma 3.4 and Proposition 4.25 are compressed as 'following the argument'; these cite a template and are less severe, but still deserve expansion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the fractional anisotropic Calderón problem with exterior data on R^n, n ≥ 2, for smooth Riemannian metrics that agree with the Euclidean metric outside a compact set. The main result, Theorem 1.1, asserts that the partial exterior Dirichlet-to-Neumann map Λ_g^{W_1,W_2}, measured on arbitrary nonempty open sets W_1,W_2 in the exterior Ω^e, determines the metric up to a C^∞ diffeomorphism fixing Ω^e, under the a priori assumption that the two metrics agree on Ω^e. Two proofs are offered. The first uses the heat semigroup representation of the fractional Laplace–Beltrami operator, a pseudodifferential analysis of (−Δ_g)^α, and a boundary-control reduction to equality of heat kernels. The second uses a variable-coefficient Caffarelli–Silvestre-type extension, Vishik–Eskin regularity estimates, and a source-to-solution recovery of the heat kernel. The paper also contains a detailed appendix proving that (−Δ_g)^α is a classical elliptic pseudodifferential operator on R^n, and an appendix with regularity estimates for the mixed Dirichlet–Neumann extension problem.","tokens_in":78874,"tokens_out":4273,"duration_ms":45705,"significance":"If the main theorem is correct, it resolves a notable open problem: the fractional anisotropic Calderón problem with exterior data in the noncompact Euclidean setting, with the expected diffeomorphism gauge that fixes the exterior. The paper is unusually detailed: it gives two independent proof strategies, a complete proof of the pseudodifferential nature of the fractional Laplace–Beltrami operator, explicit heat-kernel estimates, and a full treatment of the reduction from partial exterior Dirichlet-to-Neumann data to equality of heat kernels in the exterior. The authors are explicit about the two possible routes (heat semigroup and degenerate elliptic extension), and Appendix B is a genuine contribution that may be of independent interest. The main caveat is that the final exterior-fixing step of the uniqueness argument is imported from a private communication without proof; this is load-bearing for the statement of Theorem 1.1. With that gap closed, the paper would be a significant advance in the fractional Calderón theory.","major_comments":[{"comment":"The exterior-fixing conclusion Φ(x)=x for x∈Ω^e is load-bearing for Theorem 1.1, but it is not proved in the manuscript. Theorem 3.5 is quoted from [41, Theorem 1.5], and Remark 3.6 admits that [41, Theorem 1.5] did not contain the pointwise fixing claim and that it follows only from a 'small modification' of [59, Theorem 2] communicated privately by T. Saksala. No statement of the modified theorem, no proof, and no public reference are given. Because the gauge obstruction is exactly diffeomorphisms that may move points in Ω^e while preserving the equality g_1=g_2 there, the current text does not establish Theorem 1.1 as stated. This issue affects both proofs, since Section 3.5 and the conclusion of Section 4.2.3 both invoke Theorem 3.5. Please add a complete proof, or a precise statement with a citable public source, of the exterior-fixing version of the heat-kernel rigidity result.","section":"Section 3.5 / Remark 3.6"},{"comment":"The key new implication L_{g_1}^{Ω^e,Ω^e}=L_{g_2}^{Ω^e,Ω^e} ⇒ e^{tΔ_{g_1}}(x,y)=e^{tΔ_{g_2}}(x,y) for all x,y∈Ω^e is not actually proved. After equations (3.54)–(3.55), the text says 'Following the argument in the proof of Lemma 3.1 (see also [41]), we complete the proof.' The step from the integrated moment identities to pointwise equality of heat kernels requires a Mellin-inversion argument and a passage from identities tested against smooth compactly supported F to pointwise identities away from the diagonal; this is the central new reduction of the heat-semigroup proof. Please expand the argument, or give an explicit reduction to a stated lemma with all hypotheses verified.","section":"Section 3.4 / Lemma 3.4"},{"comment":"The final paragraph of Proposition 4.25 repeats the same omission in the second proof: after deriving the moment equalities, it concludes with 'Relying on an argument as in (3.33) and following, this then implies...' that the heat kernels agree pointwise. Since this is the concluding step of the second proof of Theorem 1.1, the moment-to-pointwise step should either be proved directly or explicitly reduced to a fully stated and proved lemma, for example to a completed version of Lemma 3.4.","section":"Section 4.2.3 / Proposition 4.25"}],"minor_comments":[{"comment":"In the first paragraph of the introduction, the sentence 'Then the n( Rn,g ) is a complete Riemannian manifold' appears to contain a typo ('the n'); it should probably read 'Then (R^n,g) is a complete Riemannian manifold.'","section":"Introduction"},{"comment":"In the final display of Case I, the unique continuation conclusion is written as 'U^{(1)}(t,x)=U^{(2)}(t,x)=0'; the right-hand side should be 'U^{(1)}(t,x)=U^{(2)}(t,x)', since the two heat evolutions are being shown equal, not zero.","section":"Section 3.1, proof of Lemma 3.1, Case I"},{"comment":"The justification of the convergence in (4.22) is very compressed: it cites continuity of the operators involved, but the H^{-α} continuity estimate from (4.10) applies to the generalized Dirichlet-to-Neumann maps, not directly to the normal derivative. A short indication of the approximation argument would improve readability.","section":"Section 4.1.4 / Remark 4.16"},{"comment":"The paper repeatedly says 'following the argument' or 'see also [41]' at crucial transition points. Given the length of the paper, it would be preferable to state the exact lemma being used, especially in Lemma 3.4, Proposition 4.25, and the reduction from heat-kernel equality to the metric in Section 3.5.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The dependence of the main theorem on an unproved statement communicated privately by T. Saksala should be resolved before publication: either a full proof should be included in the appendix, or a public reference should be supplied. The authors should also confirm with Saksala that the attribution is appropriate. Apart from this load-bearing gap, the paper is substantial, well organized, and likely correct in its main reduction; the two independent proofs and the detailed pseudodifferential appendix are genuine strengths."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is right: this is a serious solution of the fractional anisotropic Calderon problem with exterior data in all dimensions n≥2 and all α∈(0,1), for smooth Euclidean-asymptotic metrics. That was an open problem from Ghosh–Salo–Uhlmann, and the paper closes it. Two independent proof strategies are given, the pseudodifferential/heat semigroup route and the Caffarelli–Silvestre extension route. The authors earn credit for the reduction from partial exterior DtN data to heat kernel equality, which is the genuinely new technical core: the Vishik–Eskin estimates, the passage through full exterior DtN maps, and the exterior source-to-solution recovery are all substantial and mostly fleshed out. The appendices, especially the pseudodifferential proof that (−Δ_g)^α is classical elliptic, are detailed and useful on their own. On the citation pattern: yes, several prior papers share authors, but the imported results are independent published or preprint theorems with derivations, and the target theorem is not in them. No circularity red flag.\n\nThe soft spots are exactly where the stress-test lands. Theorem 3.5, the step from heat kernel equality on Ω^e to a diffeomorphism fixing Ω^e, is imported from [41, Thm 1.5] — but that theorem does not include the exterior-fixing conclusion. Remark 3.6 says this follows from a 'small modification' of [59, Thm 2] communicated privately by Saksala. That is load-bearing: the gauge in Theorem 1.1 is exactly Φ|Ω^e = Id, and if the constructed diffeomorphism only maps Ω^e to itself without fixing it pointwise, the main theorem does not follow. The assumption g1=g2 on Ω^e does not rule out hidden exterior isometries, e.g. translations on an unbounded Euclidean piece. The compressed 'following the argument' steps in Lemma 3.4 and Proposition 4.25 are smaller, but they sit inside the same recovery chain, so they should also be expanded or replaced by explicit references in a revision.\n\nI disagree with the stress-test only on one nuance: the claimed equivalence in Remark 1.3 is explicitly flagged as not needed and left for future work, so that is not a gap in the proof. The rest of the critique holds up. In proportion: the overall architecture is coherent, the mass of the proof is present, and the missing pieces look fillable rather than fatal. This paper deserves a serious referee: a capable referee can check the private-communication claim against [59] and either confirm the modification or find a counterexample, and can ask for the compressed steps to be expanded. I would not desk-reject this. Recommend engage, with the exterior-fixing issue as the primary request.","headline":"Solves a real open problem with two serious proofs; the main unresolved thread is a load-bearing exterior-fixing diffeomorphism claim that currently rests on a private communication.","tokens_in":79453,"tokens_out":802,"would_cite":true,"duration_ms":11920,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35R11","35S15","58J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Partial exterior Dirichlet-to-Neumann data for the fractional Laplace–Beltrami operator determine a smooth Riemannian metric up to an exterior-fixing diffeomorphism.","keywords":["fractional Calderón problem","anisotropic inverse problems","fractional Laplace–Beltrami operator","partial exterior Dirichlet–to–Neumann map","heat kernel recovery","elliptic extension","transmission-condition regularity","unique continuation"],"falsifier":"Search for two smooth complete Riemannian metrics on a connected manifold and an open set $O$ such that $e^{t\\Delta_{g_1}}(x,y)=e^{t\\Delta_{g_2}}(x,y)$ for all $t>0$ and $x,y\\in O$ but no diffeomorphism fixing $O$ maps $g_1$ to $g_2$; such a pair would disprove the imported Theorem 3.5 and, with it, the exterior-fixing conclusion of Theorem 1.1.","tokens_in":78434,"feed_emoji":"📐","tokens_out":6907,"duration_ms":66417,"temperature":0.7,"pith_summary":"This paper is trying to settle the fractional analogue of the anisotropic Calderón problem when measurements are made only outside a bounded domain in Euclidean space. Its central claim is that partial exterior Dirichlet-to-Neumann data for the fractional Laplace–Beltrami operator always determine the Riemannian metric up to a diffeomorphism that fixes the exterior, and nothing else. If true, this means that nonlocal exterior measurements carry the same geometric information as boundary measurements on a closed manifold, and that diffeomorphism gauge invariance is the only obstruction to uniqueness. The result is shown by two independent routes, one via heat-semigroup and pseudodifferential arguments and one via a variable-coefficient elliptic extension.","feed_headline":"Fractional exterior data pin the metric up to diffeomorphism","feed_subtitle":"Two proofs show outside measurements of the fractional Laplacian recover the Riemannian metric up to a gauge.","key_machinery":"The load-bearing object is the fractional Laplace–Beltrami operator $(-\\Delta_g)^\\alpha$, defined by functional calculus, together with its partial exterior Dirichlet-to-Neumann map $\\Lambda_g^{W_1,W_2}$, which records the restriction to $W_2$ of $(-\\Delta_g)^\\alpha$ applied to the unique energy solution with exterior datum supported in $W_1$. The key identity is the heat-semigroup representation $(-\\Delta_g)^\\alpha = \\frac{1}{\\Gamma(-\\alpha)}\\int_0^\\infty (e^{t\\Delta_g}-1)\\,t^{-1-\\alpha}\\,dt$, combined with a unique-continuation argument that upgrades equality on $W_2$ to equality of the heat kernels on the entire exterior. The second proof replaces this representation with a mixed Dirichlet–Neumann problem in one extra dimension whose trace is the fractional Laplacian, and uses transmission-condition regularity estimates for those solutions to replace the unique-continuation step.","core_discovery":"The paper proves Theorem 1.1: for $n\\ge 2$, smooth metrics $g_1,g_2$ on $\\mathbb{R}^n$ that agree with the Euclidean metric outside a compact set and agree on the exterior $\\Omega^e$ are forced to satisfy $g_1=\\Phi^*g_2$ on all of $\\mathbb{R}^n$ for some $C^\\infty$ diffeomorphism $\\Phi$ with $\\Phi(x)=x$ for $x\\in\\Omega^e$, whenever the partial exterior Dirichlet-to-Neumann maps $\\Lambda_{g_1}^{W_1,W_2}$ and $\\Lambda_{g_2}^{W_1,W_2}$ coincide on any nonempty open sets $W_1,W_2\\subset\\Omega^e$. The proof first converts the partial exterior data into equality of heat kernels $e^{t\\Delta_{g_1}}(x,y)=e^{t\\Delta_{g_2}}(x,y)$ for all $t>0$, $x,y\\in\\Omega^e$, and then invokes an imported theorem, stated as Theorem 3.5, that heat-kernel equality on an open set of complete manifolds implies such an exterior-fixing isometry. A second proof reaches the same heat-kernel conclusion through the elliptic extension formulation of the fractional Laplacian.","pith_inferences":["The paper conjectures in Remark 1.3 an equivalence between partial exterior Dirichlet-to-Neumann data and exterior source-to-solution data; if that equivalence is established, uniqueness would follow from either type of measurement, and one could test numerically whether the two data sets are interconvertible.","The elliptic-extension proof notes that only $C^{k,\\epsilon}$ metric regularity and $C^{1,\\epsilon}$ domains would be needed; this suggests the same uniqueness should hold with lower regularity, an extension beyond the smooth statement that could be tested directly.","Because the theorem holds for every $\\alpha\\in(0,1)$, one could probe the limit cases $\\alpha\\to0$ and $\\alpha\\to1$; the behavior of the uniqueness statement under these limits is not treated here and could be studied numerically or analytically."],"forward_implications":["Partial exterior Dirichlet-to-Neumann data on any nonempty open sets $W_1,W_2$ determine the full Riemannian metric up to the diffeomorphism gauge fixing the exterior, so small observation windows suffice.","If two metrics in the same conformal class produce equal partial exterior data, the conformal factor must be identically $1$.","Equality of heat kernels on the exterior becomes a full recovery statement: exterior measurements determine the interior geometry completely.","Both the pseudodifferential route and the elliptic-extension route yield the same theorem, so the extension formulation can serve as a primary analytic tool in related fractional inverse problems."],"supporting_citations":[{"why":"Supplies the reduction from equality of heat kernels on an open set to a diffeomorphism recovering the metric, and states the imported Theorem 3.5.","marker":"[41]"},{"why":"Provides the boundary-control theorem whose small modification, mentioned in Remark 3.6, yields a diffeomorphism fixing the exterior.","marker":"[59]"},{"why":"Introduces the fractional Calderón problem with exterior data and the partial exterior Dirichlet-to-Neumann map used throughout the paper.","marker":"[47]"},{"why":"Establishes the constant-coefficient elliptic extension of the fractional Laplacian that underlies the second proof.","marker":"[14]"},{"why":"Provides the variable-coefficient elliptic extension via functional calculus, used in the second proof of Theorem 1.1.","marker":"[100]"},{"why":"Gives the transmission-condition and factorization-index theory needed for the Vishik–Eskin type regularity estimates that rule out boundary concentration of solutions.","marker":"[54]"},{"why":"Supplies the original Vishik–Eskin estimates for elliptic pseudodifferential boundary problems, which the paper adapts to the fractional Laplace–Beltrami operator.","marker":"[104]"},{"why":"Establishes that complex powers of elliptic operators are pseudodifferential operators, supporting Theorem 2.1 on the symbol class of $(-\\Delta_g)^\\alpha$.","marker":"[97]"}],"fun_headline_variants":["Fractional exterior data pin the metric up to isometry","Partial outside data determine the metric via fractional Laplacian","Two proofs: fractional external data recover the Riemannian metric","Fractional Dirichlet-Neumann map fixes the metric up to gauge","Exterior fractional measurements reveal the metric structure"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on an imported theorem, stated as Theorem 3.5 and refined in Remark 3.6, that equality of heat kernels on an open exterior set already forces a diffeomorphism fixing that set; the exterior-fixing refinement is attributed to a private communication and is not proved in this paper, so if that modification fails the conclusion weakens to recovery up to a diffeomorphism that may move the exterior.","fun_headline_variants_meta":{"raw":{"variants":["Fractional exterior data pin the metric up to isometry","Partial outside data determine the metric via fractional Laplacian","Two proofs: fractional external data recover the Riemannian metric","Fractional Dirichlet-Neumann map fixes the metric up to gauge","Exterior fractional measurements reveal the metric structure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000225,"raw_usage":{"total_tokens":1461,"prompt_tokens":939,"completion_tokens":522,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":441}},"tokens_in":555,"tokens_out":522,"duration_ms":5792,"temperature":1.0,"reasoning_tokens":441,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T17:59:31.840880+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for two smooth complete Riemannian metrics on a connected manifold and an open set $O$ such that $e^{t\\Delta_{g_1}}(x,y)=e^{t\\Delta_{g_2}}(x,y)$ for all $t>0$ and $x,y\\in O$ but no diffeomorphism fixing $O$ maps $g_1$ to $g_2$; such a pair would disprove the imported Theorem 3.5 and, with it, the exterior-fixing conclusion of Theorem 1.1.","supporting_citations":[{"cited_title":"Differential Geom., to appear","cited_arxiv_id":null,"evidence_quote":"Supplies the reduction from equality of heat kernels on an open set to a diffeomorphism recovering the metric, and states the imported Theorem 3.5."},{"cited_title":"PDE 13 (2020), no","cited_arxiv_id":null,"evidence_quote":"Introduces the fractional Calderón problem with exterior data and the partial exterior Dirichlet-to-Neumann map used throughout the paper."},{"cited_title":"Partial Differential Equations 35 (2010), no","cited_arxiv_id":null,"evidence_quote":"Provides the variable-coefficient elliptic extension via functional calculus, used in the second proof of Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the transmission-condition and factorization-index theory needed for the Vishik–Eskin type regularity estimates that rule out boundary concentration of solutions."},{"cited_title":"Nauk 20 (1965), 89–152; English translation in: Russian Math","cited_arxiv_id":null,"evidence_quote":"Supplies the original Vishik–Eskin estimates for elliptic pseudodifferential boundary problems, which the paper adapts to the fractional Laplace–Beltrami operator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that complex powers of elliptic operators are pseudodifferential operators, supporting Theorem 2.1 on the symbol class of $(-\\Delta_g)^\\alpha$."}],"review_version":1}