{"id":"677f56bc-f797-4d32-902b-cfcf38b52bce","arxiv_id":"2606.06427","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Uniqueness theorems recover the angular density a of translation-invariant symmetric stable operators from exterior DN maps, using diagonal singularity in overlapping cases and symbol factorization or analytic continuation in separated cases.","lead":"The paper proves uniqueness results for recovering the even angular density of a symmetric stable nonlocal operator from restricted exterior Dirichlet-to-Neumann maps in both overlapping and separated measurement regimes. This advances the mathematical theory of inverse problems for nonlocal equations, with potential relevance to applications where only exterior data is available.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flags that all three results hinge on the regularity classes and the intersection/separation hypotheses; these are not hidden but are the explicit scope of the theorems. No additional load-bearing gap (e.g., circular use of the symbol or unstated boundary correction) is apparent in the central claim.","tokens_in":1785,"tokens_out":269,"duration_ms":18942,"concrete_test":"Re-derive the symbol factorization step for the finite-harmonic case from the expression of Λ_a^{W1,W2} (without assuming the full symbol is already known) and check whether the resulting algebraic system for the harmonic coefficients is square and invertible under the separation condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The three recovery arguments (diagonal singularity extraction for smooth elliptic a when W1 ∩ W2 ≠ ∅; exact symbol factorization for finite-harmonic a when sets are separated; analytic continuation plus far-field asymptotics for real-analytic a) are each conditioned on the stated regularity class and the corresponding geometric relation between W1 and W2. These conditions are explicitly required and appear to be the minimal hypotheses under which the respective techniques apply; no internal inconsistency in the outlined strategy is visible from the claim structure.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper establishes three uniqueness theorems for recovering the even angular density a of a translation-invariant symmetric stable operator L_a (defined via a principal-value integral with kernel a((x-y)/|x-y|)/|x-y|^{n+2s}) from restricted exterior Dirichlet-to-Neumann maps Λ_a^{W1,W2} on subsets W1, W2 of the exterior of a bounded domain Ω. In the overlapping regime W1 ∩ W2 ≠ ∅, the exterior diagonal singularity determines every smooth elliptic a. In the separated regime \bar W1 ∩ \bar W2 = ∅, uniqueness holds for the finite-harmonic class via exact symbol factorization and for real-analytic densities via analytic continuation of the off-diagonal kernel plus far-field asymptotics when the sets lie in the unbounded exterior component.","tokens_in":1894,"tokens_out":523,"duration_ms":17228,"significance":"If the derivations hold, the results are a significant contribution to inverse problems for nonlocal operators. They provide the first explicit recovery statements distinguishing overlapping versus separated exterior measurement regimes, with the symbol-factorization and analytic-continuation arguments offering concrete technical tools. The absence of free parameters or data-fitting steps and the explicit regularity/geometric hypotheses strengthen the claims.","major_comments":[{"comment":"The three recovery arguments are each conditioned on distinct regularity classes for a and geometric relations between W1 and W2; the manuscript should include a clear statement (near the main theorems) confirming that these hypotheses are sharp and that no single argument extends to the other classes without additional work.","section":"Main theorems (statements following the abstract)"},{"comment":"In the separated-regime finite-harmonic case, the exact factorization of the stable symbol must be shown to be unaffected by the principal-value regularization; a brief verification that lower-order terms do not enter the leading symbol would strengthen the uniqueness claim.","section":"Separated-regime factorization argument"}],"minor_comments":[{"comment":"The notation \bar W1 ∩ \bar W2 = ∅ versus W1 ∩ W2 ≠ ∅ should be used consistently in all theorem statements to avoid ambiguity about closure.","section":null},{"comment":"A short remark on the relation between the three regularity classes (smooth elliptic, finite harmonic, real-analytic) would help readers understand why separate arguments are needed.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and will revise the manuscript accordingly.","responses":[{"response":"We agree that an explicit statement on the distinctness of the hypotheses would improve clarity. In the revised manuscript we will insert a short remark immediately after the statements of the three main theorems, noting that each result relies on a specific combination of regularity on a and geometric separation between W1 and W2, and that the arguments do not extend to the remaining regimes without additional technical work.","revision_made":"yes","referee_comment":"[Main theorems (statements following the abstract)] The three recovery arguments are each conditioned on distinct regularity classes for a and geometric relations between W1 and W2; the manuscript should include a clear statement (near the main theorems) confirming that these hypotheses are sharp and that no single argument extends to the other classes without additional work."},{"response":"We will add a brief verification paragraph in the separated-regime section. The principal symbol of the pseudodifferential operator is extracted from the homogeneous degree -(n+2s) kernel; the principal-value regularization modifies only the lower-order terms in the symbol expansion, leaving the leading homogeneous symbol unchanged. This is standard in the calculus of singular integral operators and will be recorded explicitly.","revision_made":"yes","referee_comment":"[Separated-regime factorization argument] In the separated-regime finite-harmonic case, the exact factorization of the stable symbol must be shown to be unaffected by the principal-value regularization; a brief verification that lower-order terms do not enter the leading symbol would strengthen the uniqueness claim."}],"tokens_in":1461,"tokens_out":367,"duration_ms":13790,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this work proves three distinct recovery results for the even angular density a in translation-invariant symmetric stable operators, based on restricted exterior Dirichlet-to-Neumann maps. In the overlapping case the diagonal singularity recovers smooth elliptic a. When the sets are separated it gets uniqueness for finite harmonics via symbol factorization and for real-analytic densities via off-diagonal kernel continuation plus far-field asymptotics.\n\nThe paper does well by keeping the cases cleanly separated and tying each argument to the operator symbol or DN map properties without obvious circular steps or fitted parameters. The conditions on regularity of a and the geometry between W1 and W2 are stated explicitly and match the techniques.\n\nThe soft spot is that the abstract does not show the full derivations, so any gaps in the principal-value handling or the analytic continuation step would only surface in the manuscript. From what is visible the strategy looks consistent and the claims do not reduce to prior results.\n\nThis is for specialists in nonlocal inverse problems. A reader working on fractional operators or exterior Calderón problems would find the regime-specific statements useful. It deserves a serious referee because the results are new and the approach is technically grounded.","headline":"The paper gives three regime-specific uniqueness theorems for the angular density a of stable operators from exterior DN maps, using singularity, factorization, and continuation arguments.","tokens_in":2363,"tokens_out":309,"would_cite":false,"duration_ms":15959,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The restricted exterior Dirichlet-to-Neumann map determines the angular density a of a stable operator under overlap or separation conditions on the measurement sets.","keywords":["stable operators","inverse problems","Dirichlet-to-Neumann map","angular density","nonlocal operators","exterior measurements","uniqueness"],"falsifier":"Finding two distinct smooth elliptic angular densities that produce identical exterior DN maps on overlapping sets, or two different finite harmonic densities with the same separated DN map.","tokens_in":2687,"feed_emoji":"","tokens_out":431,"duration_ms":24344,"temperature":0.7,"pith_summary":"The paper proves that the angular density defining a translation-invariant symmetric stable operator can be recovered from restricted exterior Dirichlet-to-Neumann maps. When the source and observation sets overlap, the diagonal singularity of the kernel identifies any smooth elliptic density. When the sets are separated, uniqueness follows from factorization of the stable symbol for finite harmonic densities and from analytic continuation plus far-field analysis for real-analytic densities. These results matter because they show how to identify the nonlocal interaction kernel using only exterior data, which is practical when interior access is limited.","feed_headline":"Stable angular densities recovered from exterior measurements","feed_subtitle":"Overlapping sets capture the kernel singularity while separated sets use symbol factorization or analytic continuation.","key_machinery":"The restricted exterior Dirichlet-to-Neumann maps Λa^{W1,W2} with data supported in W1 and observed in W2, which carry the recovery via singularity analysis or symbol factorization.","core_discovery":"In the overlapping regime the exterior diagonal singularity determines every smooth elliptic angular density; in the separated regime uniqueness holds for the finite harmonic angular class by exact factorization of the stable symbol and for real-analytic densities via analytic continuation of the off-diagonal kernel and far-field asymptotics.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Recover stable angular densities from exterior data","Angular densities recovered via exterior DtN maps","Stable kernels identified from exterior measurements","Uniqueness for angular densities in separated regimes"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The angular density a must belong to one of the three regularity classes (smooth elliptic, finite harmonic, or real-analytic) and the measurement sets must satisfy the intersection or separation conditions.","fun_headline_variants_meta":{"raw":{"variants":["Recover stable angular densities from exterior data","Angular densities recovered via exterior DtN maps","Stable kernels identified from exterior measurements","Uniqueness for angular densities in separated regimes"]},"model":"grok-4.3","cost_usd":0.00895,"raw_usage":{"total_tokens":4028,"prompt_tokens":681,"num_sources_used":0,"completion_tokens":50,"cost_in_usd_ticks":89499500,"prompt_tokens_details":{"text_tokens":681,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3297,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":681,"tokens_out":50,"duration_ms":27633,"temperature":1.0,"reasoning_tokens":3297,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T00:24:31.579384+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding two distinct smooth elliptic angular densities that produce identical exterior DN maps on overlapping sets, or two different finite harmonic densities with the same separated DN map.","supporting_citations":[],"review_version":1}