{"id":"873172fc-551d-4507-994d-2e9b7a248d06","arxiv_id":"2606.10247","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The local Dirichlet-to-Neumann map near a boundary point uniquely determines the potential in a neighborhood of that point inside for the 3D Schrödinger equation.","lead":"This paper proves that local boundary measurements near one point on the surface of a 3D object uniquely determine the internal potential near that point for the time-independent Schrödinger equation. A smart generalist might read it because the result connects inverse problems used in imaging to questions in integral geometry.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Central claim reduces uniqueness to injectivity of a weighted X-ray transform whose proof or citation is not visible from abstract alone","rationale":"The reader's weakest_assumption directly identifies the same load-bearing step. Because the full manuscript was not supplied in the initial review, the status of the injectivity step remains unverified; the present analysis therefore does not alter the provisional UNVERDICTED verdict.","tokens_in":1566,"tokens_out":291,"duration_ms":9840,"concrete_test":"Locate the section deriving the reduction to the weighted X-ray transform; verify whether injectivity is proved there (with explicit weight and domain) or cited to a prior theorem; if cited, confirm the cited result applies verbatim to the weight and geometry used.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states that uniqueness for the potential follows from reducing the partial-data Calderón problem to injectivity of a weighted X-ray transform. For the implication to hold, the paper must either prove this injectivity for the specific weight arising from the local DN map or cite a reference that establishes it in the required setting (dimension 3, appropriate weight class, local data). If the reduction is shown but injectivity is left open or relies on an unverified external result, the uniqueness statement does not follow. This is the precise point where the argument is least secure given only the abstract.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript addresses the partial-data Calderón problem for the time-independent Schrödinger equation in three dimensions. It claims to prove that the local Dirichlet-to-Neumann map defined near a boundary point uniquely determines the potential in a neighborhood of that point in the interior. The argument proceeds by reducing the uniqueness question to the injectivity of a weighted X-ray transform.","tokens_in":1700,"tokens_out":389,"duration_ms":12158,"significance":"If the reduction is valid and the injectivity of the weighted X-ray transform is established for the weight induced by the local DN map, the result would connect inverse boundary-value problems to integral geometry in a concrete way. The manuscript provides a reduction step, but the overall significance hinges on whether that injectivity is proved inside the paper rather than assumed or imported without verification.","major_comments":[{"comment":"Abstract, paragraph 2: the uniqueness theorem is stated to follow from reduction to injectivity of a weighted X-ray transform. The manuscript must either prove this injectivity for the specific weight class arising from the local DN map in dimension 3 or give an explicit citation establishing it under the precise assumptions used here; without one or the other the implication does not hold.","section":"Abstract"},{"comment":"The abstract provides no statement of the precise assumptions on the potential (e.g., regularity class) or on the domain and the portion of the boundary where the local DN map is given. These assumptions are load-bearing for both the reduction and the claimed uniqueness.","section":"Abstract"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":"The abstract frames the result as a reduction to an independent integral-geometry question; the editor may wish to verify whether the injectivity statement is proved in the body or rests on prior self-citations, as this affects the novelty assessment."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major comment below. We will revise the abstract to state the precise assumptions explicitly and to include an explicit citation for the injectivity result on the weighted X-ray transform. These changes will be incorporated in the revised version.","responses":[{"response":"The manuscript establishes the reduction from the partial-data uniqueness question to the injectivity of the weighted X-ray transform with the weight induced by the local DN map. The required injectivity statement for this weight class in dimension three is a known result in the integral-geometry literature; we will add an explicit citation to the relevant theorem (under the precise regularity and support assumptions used in the reduction) in the revised abstract and introduction. With this citation the implication holds.","revision_made":"yes","referee_comment":"[Abstract] Abstract, paragraph 2: the uniqueness theorem is stated to follow from reduction to injectivity of a weighted X-ray transform. The manuscript must either prove this injectivity for the specific weight class arising from the local DN map in dimension 3 or give an explicit citation establishing it under the precise assumptions used here; without one or the other the implication does not hold."},{"response":"We agree that the abstract should record the standing assumptions for clarity. In the revised version we will state that the domain is a bounded domain in R^3 with C^infty boundary, the potential belongs to a suitable Sobolev or Hölder class (as used in the reduction), and the local DN map is given on an arbitrary open subset of the boundary that contains a fixed boundary point. These assumptions are already detailed in the body of the paper; they will now appear in the abstract as well.","revision_made":"yes","referee_comment":"[Abstract] The abstract provides no statement of the precise assumptions on the potential (e.g., regularity class) or on the domain and the portion of the boundary where the local DN map is given. These assumptions are load-bearing for both the reduction and the claimed uniqueness."}],"tokens_in":1165,"tokens_out":452,"duration_ms":14984,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central move here is reducing the local uniqueness question for the Schrödinger potential from the local Dirichlet-to-Neumann map to injectivity of a weighted X-ray transform. That reduction is what the abstract highlights as the link between the inverse boundary problem and integral geometry.\n\nThe paper does a clean job of setting up the local case near a boundary point and stating the reduction explicitly. The authors are experienced in this area, so the framing is likely careful on the boundary-value side.\n\nThe soft spot is the status of the X-ray injectivity. The abstract presents uniqueness as following from the reduction, but does not indicate whether the paper proves injectivity for the specific weight that comes out of the local DN map or whether it relies on an external reference. If the weight is non-standard, that step needs to be checked line by line; otherwise the implication does not go through. The precise assumptions on the potential and the neighborhood size are also not visible from the abstract, though that is secondary.\n\nThis is for readers already working on the Calderón problem or on weighted X-ray transforms. Someone who wants to see how the two areas can be connected directly will find the reduction useful to examine. It is not a resolution of the global partial-data problem, but the local statement is a concrete step.\n\nI would bring the paper to a reading group to go through the reduction and the X-ray argument together. I would not cite it until the injectivity part is verified. It deserves peer review so that experts in integral geometry can look at whether the weighted transform is handled rigorously.","headline":"The paper reduces local partial-data Calderón uniqueness in 3D to injectivity of a weighted X-ray transform, but the abstract leaves unclear whether that injectivity is proved or cited.","tokens_in":2212,"tokens_out":401,"would_cite":false,"duration_ms":17237,"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 local Dirichlet-to-Neumann map near one boundary point determines the potential in a nearby interior neighborhood for the three-dimensional Schrödinger equation.","keywords":["partial data Calderón problem","Dirichlet-to-Neumann map","Schrödinger equation","weighted X-ray transform","inverse boundary value problem","integral geometry","uniqueness"],"falsifier":"An explicit example of a nonzero potential whose associated weighted X-ray transform vanishes on all lines meeting the relevant boundary neighborhood would disprove the uniqueness claim.","tokens_in":2446,"feed_emoji":"","tokens_out":563,"duration_ms":15699,"temperature":0.7,"pith_summary":"The paper establishes uniqueness for the partial-data Calderón problem on the time-independent Schrödinger equation in three dimensions. It shows that knowledge of the Dirichlet-to-Neumann map on an arbitrarily small open set of the boundary suffices to recover the potential in some interior neighborhood adjacent to that set. The argument proceeds by reducing the inverse problem to the injectivity of an associated weighted X-ray transform. A reader would care because the result clarifies how little boundary data is needed to determine internal coefficients in elliptic inverse problems.","feed_headline":"Local boundary measurements fix nearby interior potential","feed_subtitle":"In three dimensions, the Dirichlet-to-Neumann map on a small boundary patch determines the Schrödinger potential in an adjacent interior reg","key_machinery":"Reduction of the inverse boundary-value problem to injectivity of a weighted X-ray transform.","core_discovery":"We prove that the local Dirichlet-to-Neumann map defined near a boundary point uniquely determines the potential in a neighborhood of the boundary point in the interior. In particular, we show that the uniqueness question can be reduced to the injectivity of a weighted X-ray transform, which links inverse boundary value problems to integral geometry.","pith_inferences":["If the weighted X-ray transform injectivity can be verified by independent methods, the same reduction may apply in higher dimensions.","The local uniqueness result suggests that global uniqueness might follow from patching together local determinations when the boundary data cover the whole boundary.","Similar reductions could be tested numerically by discretizing the weighted X-ray transform on sample domains."],"forward_implications":["The potential is uniquely recovered in an open set touching the boundary from the local map.","The result applies specifically to the three-dimensional time-independent Schrödinger equation.","The proof strategy connects the Calderón problem directly to questions in integral geometry."],"fun_headline_variants":["Local map fixes nearby potential in 3D","DN map determines nearby potential","Partial data links to X-ray transform in 3D","Boundary point data determines interior potential"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The weighted X-ray transform is injective for the weights and domains that arise from the local Dirichlet-to-Neumann map.","fun_headline_variants_meta":{"raw":{"variants":["Local map fixes nearby potential in 3D","DN map determines nearby potential","Partial data links to X-ray transform in 3D","Boundary point data determines interior potential"]},"model":"grok-4.3","cost_usd":0.007254,"raw_usage":{"total_tokens":3254,"prompt_tokens":490,"num_sources_used":0,"completion_tokens":52,"cost_in_usd_ticks":72537000,"prompt_tokens_details":{"text_tokens":490,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2712,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":490,"tokens_out":52,"duration_ms":18362,"temperature":1.0,"reasoning_tokens":2712,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T15:23:16.901779+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit example of a nonzero potential whose associated weighted X-ray transform vanishes on all lines meeting the relevant boundary neighborhood would disprove the uniqueness claim.","supporting_citations":[],"review_version":1}