{"id":"676fb333-6ec8-48ed-ae55-75b01444d76e","arxiv_id":"2607.22079","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A countable family of scaled exterior measurements of the fractional Schrödinger obstacle problem determines the nonnegative potential throughout the domain.","lead":"Exterior measurements of the fractional Schrödinger equation with an obstacle determine the unknown potential, even though the contact region inside is unknown. A countable family of scaled exterior data is enough to recover the potential globally when it is nonnegative.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof is coherent and the external strong-maximum-principle input appears appropriately applied.","rationale":"The reader's weakest assumption correctly identifies Lemma 5.2 as the ingredient that guarantees geometric coverage rather than algebraic recovery. However, after tracing the proof, the reliance on [JW19] is justified: the operator satisfies the hypotheses of a standard strong maximum principle for nonlocal operators, and the equation h=0 vs h>0 dichotomy is exactly what the coverage argument needs. The remaining steps (antilocality, subtraction of equations, measurable UCP, continuous-potential branch, countable-union coverage) are internally consistent and each relies on explicitly stated, standard theorems. The paper does not claim to prove [JW19] or the measurable UCP, and in the intended readership these are accepted tools. The only point worth a careful check is whether [JW19] is stated for H^s weak solutions or only for bounded solutions; the manuscript gives no additional argument. Since this is a technical verification rather than a discovered contradiction, I do not see a reason to downgrade the verdict. The reader's ACCEPT with moderate confidence remains appropriate; a stronger verdict would require formal verification, which the reader already did not claim.","tokens_in":25505,"tokens_out":23143,"duration_ms":220850,"concrete_test":"Inspect [JW19, Theorem 1.1] and verify that its hypotheses cover weak solutions h∈H^s(R^n) (possibly unbounded) of (-Δ)^s h = -q h with q∈L∞(Ω), q≥0, and nonnegative exterior data. If the theorem instead requires h∈L∞(R^n), test whether the strong positivity still follows by approximating f0 with truncated L∞∩H^s data, solving for h_k, and checking that the H^s limit h satisfies h>0 a.e. via the strong maximum principle for each h_k.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most external ingredient is Lemma 5.2, which asserts h>0 a.e. for the unconstrained solution of ((-Δ)^s+q)h=0 with nonnegative exterior datum f0, imported from [JW19, Theorem 1.1]. The coverage theorem (Prop 5.4) and hence the global uniqueness theorem (Thm 5.5) depend on this. I checked the surrounding argument: Lemma 5.1 gives h≥0, the equation is (-Δ)^s h = -q h with -q∈L∞(Ω), -q≤0, and the boundary data are nonnegative and nontrivial. [JW19] is a standard strong maximum principle for nonlocal operators of this type, and it is routinely cited for fractional Schr\\\"odinger operators with L∞ potentials. The comparison lemma 5.3 and the countable-family coverage argument then work as written. No internal inconsistency or missing step was found. The only residual risk is a technical mismatch if [JW19, Theorem 1.1] is stated for bounded (or viscosity) solutions rather than weak H^s solutions, which can be unbounded. If so, a density argument is needed, but this is a verification detail, not a flaw in the central claim. The measurable UCP restriction s∈[1/4,1) is explicitly handled by the continuous-potential branch for s<1/4, and the statement of Theorem 5.5 is properly conditional on the cited theorems.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper treats the inverse obstacle problem for the fractional Schrödinger operator ((-Δ)^s+q), 0<s<1, in a bounded domain Ω. For a fixed obstacle ψ and exterior Dirichlet datum f, the state is forced to stay above ψ, and the measured quantity is the nonlinear exterior Dirichlet-to-Neumann map Λ_{q,ψ}(f)|_W = (-Δ)^s u|_W on a nonempty open exterior set W. The main theorem (Theorem 5.5) states that for nonnegative potentials q1,q2∈L∞(Ω), with either s∈[1/4,1) or q1,q2 continuous, equality of Λ_{q1,ψ}(t f0)|_{W2} and Λ_{q2,ψ}(t f0)|_{W2} for all t∈Q^+ (with one fixed nontrivial nonnegative C_c∞ exterior datum f0) implies q1=q2 a.e. in Ω. The proof has four steps: (i) antilocality shows a single equality of DN data forces the two obstacle states to coincide throughout R^n; (ii) subtracting the equations in the common noncontact set gives (q1-q2)u=0; (iii) measurable UCP (for s≥1/4) or continuity plus standard UCP (for continuous potentials) recovers q on the exposed set; (iv) a coverage theorem uses the nonlocal strong maximum principle to show that rational positive scalings of a nonnegative exterior datum expose all of Ω up to a Lebesgue null set. The paper also develops the direct obstacle problem in detail: well-posedness in the \tilde H^s setting, capacitary formulation, an abstract Lewy–Stampacchia estimate, and a sufficient condition for openness of the noncontact set under compatible smooth data.","tokens_in":25761,"tokens_out":11992,"duration_ms":122080,"significance":"If correct, this is a significant new result in inverse problems for nonlocal operators. It provides the first global uniqueness theorem for a fractional inverse obstacle problem that avoids linearization, free-boundary regularity, and strict-complementarity conditions; only a countable one-parameter family of nonlinear measurements is required. The argument is modular and transparent, reducing each step to a clearly identified external theorem (GSU antilocality, GRSU measurable UCP, JW19 strong maximum principle). The paper also makes a useful technical contribution by setting up the capacitary obstacle problem and using a Lewy–Stampacchia estimate to obtain openness of the noncontact set. The main limitations—nonnegative potentials, smooth compatibility of the data, and the restriction s∈[1/4,1) in the rough-potential branch—are stated explicitly. I found no circularity and no fitted parameters; the result is genuinely conditional on the cited theorems, as is normal for this area.","major_comments":[],"minor_comments":[{"comment":"The proof invokes [JW19, Theorem 1.1] for a weak H^s solution h, but the cited strong maximum principle is often stated for bounded (or at least locally bounded) solutions. In the setting of Theorem 5.5, h is in fact bounded: since f0∈L∞ and q≥0, the maximum principle (as in Lemma 5.1) applied to ∥f0∥∞−h gives 0≤h≤∥f0∥∞. Adding this one sentence would make the application fully transparent.","section":"Lemma 5.2"},{"comment":"The paper is commendably explicit about the range restriction s∈[1/4,1) for the L∞ measurable UCP and provides a continuous-potential alternative for s<1/4. In Theorem 5.5, the phrasing 'either s∈[1/4,1) or q1,q2 have continuous representatives' could be misread as requiring s≥1/4 in the continuous branch; a short parenthetical clarifying that the continuous branch works for all s∈(0,1) would prevent confusion.","section":"Remark 3.3 / Theorem 5.5"},{"comment":"The reference list contains duplicates: [GSU20a] and [GSU20b] are the same paper, as are [FKU24a] and [FKU24b]; [GRSU18] and [GRSU20] are the preprint and published versions. Consolidating these entries would improve readability.","section":"References"}],"recommendation":"accept","confidential_remarks":"The paper is well within the scope of the journal and the result is of high quality. The only caveat I weighed was whether the external strong maximum principle [JW19, Theorem 1.1] applies to the H^s weak solution h used in the coverage theorem; on reading the paper, boundedness of h follows immediately from the data assumptions and the maximum principle, so this is a verification detail rather than a flaw. I recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper proves the first global uniqueness theorem for an inverse obstacle problem for the fractional Schrödinger operator. A countable one-parameter family of nonlinear exterior measurements determines a nonnegative L∞ potential. The proof is transparent and the analytical foundations are spelled out in unusual detail. I think it deserves a serious referee.\n\nWhat is new: the inverse obstacle formulation itself, and especially the coverage theorem (Prop 5.4). The single-state step—equality of one exterior DN measurement forces equality of the two states via the known antilocality theorem—is a direct application of Ghosh–Salo–Uhlmann, and the paper says so. The genuinely new machinery is the comparison and strong-positivity argument showing that rational scalings of one nontrivial nonnegative exterior datum expose the whole domain up to null set. That is a real new ingredient and it is clean.\n\nWhat the paper does well: the direct problem is set up carefully with capacity, quasicontinuous representatives, and a Lewy–Stampacchia route to openness of the noncontact set under smooth data. The hypotheses in the main theorem are explicit, and the authors are honest about the branching: measurable UCP for L∞ potentials in s≥1/4, continuous-potential branch for s<1/4. I found no circular use of the target result and no fitted parameters.\n\nSoft spots, in proportion: the coverage theorem rests on Lemma 5.2, strong positivity of the unconstrained solution, imported from Jarohs–Weth. That is a genuinely different ingredient from unique continuation—it guarantees geometric exposure rather than algebraic recovery. As the stress test notes, the only residual risk is a technical mismatch if [JW19] is stated for bounded or viscosity solutions rather than weak H^s solutions, which might need a density argument. That is a verification detail, but the referee should check it. The other caveat—measurable UCP limited to s≥1/4 for rough potentials—is explicit and handled. The reliance on cited theorems by overlapping authors is normal; the citations themselves do not contain the obstacle result.\n\nWho this is for: anyone working on nonlocal inverse problems or fractional obstacle problems. The paper is well written and the proof is easy to follow once the setup is accepted.\n\nRecommendation: send it to peer review. I would accept it with moderate confidence. If I were refereeing, my main question to the authors would be to confirm that the strong maximum principle applies to the weak H^s setting used here, and if not, to add the density argument.","headline":"New global uniqueness for a fractional Schrödinger inverse obstacle problem via a clever countable-exposure argument; the proof is sound but rests on an imported strong maximum principle worth verifying.","tokens_in":26261,"tokens_out":2728,"would_cite":true,"duration_ms":27735,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35J62","35J70"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a countable one-parameter family of nonlinear exterior obstacle measurements determines the unknown potential of a fractional Schrödinger equation everywhere in the domain, without knowing the contact set.","keywords":["fractional Schrödinger equation","obstacle problem","inverse problems","unique continuation principle","Dirichlet-to-Neumann map","nonlocal operators","free boundary","coefficient recovery"],"falsifier":"Compute the unconstrained solution h of (-Δ)^s h + q h = 0 in Ω, h=f0 outside Ω, for a smooth nonnegative potential q and a nontrivial nonnegative smooth exterior datum f0. If h vanishes on a positive-measure subset of Ω, the coverage step collapses. Alternatively, reconstruct q from a single exterior measurement by inverting the state and forming the quotient q=-( (-Δ)^s u )/u; a positive-measure disagreement with the true q in the noncontact set would falsify single-state recovery.","tokens_in":25336,"feed_emoji":"📡","tokens_out":8329,"duration_ms":82469,"temperature":0.7,"pith_summary":"This paper establishes that for the fractional Schrödinger operator (-Δ)^s+q with an obstacle constraint, an unknown nonnegative potential can be recovered globally from exterior measurements without locating the free boundary. The mechanism is the nonlocal unique continuation property: equality of one obstacle measurement on an exterior open set forces the two obstacle states to agree everywhere in R^n. Subtracting the two equations in their common noncontact set yields the scalar identity (q1−q2)u=0. Unique continuation converts this identity into recovery of q wherever the state is nonzero, and a coverage theorem shows that rational positive scalings of a single nonnegative exterior datum expose the whole domain up to a null set. If correct, this removes free-boundary regularity, linearization, and strict-complementarity conditions that a local analogue would need.","feed_headline":"Countable nonlinear obstacle readings recover the potential","feed_subtitle":"Nonlocal continuation fixes the state globally, so the hidden contact set does not block recovery.","key_machinery":"The central object is the nonlinear exterior obstacle Dirichlet-to-Neumann map Λ_{q,ψ}(f)|_W=(-Δ)^s u|_W, where u is the obstacle state constrained by ψ and solving the fractional Schrödinger equation only in the unknown noncontact set. The load-bearing identity is (q1−q2)u=0 in the common noncontact set {u>ψ}, obtained by subtracting the two equations after unique continuation forces u1=u2 everywhere. The second engine is the coverage theorem: for nonnegative potentials, the unconstrained solution h with a fixed nonnegative exterior datum is strictly positive a.e., and the comparison u_t ≥ t h shows that as t runs through positive rationals the sets {u_t>ψ} cover Ω up to a null set. This re","core_discovery":"The central claim is that equality of the nonlinear obstacle Dirichlet-to-Neumann maps Λ_{q1,ψ}(t f0)|_{W2}=Λ_{q2,ψ}(t f0)|_{W2} for every t in the positive rationals, with nonnegative potentials q1,q2 in L^∞(Ω), a smooth obstacle ψ, and a nontrivial nonnegative smooth exterior datum f0, forces q1=q2 almost everywhere in Ω. The proof first shows that a single such measurement is enough to conclude u1=u2 in all of R^n: the states agree outside Ω, equality of the measurements gives (-Δ)^s(u1−u2)=0 on an exterior open set, and nonlocal unique continuation forces global equality. In the common noncontact set {u>ψ} the two equations subtract to (q1−q2)u=0. For s∈[1/4,1) the measurable unique cont","pith_inferences":["Editorial inference: the proof isolates three reusable ingredients — exterior unique continuation, a comparison principle against the unconstrained solution, and strong positivity — so the same countable-family strategy should transfer to other nonlocal elliptic operators that satisfy these three properties.","Editorial inference: if a measurable unique continuation theorem is later proved for s<1/4 and general L^∞ potentials, Theorem 5.5 would extend verbatim to the full range 0<s<1; the paper itself flags this dependence.","Editorial inference: the identity (q1−q2)u=0 replaces linearization around an unknown free boundary by division by the state itself, which suggests a general template for inverse obstacle problems — a testable next step is whether smooth perturbations of the obstacle datum can be used instead of amplitude scaling to achieve coverage."],"forward_implications":["A single obstacle measurement determines the potential in the region exposed by the corresponding state, even though that region is not known a priori.","Global uniqueness holds without any information about the contact set, and without differentiating the nonlinear measurement map with respect to the datum.","The recovery is constructive in principle: after reconstructing the state u from the exterior measurement, q is recovered a.e. in the noncontact set by the quotient q=-( (-Δ)^s u )/u.","Countably many rational scalings of one fixed exterior datum suffice, replacing a continuum of boundary amplitudes with a countable measurement protocol.","For continuous potentials the result covers all s∈(0,1), while for rough L^∞ potentials it covers s∈[1/4,1) under the extra uniqueness-continuation input used by the paper."],"fun_headline_variants":["Fractional Schrödinger obstacles expose the potential","Nonlocal continuation recovers potential from obstacle maps","Countable obstacle measurements determine the potential","Hidden contact set no match for nonlocal uniqueness","Nonlinear obstacle data force global potential recovery"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The global conclusion rests on the strong positivity of the unconstrained solution: for a nonnegative exterior datum and nonnegative potential, the free solution is assumed to be strictly positive almost everywhere in Ω; if a positive-measure zero set existed, scaled obstacle states would fail to expose those points and recovery could leave holes.","fun_headline_variants_meta":{"raw":{"variants":["Fractional Schrödinger obstacles expose the potential","Nonlocal continuation recovers potential from obstacle maps","Countable obstacle measurements determine the potential","Hidden contact set no match for nonlocal uniqueness","Nonlinear obstacle data force global potential recovery"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000995,"raw_usage":{"total_tokens":4079,"prompt_tokens":802,"completion_tokens":3277,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":3210}},"tokens_in":546,"tokens_out":3277,"duration_ms":20866,"temperature":1.0,"reasoning_tokens":3210,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T05:51:26.285289+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the unconstrained solution h of (-Δ)^s h + q h = 0 in Ω, h=f0 outside Ω, for a smooth nonnegative potential q and a nontrivial nonnegative smooth exterior datum f0. If h vanishes on a positive-measure subset of Ω, the coverage step collapses. Alternatively, reconstruct q from a single exterior measurement by inverting the state and forming the quotient q=-( (-Δ)^s u )/u; a positive-measure disagreement with the true q in the noncontact set would falsify single-state recovery.","supporting_citations":[],"review_version":1}