{"id":"b7d139bc-4cf7-49a0-94da-38f9d5d32487","arxiv_id":"1908.11696","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Dirichlet-to-Neumann map for the fractional magnetic Schrödinger equation uniquely determines the magnetic and electric potentials up to a natural gauge.","lead":"This paper proves that exterior measurements for a fractional magnetic Schrödinger equation determine the electromagnetic potentials up to a natural gauge. The result extends the fractional Calderón problem to magnetic fields and includes a random walk interpretation of the nonlocal operator.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's final limiting step is not justified by the stated L2 Runge approximation; the magnetic term is only controlled in a weaker space.","rationale":"After reading the paper in good faith, the central theorem is plausible and supported by a coherent strategy: Alessandrini's identity, WUCP via (p5), and Runge approximation. The paper explicitly flags (p5) as a removable technical assumption; that is a limitation, not an internal defect. The weakest point I find is not (p5) but the limiting procedure after equation (23). The manuscript's Runge approximation is stated and proved only as density in L²(Ω), while the magnetic integrand is naturally an L^r function with r<2, dual to L^{r'} with r'>2. Therefore the phrase 'taking the limit as k→∞' requires either F∈L²(Ω) or a Runge property in a stronger space; neither is proved in the text. This is load-bearing because the recovery of Q and of σ_1-σ_2 both pass through these limits. The same Hahn-Banach argument in Lemma 3.15 appears to give the stronger H^s-density version at little extra cost, so the defect is likely repairable; hence the conditional verdict already given by the reader remains appropriate rather than a rejection. I partially agree with the reader's identification: the reader focused on (p5) and on the WLOG disjoint-open-set step, but the more serious locality issue is the mismatch between the L² Runge approximation and the weak L^r regularity of the magnetic term. Other sections (random walk interpretation, FMCE reduction) do not affect the main theorem, though the Section 6 claim of an analogous inverse problem is indeed an overclaim as no theorem or proof is supplied there. No ad hominem is intended; the concern is strictly about the proof's written justification.","tokens_in":21411,"tokens_out":33654,"duration_ms":327926,"concrete_test":"Repeat Lemma 3.15 with an arbitrary v∈H^{-s}(Ω) to see whether the Runge approximations can be made H^s(Ω)-dense; if the WUCP argument still forces v=0, re-run the limits in Theorem 1.1 with u_i^(k)|Ω→f in H^s(Ω) and verify the magnetic term tends to 0 and F(x)=∫Ω((A_1)_a‖-(A_2)_a‖)(x,y)·∇^s u_1(x,y)dy lies in H^{-s}(Ω). If the stronger Runge property fails, or if F is only in L^r with r<2 and not in H^{-s}, the theorem as stated is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 hinges on passing to limits in equation (23) after localizing to Ω by (p5). The stated Runge approximation property (Lemma 3.15) only gives density of the solution restrictions in L²(Ω), and the proof uses exactly that: u_1^(k)|Ω→1 and u_2^(k)|Ω→f in L². But the magnetic term, written as T_k=∫Ω u_2^(k)(x)[∫Ω ((A_1)_a‖-(A_2)_a‖)·∇^s u_1^(k) dy] dx, is controlled only by the estimate |∫_Ω A·∇^s u_1 dy|_Lr ≤ C ||J_2 A||_{L^{2p}} ||J_2(∇^s u_1)||_{L²}, with r=2p/(p+1)<2 (and r=2n/(n+2s) in the basic estimate). Since r' > 2, convergence of u_2^(k) to f in L² does not imply convergence in L^{r'}, and F=∫_Ω ΔA_a‖·∇^s u_1 dy is not shown to be in L². Consequently, the passage from equation (23) to ∫_Ω f(Q_1-Q_2)=0, and later to the pointwise identity for σ_1-σ_2, is not justified by the stated L²-RAP: a nonzero F supported on a set where the approximating u_2^(k) oscillate in L² but not in L^{r'} could escape the limit. This is a genuine gap in the written proof, though it appears repairable: the same Hahn-Banach/WUCP argument in Lemma 3.15 can likely be run with v∈H^{-s}(Ω), giving density in H^s(Ω); then the magnetic term tends to 0 and F, viewed in H^{-s}, can be tested against H^s limit functions. As written, however, the central uniqueness proof relies on an unjustified L²-limit step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies an inverse boundary value problem for a fractional magnetic Schrödinger equation (FMSE), defined through nonlocal fractional gradient and divergence operators. It proves (Theorem 1.1) that if two admissible pairs (A_i,q_i) produce Dirichlet-to-Neumann maps that agree only on exterior test data supported in an open set W1 and measured on another open set W2, then the pairs are equivalent up to a natural gauge relation ∼. The proof combines an Alessandrini-type integral identity, a weak unique continuation property obtained by reducing to the fractional Laplacian under the compact-support assumption (p5), and a Runge approximation property. The paper also gives a random walk interpretation of the leading term and reduces a fractional magnetic conductivity equation to the FMSE.","tokens_in":21869,"tokens_out":12751,"duration_ms":124046,"significance":"If the proof is completed, the result is a valuable generalization of the fractional Calderón problem to first-order magnetic terms: it would be a global uniqueness statement for the fully fractional magnetic Schrödinger equation, and it correctly identifies the gauge freedom. The operator decomposition in Lemma 3.3, the gauge analysis in Lemmas 3.8–3.10, the reduction of the weak unique continuation property to the known fractional Laplacian case, and the random walk interpretation are concrete strengths. The author is also transparent about the compact-support assumption (p5) and where it is used. The manuscript is not circular: the main theorem is not used as an input, and the external WUCP and well-posedness results are standard tools. However, one limit step in the proof of Theorem 1.1 is not justified as written; the gap appears repairable by strengthening the Runge approximation to H^s.","major_comments":[{"comment":"The passage to the limit after Eq. (23) uses only the L² Runge approximation from Lemma 3.15, but the magnetic term is not controlled in L². For sequences u_1^{(k)}|_Ω → 1 and u_2^{(k)}|_Ω → f in L²(Ω), the term M_k = 2∫_{R^n} (u_2^{(k)}|_Ω)(x) ∫_{R^n} ((A_1)_{a‖}-(A_2)_{a‖})·∇^s u_1^{(k)} dy dx involves the inner integral F_k(x)=∫_{R^n} ((A_1)_{a‖}-(A_2)_{a‖})·∇^s u_1^{(k)} dy, which by the estimate used in Lemma 3.3 lies only in L^r(Ω) with r=2p/(p+1)<2, not in L²(Ω). Since r'>2, convergence of u_2^{(k)} to f in L² does not imply ∫ u_2^{(k)}F_k → 0, and L² convergence of u_1^{(k)} to 1 does not imply F_k → 0 because ∇^s u_1^{(k)} is controlled by the H^s norm, not the L² norm. The same objection applies to the later step in which an arbitrary f∈L² is used to force the inner integral involving σ_1−σ_2 to vanish. This is a load-bearing gap in the written proof. It appears repairable: running the Hahn-Banach argument of Lemma 3.15 with v∈H^{-s}(Ω) should give the Runge approximation property in H^s(Ω), and then the H^s convergence of u_1^{(k)} and u_2^{(k)} makes the magnetic term tend to zero and justifies both limit passages. The proof should be revised accordingly.","section":"§4, proof of Theorem 1.1, passage from Eq. (23) to the conclusions"}],"minor_comments":[{"comment":"The phrase 'Without loss of generality, let W1∩W2=∅' needs justification. If W1 and W2 overlap, one can shrink to disjoint nonempty open subsets, so a short explanatory sentence should be added.","section":"§4, proof of Theorem 1.1"},{"comment":"The proof refers to 'Lemmas 3.18 and 3.19' for the RAP and WUCP, but the correct references appear to be Lemma 3.15 and Lemma 4.1.","section":"§4, proof of Theorem 1.1"},{"comment":"The statement that 'we can consider and solve an analogous inverse problem' for the fractional magnetic conductivity equation is unsupported: no uniqueness theorem or proof is given for the FMCE inverse problem. The sentence should either be removed, marked as a conjecture, or backed by a stated theorem.","section":"§6, final paragraph"},{"comment":"The theorem should specify that W1 and W2 are nonempty open sets; otherwise the statement is vacuous if either set is empty.","section":"Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is likely correct, and the main gap is repairable by upgrading the Runge approximation property from L² to H^s. I recommend major revision rather than rejection. The Section 6 claim about an FMCE inverse problem should be scaled back unless a proof is supplied. The paper is within scope for math.AP."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a real extension of the fractional Calderón program. Covi defines a fractional magnetic Laplacian from nonlocal vector calculus, works out its algebraic structure, proves the relevant Runge approximation and WUCP under a compact-support assumption on A, and gets global uniqueness up to the natural gauge. The gauge analysis is the most interesting part: the FMSE has a gauge ∼ that the local MSE does not, and lacks the gauge ≈ that the local MSE has; Lemmas 3.8–3.10 are clean and make the nonlocal-versus-local contrast precise. The random walk interpretation in Section 5 is a nice bonus, though informal. Citation practice is fine; the earlier fractional conductivity paper is used as a toolbox, not as an input to the main theorem.\n\nThe main soft spot is the final limit in Theorem 1.1. The stated Runge approximation property only gives density of solution restrictions in L²(Ω), but the magnetic term in Alessandrini’s identity is controlled only in a space L^r with r<2 (or equivalently as a functional in H^{-s}), so L² convergence of u_2^{(k)} to f does not justify passing to the limit in (23). As written, the recovery of Q1=Q2 and then σ1=σ2 is not fully proved. I don’t think this is fatal—the same WUCP/Hahn-Banach argument should give density in H^s(Ω), or one can work directly with the H^{-s} pairing—but it needs to be fixed before the proof is complete. This is a genuine gap, not a manufactured one, and I agree with the stress-test note here.\n\nTwo smaller issues. The “without loss of generality W1∩W2=∅” step is asserted without justification; it is fixable by passing to smaller disjoint balls inside the two exterior sets, but as written it needs a sentence. And Section 6 claims an inverse problem for the fractional magnetic conductivity equation is solved, when only a reduction formula is proved; the DN-map correspondence for the FMCE is not established. That is an overclaim, though it does not affect the main theorem.\n\nIf the L²-limit gap is repaired, this is a solid contribution. The result settles a natural open problem, introduces the right operator class, and the main theorem is almost certainly true with the compact-support assumption. I would send it to a serious referee, and after the limit argument and the Section 6 claim are fixed I would be happy to see it published.","headline":"A solid, genuinely new fractional magnetic Calderón result whose written proof has one repairable gap in the final limiting argument.","tokens_in":22336,"tokens_out":3931,"would_cite":true,"duration_ms":42242,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","35R30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the vector and scalar potentials of a fractional magnetic Schrödinger equation are uniquely determined, up to their natural gauge, by Dirichlet-to-Neumann measurements taken on arbitrary open subsets of the exterior…","keywords":["fractional magnetic Schrödinger equation","inverse problems","Calderón problem","non-local operators","gauge invariance","Dirichlet-to-Neumann map","unique continuation","Runge approximation"],"falsifier":"Take a vector potential $A$ supported in $\\Omega^2$ whose antisymmetric-parallel part is nonzero, and compare the exterior DN map of $(-\\Delta)^s_A + q$ with that of $(-\\Delta)^s + q'$ where $q'$ is chosen so that the effective potentials $Q$ match; the paper's identity predicts the two DN maps are identical. A numerical or analytical computation exhibiting any difference between these maps would refute Theorem 1.1's gauge classification, while exact agreement would confirm the load-bearing identity.","tokens_in":21235,"feed_emoji":"🧲","tokens_out":9346,"duration_ms":83880,"temperature":0.7,"pith_summary":"This paper proves that a bounded electromagnetic medium can be fully identified from nonlocal exterior measurements, up to precisely the gauge freedom that the physics forces. The object is the fractional magnetic Schrödinger equation $( -\\Delta)^s_A u + q u = 0$ on a bounded domain $\\Omega$, with exterior Dirichlet data; the paper shows that equality of the associated Dirichlet-to-Neumann maps on arbitrary open subsets of the exterior implies that the two vector potentials and scalar potentials are gauge-equivalent. This extends the fractional Calderón problem, where only a scalar potential was recovered, to first-order magnetic terms, where a gauge obstruction is unavoidable. The result matters because magnetic fractional models describe anomalous diffusion in media with both electric and magnetic structure, and the theorem says that boundary measurements actually determine that structure.","feed_headline":"Unknown fractional magnetic potentials recover up to gauge","feed_subtitle":"Exterior measurements of the nonlocal equation pin down both potentials up to the unavoidable gauge freedom.","key_machinery":"The machinery is Alessandrini's identity for the magnetic fractional Laplacian. The Hamiltonian enters through the magnetic fractional gradient $\\nabla^s_A u = \\nabla^s u + A(x,y)u(x)$; a decomposition of the two-point vector potential into symmetric/antisymmetric and parallel/perpendicular parts shows that only $A_{a\\|}$ couples to the fractional gradient, because $\\nabla^s u$ is a.e. parallel. This yields the identity $\\langle(\\Lambda^s_{A_1,q_1}-\\Lambda^s_{A_2,q_2})f_1,f_2\\rangle = 2\\langle\\int((A_1)_{a\\|}-(A_2)_{a\\|})\\cdot\\nabla^s u_1\\,dy, u_2\\rangle + \\langle(Q_1-Q_2)u_1,u_2\\rangle$, which converts equality of DN maps into a distributional equation for $A_{a\\|}$ and $Q$. The proof then uses the weak unique continuation property, obtained by reducing to the fractional Laplacian under the support assumption, to get the Runge approximation property, which allows arbitrary $L^2$ interior probes; varying the probes isolates $Q$ first, then $\\sigma(x,y)$, pointwise in $\\Omega$. The support condition $\\operatorname{supp}(A) \\subseteq \\Omega^2$ is what forces $\\sigma \\equiv 1$ outside $\\Omega^2$ and makes the reduction to the known WUCP work.","core_discovery":"In the paper's own terms, the discovery is Theorem 1.1: for $n \\ge 2$, $s \\in (0,1)$, and potential pairs $(A_i,q_i)$ in the admissible class $P$, if the DN maps satisfy $\\Lambda^s_{A_1,q_1}[f]|_{W_2} = \\Lambda^s_{A_2,q_2}[f]|_{W_2}$ for all smooth $f$ supported in an open exterior set $W_1$ and all measurements taken in another open exterior set $W_2$, then $(A_1,q_1) \\sim (A_2,q_2)$. The gauge $\\sim$ means that the two operators $(-\\Delta)^s_A + q$ coincide, which Lemma 3.8 identifies explicitly: the measurable content of the vector potential is only its antisymmetric-parallel part $A_{a\\|}$, and the measurable content of the scalar potential is the effective potential $Q = q + \\int |A|^2\\,dy + (\\nabla\\cdot)^s A_{s\\|}$. The proof shows that $Q$ and $A_{a\\|}$ (equivalently the symmetric weight $\\sigma(x,y)$) are uniquely recovered, and that all remaining freedom in $A$ is gauge, not physics.","pith_inferences":["An extension the paper leaves implicit is that the invisible components of $A$ can be chosen arbitrarily and absorbed into a modified scalar potential, so any numerical reconstruction should output only the pair $(A_{a\\|}, Q)$; other choices of $A$ in the same gauge class are artifacts of the gauge, not of the data.","If the flagged support assumption $\\operatorname{supp}(A) \\subseteq \\Omega^2$ is eventually removed, the same detection result should hold for vector potentials supported further out; a natural route is to prove the WUCP for $(-\\Delta)^s_A$ directly rather than by reduction to the fractional Laplacian.","The random walk model suggests a concrete experiment: fit the jump kernel of a particle in a heterogeneous medium and compare the inferred $\\sigma(x,y)$ with a reconstruction from exterior voltage-current-type measurements; the theorem implies the fitted jump kernel should be identifiable from those measurements.","Because the theorem allows the exterior open sets $W_1$ and $W_2$ to be disjoint and arbitrary, the same proof should extend in spirit to remote-sensor configurations, pointing toward nonlocal tomography with separated detectors."],"forward_implications":["Equality of all exterior measurements, taken on arbitrarily small open patches outside $\\Omega$, pins down the whole electromagnetic configuration inside $\\Omega$ up to the gauge $\\sim$; no measurement on the boundary $\\partial\\Omega$ itself is needed.","The DN map determines the antisymmetric-parallel part $A_{a\\|}$ of the vector potential and the effective potential $Q$; the components $A_{s\\|}$, $A_{a\\perp}$, and $A_{s\\perp}$ are invisible except through gauge-equivalent changes of $q$.","The FMSE has a gauge $\\sim$ but no multiplicative gauge $\\approx$, the reverse of the local magnetic Schrödinger equation; any recovery algorithm must fix the $\\sim$-class, and cannot be misled by multiplying solutions by nontrivial exterior-trivial functions.","The fractional magnetic conductivity equation is conjugate to an FMSE, so the same uniqueness theorem solves an analogous inverse problem for FMCE.","The long-jump random walk with position-dependent weight $\\sigma(x,y)$ has the FMSE leading term as its continuous-time limit, so exterior measurements carry information about a particle-jump environment whose conductivity at a point depends on where the walker currently is."],"supporting_citations":[{"why":"Defines the local magnetic Schrödinger inverse problem and its gauge identifiability, the classical result this paper generalizes to $s \\in (0,1)$.","marker":"[32]"},{"why":"Supplies the fractional Calderón problem strategy of integral identity plus WUCP plus Runge approximation on which the proof is built.","marker":"[16]"},{"why":"Provides the weak unique continuation property for the fractional Laplacian with low regularity, the known result used to obtain WUCP for FMSE via the support assumption.","marker":"[37]"},{"why":"Gives unique continuation for fractional Schrödinger equations with rough potentials, supporting the WUCP step and the low-regularity framework.","marker":"[36]"},{"why":"Introduces the fractional conductivity equation and its long-jump random walk with weight $\\gamma^{1/2}$, the model that Section 5 adapts to the magnetic weight $\\sigma(x,y)$.","marker":"[9]"},{"why":"Shows the fractional Laplacian as a continuous limit of a long-jump random walk, the probabilistic template for Section 5.","marker":"[42]"}],"fun_headline_variants":["Fractional magnetic potentials fixed up to gauge","Exterior probes pin down fractional magnetic fields","Nonlocal magnetic potentials: unique up to gauge","Gauge-only ambiguity in fractional magnetic inverse problem","Fractional magnetic Schrödinger: exterior data fixes fields"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the vector potential vanishes outside the product domain $\\Omega \\times \\Omega$ ($\\operatorname{supp}(A) \\subseteq \\Omega^2$), which lets the proof replace the magnetic operator by the usual fractional Laplacian in the exterior and invoke the known unique continuation result; the author explicitly suspects this condition is unnecessary.","fun_headline_variants_meta":{"raw":{"variants":["Fractional magnetic potentials fixed up to gauge","Exterior probes pin down fractional magnetic fields","Nonlocal magnetic potentials: unique up to gauge","Gauge-only ambiguity in fractional magnetic inverse problem","Fractional magnetic Schrödinger: exterior data fixes fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000906,"raw_usage":{"total_tokens":3870,"prompt_tokens":890,"completion_tokens":2980,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":2910}},"tokens_in":506,"tokens_out":2980,"duration_ms":20092,"temperature":1.0,"reasoning_tokens":2910,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:08:55.891928+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a vector potential $A$ supported in $\\Omega^2$ whose antisymmetric-parallel part is nonzero, and compare the exterior DN map of $(-\\Delta)^s_A + q$ with that of $(-\\Delta)^s + q'$ where $q'$ is chosen so that the effective potentials $Q$ match; the paper's identity predicts the two DN maps are identical. A numerical or analytical computation exhibiting any difference between these maps would refute Theorem 1.1's gauge classification, while exact agreement would confirm the load-bearing identity.","supporting_citations":[{"cited_title":"Global identifiability for an inverse problem for the Schr¨ odinger equation in a magnetic field","cited_arxiv_id":null,"evidence_quote":"Defines the local magnetic Schrödinger inverse problem and its gauge identifiability, the classical result this paper generalizes to $s \\in (0,1)$."},{"cited_title":"The Calder\\'on problem for the fractional Schr\\\"odinger equation","cited_arxiv_id":"1609.09248","evidence_quote":"Supplies the fractional Calderón problem strategy of integral identity plus WUCP plus Runge approximation on which the proof is built."},{"cited_title":"The fractional Calder\\'on problem: low regularity and stability","cited_arxiv_id":"1708.06294","evidence_quote":"Provides the weak unique continuation property for the fractional Laplacian with low regularity, the known result used to obtain WUCP for FMSE via the support assumption."},{"cited_title":"Unique continuation for fractional Schr¨ odinger equations with rough potentials","cited_arxiv_id":null,"evidence_quote":"Gives unique continuation for fractional Schrödinger equations with rough potentials, supporting the WUCP step and the low-regularity framework."},{"cited_title":"Inverse problems for a fractional conductivity e quation","cited_arxiv_id":null,"evidence_quote":"Introduces the fractional conductivity equation and its long-jump random walk with weight $\\gamma^{1/2}$, the model that Section 5 adapts to the magnetic weight $\\sigma(x,y)$."},{"cited_title":"From the long jump random walk to the fractio nal Laplacian","cited_arxiv_id":null,"evidence_quote":"Shows the fractional Laplacian as a continuous limit of a long-jump random walk, the probabilistic template for Section 5."}],"review_version":1}