{"id":"3c7bf048-1e78-4fc3-82c3-7c59cad8b971","arxiv_id":"2505.18567","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Partial exterior measurements stably determine the fractional conductivity, with logarithmic (resp. log-log) stability when conductivities agree in the exterior (resp. when their difference has compact support).","lead":"This mathematics paper proves the first partial-data stability estimates for a nonlocal version of the electrical impedance tomography problem, the fractional conductivity inverse problem. It shows that small differences in exterior measurements imply small differences in the reconstructed conductivity, with error growing only logarithmically (or log-log) in the measurement noise.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 invokes CRTZ24 Prop. 4.1 under a regularity hypothesis that is not implied by the stated assumption m_i in H^{2s+epsilon,n/s}; a scaling argument shows the relevant Sobolev spaces are not nested.","rationale":"The paper's central contribution is a partial-data stability theory for the inverse fractional conductivity problem. For Theorem 1.1 the route is coherent: Lemma 3.3 and Lemma 3.6 reduce conductivity measurements to Schrodinger measurements, Theorem 3.1 (RS20) gives logarithmic stability from partial data for the reduced potentials, and Lemma 3.8 converts the potential difference back to the conductivity difference. I found no fatal objection in that chain. The product estimate in Lemma 3.4 is misstated as written, but it is easily repaired because the cut-off is C_c^infty, so multiplication by psi is bounded on H^s. The missing verification that zero is not a Dirichlet eigenvalue for the reduced operators also follows from the coercivity of the conductivity form together with the Liouville identity, so it is a fixable omission rather than a fatal flaw.\n\nThe load-bearing concern is in Theorem 1.2. The proof's key step is the estimate ||Lambda_{q1}-Lambda_{q2}|| <= omega(||Lambda_{gamma1}-Lambda_{gamma2}||_W), obtained by combining Theorem 3.1 and CRTZ24 Prop. 4.1. Proposition 4.1 has a regularity hypothesis involving the Besov-style space H^{(2s+epsilon)/theta0, theta0 n/s}, whereas the theorem only assumes H^{2s+epsilon,n/s}. These are not nested: the target space asks for more smoothness and weaker integrability simultaneously, and a scaling computation shows that no embedding H^{2s+epsilon,n/s}(R^n) -> H^{(2s+epsilon)/theta0, theta0 n/s}(R^n) exists for theta0<1. The proof does not acknowledge this mismatch or provide a replacement argument. This affects the log-log claim in Theorem 1.2, one of the two advertised main results.\n\nI want to be clear that this is a proof gap, not evidence that the theorem is false. The stability result may well hold under the stronger CRTZ24 regularity condition, or Prop. 4.1 may admit a proof under the weaker H^{2s+epsilon,n/s} assumption using the same ideas as Lemma 3.2. The omission is of the kind that can be corrected by adding condition (iii') to Theorem 1.2 and rerunning the same argument. Therefore the reader's CONDITIONAL verdict remains the right editorial position; my analysis does not move the verdict. I agree with the reader that the regularity mismatch is a central weak assumption, and I have isolated it as the single most load-bearing gap.","tokens_in":14260,"tokens_out":21375,"duration_ms":176626,"concrete_test":"Fix concrete parameters, e.g. n=3, s=1/4, epsilon=1/4, so Theorem 1.2 assumes m_i in H^{0.75,12}(R^3) while Prop. 4.1 with theta0=1/2 requires m_i in H^{1.5,6}(R^3). To settle the gap, test the embedding claim directly: for any nonzero m in the source space, consider m_lambda(x)=m(lambda x) and compare the homogeneous H^{a,p} and H^{b,q} norms as lambda varies. The ratio scales as lambda^{-(s+epsilon)(1/theta0-1)}, which is unbounded, so the claimed embedding fails. The authors would therefore need either to exhibit a different argument proving Prop. 4.1 with only H^{2s+epsilon,n/s} regularity, or to strengthen assumption (iii) of Theorem 1.2 to the CRTZ24 hypothesis and restate the theorem accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.2 depends on the comparison estimate [CRTZ24, Prop. 5.1], restated as Prop. 4.1, which transfers the measured conductivity DN-map difference to the reduced Schrodinger DN-map difference. Prop. 4.1 requires, for some s/n < theta0 < 1 and some epsilon > 0, that the background deviations satisfy m_i in H^{(2s+epsilon)/theta0, theta0 n/s}(R^n). Theorem 1.2 only assumes m_i in H^{2s+epsilon,n/s}(R^n). These spaces are not comparable in the needed direction: writing a = 2s+epsilon and p = n/s, the target has smoothness b = a/theta0 > a and integrability q = theta0 n/s < p. On the level of homogeneous norms, H^{a,p} scales like lambda^{-n/p+a}, while H^{b,q} scales like lambda^{-n/q+b}, and here b - n/q = (s+epsilon)/theta0 > s+epsilon = a - n/p. Hence no embedding H^{a,p}(R^n) -> H^{b,q}(R^n) exists; the CRTZ24 hypothesis is strictly stronger. The paper does not supply a substitute argument: the sentence 'Repeating the argument in the proof of Proposition 3.7' only yields the H^{delta,n/(2s)}(Omega) bound on q_j, not the required H^{(2s+epsilon)/theta0, theta0 n/s} regularity of m_i. Since Prop. 4.1 is the only bridge from conductivity data to Schrodinger-potential data in Theorem 1.2, the log-log estimate is not established under the stated hypotheses. The omission is probably repairable by strengthening assumption (iii) or by proving Prop. 4.1 under the weaker H^{2s+epsilon,n/s} condition, but as written the proof has a real gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the inverse fractional conductivity problem with partial exterior measurements. The authors prove two stability estimates: a logarithmic bound for conductivities that agree a priori in the whole exterior, and a log-log bound when the difference has compact support and measurements are taken on a single exterior set. The proofs use the fractional Liouville reduction to relate the conductivity DN map to the fractional Schrödinger DN map, then invoke known stability results of Rüland–Salo and of Covi–Railo–Tyni–Zimmermann. The paper also contains a self-contained proof that multiplication by gamma^{1/2} and gamma^{-1/2} preserves the space tilde H^s(U).","tokens_in":14583,"tokens_out":6477,"duration_ms":50402,"significance":"If the stated estimates hold, these are the first partial-data stability results for the fractional conductivity problem with the expected logarithmic and log-log rates, extending the full-data result of CRTZ24. The reduction strategy is elegant and Lemma 3.3 is a useful independent technical contribution. However, the proof of Theorem 1.2 relies on a regularity hypothesis in a quoted proposition that is not implied by the theorem's assumptions, and the non-eigenvalue condition required by the main Schrödinger stability theorem is never checked. Both issues are repairable, but as written the main claims are not fully established.","major_comments":[{"comment":"The proof of Theorem 1.2 invokes Proposition 4.1, quoted from CRTZ24 Proposition 5.1, to bound the Schrödinger DN-map difference in terms of the conductivity DN-map difference. Proposition 4.1 requires the background deviations to satisfy m_i in H^{(2s+epsilon)/theta_0, theta_0 n/s}(R^n) for some s/n < theta_0 < 1, whereas Theorem 1.2 assumes only m_i in H^{2s+epsilon, n/s}(R^n). These spaces are not nested in the needed direction: on the level of homogeneous norms, H^{a,p} scales as lambda^{-n/p+a} and H^{b,q} as lambda^{-n/q+b}, and here b - n/q = (s+epsilon)/theta_0 is larger than a - n/p = s+epsilon, so no embedding H^{2s+epsilon,n/s} into H^{(2s+epsilon)/theta_0, theta_0 n/s} exists. The sentence 'Repeating the argument in the proof of Proposition 3.7' only yields the H^{delta, n/(2s)} bound on q_j, not the required Sobolev regularity of the background deviations m_i. Since Proposition 4.1 is the only bridge from the conductivity data to the Schrödinger-potential data in this proof, the log-log stability estimate is not established under the stated hypotheses. This gap is repairable by either strengthening assumption (iii) of Theorem 1.2 to the CRTZ24 condition or by proving the comparison estimate under the weaker H^{2s+epsilon,n/s} regularity.","section":"Section 3, Proposition 3.7 and Theorem 3.1"},{"comment":"Theorem 3.1, which is the key input for Proposition 3.7, requires that zero is not a Dirichlet eigenvalue of the exterior value problem (-Delta)^s u + q_j u = 0 in Omega with u = 0 in Omega^e, for j = 1, 2. Proposition 3.7 states that 'It suffices to show that the conditions of Theorem 3.1 are met' and then verifies only the H^{delta,n/(2s)} bound on q_j. The non-eigenvalue condition is never stated, checked, or referenced for the reduced potentials q_j = -((-Delta)^s m_j)/gamma_j^{1/2}. Without this condition the DN maps Lambda_{q_j} are not known to be well-defined, and the stability estimate of Theorem 3.1 does not apply. Since the paper aims to use Theorem 3.1 as a black box, the missing verification is a load-bearing gap, though it may be filled by a perturbation argument or by imposing a smallness condition on the potentials.","section":"Section 3, Proposition 3.7"}],"minor_comments":[{"comment":"In the abstract, 'shaper' should be 'sharper'.","section":"Abstract"},{"comment":"The word 'furhter' should be 'further'.","section":"Introduction, page 1"},{"comment":"The typo 'non-emtpy' should be 'non-empty'.","section":"Section 2, Theorem 3.1 statement (page 7)"},{"comment":"The typo 'becuse' should be 'because'.","section":"Section 4, proof of Theorem 1.2 (page 13)"},{"comment":"The estimate ||phi_k^pm psi - gamma^{pm 1/2} psi||_{H^s} <= ||phi_k^pm - gamma^{pm 1/2} chi||_{H^s} ||psi||_{H^s} is not a direct consequence of the H^s multiplication because H^s is not an algebra. Since psi is fixed, smooth, and compactly supported, multiplication by psi is a bounded operator on H^s, so the argument survives with a constant depending on psi. The authors should state this to avoid giving the impression that the displayed inequality holds as written.","section":"Section 3, proof of Lemma 3.4, equation (5)"},{"comment":"The assumption supp(gamma_1 - gamma_2) = Sigma is an equality; if the authors intend only that the support is contained in Sigma, the condition should be stated as supp(...) subset Sigma to avoid measure-theoretic pedantry.","section":"Section 4, Theorem 1.2 condition (i)"},{"comment":"The proof works with p = 2n/(n-2s') for s' < s; the connection to an arbitrary p in the range stated in the theorem should be made explicit by choosing s' such that p < 2n/(n-2s') and then adjusting the constants.","section":"Section 4, proof of Theorem 1.2 (parameter p)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution to the inverse fractional conductivity literature. The main concerns are the regularity mismatch in the proof of Theorem 1.2 and the unchecked non-eigenvalue condition in the application of Theorem 3.1. Both are repairable within the manuscript's scope, so I recommend major revision rather than rejection. The paper is within the journal's scope and the references are appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a genuine step forward. It proves the first partial-data stability estimates for the fractional conductivity equation, with a logarithmic modulus under exterior agreement and log-log under compact support. The Liouville reduction chain is standard and mostly clearly written. The new Lemma 3.3 on multiplication preserving \\tilde H^s is useful, and the reliance on RS20 and CRTZ24 is legitimate — those are independent published results. I agree with the reader's overall assessment: the main ideas are right, and the gaps are repairable.\n\nNow the soft spots. In Proposition 3.7, the proof applies Theorem 3.1 (RS20) but never checks the condition that zero is not a Dirichlet eigenvalue for (−Δ)^s+q_j. That is a genuine hypothesis of the theorem, and the paper just asserts the reduced potentials satisfy the bounds. For a fixed conductivity this may be generic, but the theorem as stated covers all gamma in the class, so the proof is incomplete. This should be fixable by adding the non-eigenvalue assumption to the hypotheses or by an approximation argument.\n\nThe larger problem is in Theorem 1.2. The proof uses CRTZ24, Prop. 5.1 (restated as Prop. 4.1), which requires m_i ∈ H^{(2s+ε)/θ_0, θ_0 n/s} for some s/n<θ_0<1. The theorem only assumes m_i ∈ H^{2s+ε, n/s}. The stress-test note is right: the scaling exponents give (s+ε)/θ_0 > s+ε, so H^{2s+ε,n/s} is strictly larger than the needed space; there is no embedding in the right direction. The sentence “Repeating the argument in the proof of Proposition 3.7” only controls q_j, not m_i in the stronger space. Since this proposition is the only bridge from conductivity data to Schrödinger data in that theorem, the log-log estimate is not established under the stated hypotheses. The likely fix is to strengthen assumption (iii) or prove a variant of the comparison estimate under the weaker condition, but as written the proof has a real hole.\n\nMinor: Lemma 3.4 uses an inequality ‖φψ‖ ≤ ‖φ‖ ‖ψ‖ in H^s, which fails for s