{"id":"ead956ed-dedf-40f3-9a0d-b3b1e966e196","arxiv_id":"2606.29233","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a class of warped-product manifolds, agreement of the boundary Dirichlet-to-Neumann data on any open patch forces agreement of the full Dirichlet-to-Neumann maps.","lead":"This paper proves that for certain curved spaces, if boundary measurements of two geometries agree on a small patch of the edge, they must agree on the entire edge. This 'local-to-global' rule gives a new way to attack inverse problems, where the goal is to recover the inside of an object from surface data.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.5's proof rests on unverified applicability of [4]'s Weyl–Titchmarsh/Volterra estimates to C^m, non-fillable warped metrics; if [4] needs C∞ or a regular fill-in, the chain (5.47)→(6.10)→Prop 4.1 breaks.","rationale":"I read Theorem 1.5's proof line by line. The algebra in §5.3–5.6 and §6 is coherent: (5.10)–(5.13) correctly separate variables, (5.27)/(5.30) give the diagonal representation, Lemma 5.1 and Lemma 6.1 are sound, and the polynomial Weyl factor d_k^{1/2} can be absorbed into θ because ∫θ/t=∞ forces tθ(t)/log t→∞. The Ganguly–Thangavelu theorem is used as a black box in the intended way. The real soft spot is the imported analytic machinery from [4]: the proof of the central decay estimate (5.47) consists of a citation to Lemmas 4.4–4.5 and Prop. 4.6, plus an assertion of self-adjointness/compact resolvent in §5.2 that is justified only in the regular fill-in case. Theorem 1.5 explicitly wants C^m (m≥2) conformal factors and symmetric spaces that may not admit a smooth compact completion, so the cited lemmas' hypotheses must be checked before the central claim is accepted. This is a genuine condition on the argument, not a disagreement with existing consensus. The abstract overclaim (four vs. three results) and Theorem 1.2's reliance on the unstated [18, Prop. 5.6] are secondary issues that do not affect my assessment of the main theorem.","tokens_in":15512,"tokens_out":15622,"duration_ms":133126,"concrete_test":"Check the hypotheses of [4, Lemmas 4.4–4.5 and Prop. 4.6] against c∈C^2 with K=T^d (non-fillable): independently re-derive (5.36) and the kernel bound (5.34) for this case, and verify explicitly whether the DN map defined by the Green identity on the incomplete cone equals the diagonal operator (5.30). If any step requires c∈C∞ or a smooth compact completion of M, then (5.47) has no valid proof and Theorem 1.5 needs an additional regularity/fillability hypothesis or a new derivation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 1.5 is deduced from the decay estimate (5.47) through the diagonalization (5.30)/(6.4) and Proposition 4.1. The proof imports three results from [4] — the Volterra kernel bound (5.34), the Laplace-transform representation (5.36), and the H^δ isomorphism of B ([4, Prop. 4.6]) — and applies them to conformal factors c, \\tilde c that are only C^m (m≥2) on a cone-end manifold M=(0,1]×K that generally has no smooth compact completion (e.g., K=T^d or P^2(C)). The paper does not verify that the hypotheses of these lemmas are satisfied here. Its only justification for the analytic framework (§5.2) is the regular case where r=0 is a removable point of a smooth compact completion; that case is explicitly unavailable for non-fillable K and is not the setting of Theorem 1.5. If [4]'s Volterra estimates require C∞ conformal factors or a smooth fill-in, the decay estimate (5.47) is unsupported, and the later pointwise spectral decay (6.10) — and hence the application of Ganguly–Thangavelu — has no basis. The self-adjointness of Λ_g used in the final propagation step is likewise asserted rather than proved for singular cone ends. This is a regularity/geometric-regime gap, not a disagreement with consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies local-to-global propagation of equality of Dirichlet-to-Neumann maps. Theorem 1.1 shows that under a collar coincidence assumption, equality of local DN maps on a nonempty open boundary subset propagates to the whole boundary component. Theorem 1.2 replaces the collar assumption by an exponential spectral decay assumption on the difference of the global DN maps and invokes spectral unique continuation. Theorem 1.5, the main result, concerns conformally warped metrics g=c(r)^4(dr^2+r^2 g_K) on M=(0,1]×K with K a compact symmetric space; it asserts that a quasi-analytic boundary closeness of the conformal factors, together with local DN equality on any nonempty open O⊂K, forces global DN equality. The proof separates variables, relates the DN eigenvalues to a Weyl-Titchmarsh function, derives a quantitative decay estimate for the difference via Laplace-transform and Volterra bounds imported from [4], and finally applies the Ganguly-Thangavelu quasi-analytic propagation theorem. The paper also advertises a fourth result for compact quasi-analytic manifolds that does not appear in the body.","tokens_in":15775,"tokens_out":19139,"duration_ms":180138,"significance":"If the gaps are closed, the propagation mechanism proposed here is genuinely novel: instead of Carleman estimates or classical unique continuation, the proof uses quasi-analytic decay of spectral projections together with the Ganguly-Thangavelu propagation theorem. Theorem 1.5, if fully justified, gives a quantitatively sharp boundary-closeness condition under which local DN measurements determine the full DN map for a class of cone-end warped metrics, going beyond the exponential regime of the local Borg-Marchenko theory. The paper is clearly organized; Lemma 5.1 is elementary and correct, and the algebraic chain (5.44)-(5.47)-(6.4)-(6.10) is coherent. However, the proof rests on substantial imported results whose hypotheses are not verified in the singular C^m setting, and the advertised fourth result is missing.","major_comments":[{"comment":"The proof of Theorem 1.5 relies on self-adjointness with compact resolvent of Λ_g for C^m cone-end metrics (5.2), but §5.2 justifies this only in the regular case where r=0 is a removable point of a smooth compact completion, via (5.7). The paper explicitly notes this case is unavailable for non-fillable K such as P^2(C). The imports from [4]—the Volterra kernel bound (5.34), the Laplace-transform representation (5.36), and the H^δ isomorphism of B ([4, Prop. 4.6])—are applied to c,\\tilde c∈C^m without verifying their hypotheses. These feed directly into the decay estimate (5.47), which is load-bearing for (6.10) and Theorem 1.5. This gap must be closed by proving the statements in the singular C^m setting or by restricting Theorem 1.5 to a regime where [4] is known to apply.","section":"§5.2, §5.5"},{"comment":"The application of Le Rousseau-Lebeau [18, Prop. 5.6] is incomplete as written. In its standard form this proposition is a quantitative interpolation inequality for eigenfunction sums with a constant C_O depending on the observation set O; to infer Aψ=0 on K from the decay (3.6), the rate ε must exceed C_O. The theorem allows arbitrary ε>0 and the proof does not state the proposition or check the required lower bound. Without this, the step 'all assumptions of Proposition 5.6 are fulfilled' is unjustified. Either Theorem 1.2 should include an explicit lower bound ε>ε_0(K,O), or a separate argument is needed to handle small ε.","section":"§3 (Theorem 1.2)"},{"comment":"The manuscript advertises four local-to-global results, including one for compact quasi-analytic manifolds via Bhowmik-Pradhan, but the body contains only Theorems 1.1, 1.2, and 1.5. No quasi-analytic-manifold theorem is stated or proved, and Remark 1.6 says no such analogue is currently available. The abstract in the full text says 'three' while the abstract supplied with the manuscript says 'four'. This mismatch must be corrected: either add the missing theorem or revise the claims.","section":"Abstract/Introduction"}],"minor_comments":[{"comment":"The absorption of the polynomial factor by 'possibly replacing θ by a smaller function' should be stated as a short lemma, since θ must remain decreasing, positive, and satisfy the divergent integral condition. The step is plausible but deserves a proof.","section":"§6, (6.9)-(6.10)"},{"comment":"Typo in reference [22]: 'Chacteristic classes' should read 'Characteristic classes'.","section":"References"},{"comment":"Typo: 'il (ϕℓ)' should read 'if (ϕℓ)'.","section":"Remark 1.4"},{"comment":"The statement I(ρ)≤e^{-ρθ(ρ)} should include the factor ε (or an explicit constant C_ε), since integrating the pointwise bound over (0,ε) yields ε e^{-ρθ(ρ)}. The final use is harmless, but the displayed estimate is formally missing this factor.","section":"Lemma 5.1"}],"recommendation":"major_revision","confidential_remarks":"The core idea is interesting and probably correct, but the submitted version is not yet publishable. The use of [4] in the singular C^m setting and the use of [18, Prop. 5.6] in Theorem 1.2 need explicit verification; the missing fourth result advertised in the abstract and introduction must also be reconciled. I would be willing to review a revised version that addresses these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the core idea is new: instead of the usual Carleman/unique-continuation machinery, the paper propagates local DN-map equality through quasi-analytic spectral decay, using Ganguly–Thangavelu in the symmetric-space setting. That is a real departure, and it works on paper. Second, the main theorem is not yet airtight. The proof imports Weyl–Titchmarsh and Volterra estimates from the authors' earlier paper [4] and applies them to C^m conformal factors on cone-end manifolds that generally have no smooth compact completion. The paper only justifies the analytic framework in the regular, fillable case, which explicitly excludes settings like K = T^d or P^2(C). If those lemmas require C^∞ or a smooth fill-in, the chain (5.47)→(6.10)→Prop 4.1 breaks. Self-adjointness of Λ_g with compact resolvent is also asserted for the singular ends, not proved. This is the load-bearing soft spot, and it is real, not manufactured.\n\nWhat the paper does well: Theorem 1.5 is new and the deduction is coherent. I checked the key estimates – Lemma 5.1 (the weighted Laplace bound), the symmetric-space projector kernel bound (5.52), the absorption of the polynomial Weyl factor – and they hold. The authors are honest about what they cannot prove: they explicitly say no analogue of Ganguly–Thangavelu is available for general quasi-analytic manifolds, despite the recent Bhowmik–Pradhan work. That honesty counts.\n\nMinor issues, in proportion. The arXiv metadata announces four theorems; the body abstract says three, and the introduction disclaims the fourth. Fix that. More substantively, Theorem 1.2 depends on an unstated Proposition 5.6 from Le Rousseau–Lebeau. If that is the standard interpolation inequality, the exponential rate likely has to exceed a constant depending on the observation open set; the theorem as stated for arbitrary ε > 0 on general K may be stronger than the cited tool actually gives. That needs checking, but it is not the paper's main contribution.\n\nThe heavy dependence on the authors' own [4] is not itself a flaw – the cited lemmas are specific and reproducible – but the onus is on them to verify hypotheses rather than assert applicability.\n\nBottom line: this deserves a serious referee. The mechanism is important enough and the proof is checkable enough that the right response is to send it out, with the expectation of careful revision on the singular-end functional analysis and the Le Rousseau–Lebeau hypothesis. If the regularity gap is patchable, this will be a solid contribution to the inverse boundary value literature.","headline":"A genuinely new spectral propagation mechanism for partial-boundary Calderón problems, with a real regularity gap at the singular end that needs fixing before the main theorem is fully supported.","tokens_in":16426,"tokens_out":1721,"would_cite":true,"duration_ms":17767,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","58J50","35J25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for a class of warped-product manifolds, equality of the local Dirichlet-to-Neumann maps on any nonempty open boundary patch forces equality of the global maps and hence of the metric, under a quasi-analytic boundary","keywords":["Dirichlet-to-Neumann map","local-to-global propagation","inverse boundary value problem","quasi-analytic","warped product metric","Weyl-Titchmarsh theory","compact symmetric space","partial boundary data"],"falsifier":"Take a round sphere $K=S^2$ and two distinct $C^2$ radial conformal factors $c$ and $\\tilde c$ on the unit ball whose derivatives up to order 2 satisfy the quasi-analytic closeness bound with $\\theta(t)=1/\\log t$ but which are not identical (for instance, flat but non-zero difference). If the local DN maps on some small open cap coincide, Theorem 1.5 says the global DN maps must coincide; a numerical or rigorous demonstration that they differ would falsify the claim. Alternatively, a direct check of whether the Weyl-Titchmarsh difference decays like $e^{-\\rho \\theta(\\rho)}$ for such a potential difference would test the k","tokens_in":15208,"feed_emoji":"🎯","tokens_out":8294,"duration_ms":73471,"temperature":0.7,"texified_at":"2026-08-05T21:12:21.021415+00:00","pith_summary":"The paper establishes a local-to-global propagation principle for Dirichlet-to-Neumann (DN) maps: if two metrics yield the same DN map on some nonempty open subset of the boundary, then under suitable conditions they yield the same DN map on the whole boundary component. The authors prove this in three tiers: first when the metrics agree in a collar of the boundary, second under an exponential decay assumption on the spectral projections of the DN map difference, and third—the main result—for warped product metrics over a compact symmetric space when the conformal factors are quasi-analytically close at the boundary. The crucial point is that the propagation does not use unique continuation for PDEs; it is a purely spectral mechanism combining separation of variables, Weyl-Titchmarsh asymptotics, and a quasi-analytic propagation theorem. If correct, this means local boundary measurements determine the full DN map, and thereby the metric, under a quantitatively sharp flatness assumption that allows non-analytic and only $C^m$ conformal factors.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":8964,"prompt_tokens":938,"completion_tokens":8026,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":938,"completion_tokens_details":{"reasoning_tokens":7111}},"feed_headline":"Local boundary patch fixes the full Dirichlet-to-Neumann map","feed_subtitle":"For warped-product manifolds, equality on any open set propagates to the whole boundary under a sharp quasi-analytic flatness condition.","key_machinery":"The central tool is the diagonalization of the DN map on warped products: after separation of variables, $\\Lambda_g$ acts on each eigenspace of the Laplace-Beltrami operator of $K$ by scalar multiplication, with scalar given by a term involving the Weyl-Titchmarsh function $M(-\\rho_k^2)$ evaluated at the shifted eigenvalues $\\rho_k^2=\\lambda_k+\\frac{(d-2)^2}{4}$. The difference of two such functions admits a Laplace-transform representation with a Volterra-type integral kernel (a triangular kernel that encodes the difference of the effective potentials), yielding the decay bound (5.47). The second key ingredient is a pointwise spectral projector estimate valid on symmetric spaces: the sum of squared eigenfunctions at any poi","core_discovery":"The central claim is Theorem 1.5. Let $M=(0,1]\\times K$ where $K$ is a compact Riemannian symmetric space, and let $g=c(r)^4(dr^2+r^2 g_K)$ and $\\tilde g=\\tilde c(r)^4(dr^2+r^2 g_K)$ be two conformally warped metrics. In the logarithmic coordinate $x=-\\log r$, write $f(x)=e^{-x/2}c(e^{-x})$ and similarly $\\tilde f$. Suppose the two conformal factors satisfy the boundary closeness estimate $|f^{(j)}(x)-\\tilde f^{(j)}(x)|\\le C e^{\\Phi(x)}$ for $j=0,1,2$ and all small $x$, where $\\Phi(x)=\\inf_{t\\ge T}(2xt - t \\theta(t))$ and $\\theta$ is a decreasing positive function with $\\int_T^\\infty \\frac{\\theta(t)}{t}\\,dt=\\infty$. If the local DN maps agree on any nonempty open set $O\\subset K$, then the global DN maps agree on all of $K$; by the authors' earlier uniqueness result, this implies $g=\\tilde g$.","pith_inferences":["One can test numerically on a round sphere: construct two C^2 conformal factors that are flat (all derivatives vanish) at the boundary but not identical, and check whether local DN equality on a small cap forces global equality; the theorem predicts it does.","The quasi-analytic boundary closeness is a condition on the difference of the conformal factors, not on each factor individually, so the result holds even when the individual factors are only C^m and vanish on an interior region; this suggests the mechanism is about the difference's spectral decay rather than regularity.","The same spectral propagation template might work for other boundary operators (e.g., Robin-to-Neumann) or for metrics that are not symmetric spaces, provided a pointwise bound on spectral projections is available.","If the boundary symmetric-space assumption were relaxed to any compact manifold with a sufficiently strong pointwise Weyl law, the argument would carry through; the paper notes this is open."],"forward_implications":["If Theorem 1.5 is correct, the inverse Steklov problem on these warped products is uniquely solvable from partial boundary data on any nonempty open set, without requiring the conformal factors to be real-analytic.","The borderline example θ(t)=1/log t shows the propagation condition is essentially optimal: the integral ∫ θ/t diverges, while any slower decay fails, mirroring the classical boundary between quasi-analytic and non-quasi-analytic classes.","The propagation mechanism reduces the local-to-global question to a purely spectral statement, so any future improvement in quasi-analytic propagation theorems on Riemannian manifolds would immediately yield analogous uniqueness results there.","Combined with the earlier uniqueness result of the same authors, local DN equality implies full metric equality g=\\tilde g in the warped product class, extending the reach of partial-data inverse boundary value problems.","The self-adjointness trick used to pass from vanishing of the difference on test functions to vanishing of the operator is general and applies to any difference of DN maps with a common boundary metric."],"fun_headline_variants":["Local patch fixes full DN map","Boundary openness propagates to full DN map","DN map: local equality goes global","Warped metrics: local boundary patch determines DN map","From any open boundary set to entire DN map"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument's weakest point is the reliance on the previously established spectral machinery—the Laplace-transform representation of the difference of the two Weyl-Titchmarsh functions with controlled Volterra-type kernels, and the self-adjointness of the Dirichlet-to-Neumann maps at the possibly singular cone end—since the main theorem collapses if either fails.","fun_headline_variants_meta":{"raw":{"variants":["Local patch fixes full DN map","Boundary openness propagates to full DN map","DN map: local equality goes global","Warped metrics: local boundary patch determines DN map","From any open boundary set to entire DN map"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000248,"raw_usage":{"total_tokens":1491,"prompt_tokens":961,"completion_tokens":530,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":705,"completion_tokens_details":{"reasoning_tokens":462}},"tokens_in":705,"tokens_out":530,"duration_ms":5726,"temperature":1.0,"reasoning_tokens":462,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T09:46:44.467144+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a round sphere $K=S^2$ and two distinct $C^2$ radial conformal factors $c$ and $\\tilde c$ on the unit ball whose derivatives up to order 2 satisfy the quasi-analytic closeness bound with $\\theta(t)=1/\\log t$ but which are not identical (for instance, flat but non-zero difference). If the local DN maps on some small open cap coincide, Theorem 1.5 says the global DN maps must coincide; a numerical or rigorous demonstration that they differ would falsify the claim. Alternatively, a direct check of whether the Weyl-Titchmarsh difference decays like $e^{-\\rho \\theta(\\rho)}$ for such a potential difference would test the k","supporting_citations":[],"review_version":2}