{"id":"eca2eff4-991c-49df-b8fa-63448486d81e","arxiv_id":"2608.03053","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Local conditional Lipschitz stability for anisotropic EIT in a conformal class is proved in 2D and higher dimensions, with guidelines for stable training of the deep Calderón method.","lead":"This paper proves a conditional Lipschitz stability bound for recovering anisotropic conductivity changes in electrical impedance tomography, under a restrictive low-frequency condition on the perturbation. It then uses this result to argue that the deep Calderón method, a neural network approach, should be stable if trained on such perturbations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Admissible-set condition likely excludes the paper's own training data, so the claimed link to the deep Calderón method is not tested.","rationale":"The reader's weakest assumption concerned the admissible-set condition (2.4) being strong, unverifiable, and excluding realistic conductivities. I agree that this condition is the key restriction, but the more specific and load-bearing failure is that the paper's own numerical training distributions appear to violate the condition, so the experiments cannot confirm the claimed application. The 2D proof of Theorem 2.1 seems internally consistent: the estimates in Lemmas 3.1 and 3.3 are standard, the error bounds close correctly, and the absorption using C1 N^2 < 1/2 is valid. The multidimensional theorem has a secondary gap in that (4.16) yields only ‖δ‖_{L2}, not the stated ‖δ‖_{L∞}, though this is likely repairable with standard elliptic regularity. The factor errors in the Gaussian computation (§2.3) are real but affect quantitative guidance more than the theorem's validity. Overall, the mathematical stability result is a legitimate conditional contribution, but the deep-Calderón application remains heuristic and is not validated by the reported experiments, which supports the reader's conditional acceptance rather than a stronger verdict.","tokens_in":25642,"tokens_out":27126,"duration_ms":239422,"concrete_test":"For the actual training distributions of Examples 5.1 and 5.2, generate 1000 samples and compute r(δ)=‖δ‖∞/‖χ_R(D)Δδ‖_{L2} with R=1.8 and R=1.2 (for the disks, use the Gaussian-smoothed δ = g_b*f_a from §2.3 with the same shape parameters, or a narrow mollifier). Compare the empirical r values with the threshold N < (2C1)^{-1/2} obtained from the constants in Theorem 2.1's proof (Section 3). If the vast majority of samples have r values far above the threshold, the training data are outside E_{M,N}, so the experiments do not validate the stability-to-deep-learning claim. Also recompute §2.3 with the correct |ξ|^4 factor to see whether Fig. 1's recommended parameter regime changes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's advertised application is the weakest link. Theorem 2.1 is conditional on δ ∈ E_{M,N} with N small enough that C1 N^2 < 1/2, where N must be below a bound depending exponentially on the Fourier cutoff (Remark 3.4). The training data in Section 5 do not obviously live in this regime: for Example 5.1, Gaussians with b ∈ (30,50) have most Fourier mass outside the unit disk, and even with the cutoff R=1.8 the ratio ‖δ‖∞/‖χ_R(D)Δδ‖_{L2} appears to be of order 10–100 rather than exponentially small; for Example 5.2, δ is a discontinuous characteristic function, so δ ∉ C^2 and Δδ is not an L2 function, violating the hypotheses of (2.4). Consequently the numerical experiments do not test the theorem's admissible set, and the claim that the stability theory 'relates to' the deep Calderón method's robustness is not actually established by the paper. The 2D theorem itself appears internally consistent; the concern is that the central application is not supported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies local inversion for anisotropic electrical impedance tomography in a conformal class, writing the conductivity as γ=(1+δ)(I+A) with a known anisotropy matrix A. The main analytic result, Theorem 2.1 in two dimensions, asserts a conditional Lipschitz estimate ‖δ‖_{L∞(Ω)} ≤ C‖Λγ−Λγ0‖_* for perturbations δ in an admissible set E_{M,N} defined by ‖δ‖_{L∞} ≤ N‖χ(D)Δδ‖_{L2(R^2)} with N sufficiently small. Section 4 extends the result to dimensions n≥3 by transforming the conductivity equation to a Schrödinger equation and using complex geometric optics, with the admissible condition imposed on the potential q rather than on δ. The authors then relate the stability estimates to the robustness of the deep Calderón method, arguing that training data satisfying the admissible condition promote kernel awareness, and they report numerical experiments with Gaussian and piecewise-constant conductivities.","tokens_in":25826,"tokens_out":22044,"duration_ms":190282,"significance":"The two-dimensional proof of Theorem 2.1 is self-contained and appears correct; the paper establishes a genuinely infinite-dimensional, non-finite-dimensional conditional Lipschitz stability class for anisotropic perturbations, which is a useful contribution to the EIT stability literature. The explicit discussion in Remark 3.4 of how the admissible constant degenerates with the Fourier cutoff is also valuable. However, the multi-dimensional extension contains a potentially fatal vacuousness in the admissible set, the numerical experiments do not sample the admissible set of the theorem, and the advertised application to the deep Calderón method is heuristic rather than proven. If the multi-dimensional issues are resolved and the deep-learning link is properly qualified, the two-dimensional theorem alone would be a publishable contribution.","major_comments":[{"comment":"The admissible set D_N is effectively trivial for small N. For q∈D_N, the condition ‖q‖_{L∞(Ω)} ≤ N‖χ(D)q‖_{L2(R^n)} combined with the contraction property ‖χ(D)q‖_{L2(R^n)} ≤ ‖q‖_{L2(Ω)} ≤ |Ω|^{1/2}‖q‖_{L∞(Ω)} implies (1 − N|Ω|^{1/2})‖q‖_{L∞(Ω)} ≤ 0, so D_N contains only q=0 whenever N < |Ω|^{-1/2}. The proof of Proposition 4.3 chooses N sufficiently small so that N^2 C e^{Cτ} < 1/2 with τ > M/ε; for a fixed domain such as the unit ball in R^3, where |Ω|^{-1/2} ≈ 0.49, this requirement can force N below the triviality threshold, rendering the proposition vacuous. The paper must quantify the constants and either exhibit a nontrivial admissible set compatible with the smallness condition or modify the admissible condition, for example by using a high-order derivative of q in the low-frequency norm.","section":"§4, Eq. (4.7), Proposition 4.3"},{"comment":"The proof repeatedly inverts the operator −Δ_{g0}+q with zero Dirichlet boundary condition, both to define v in Lemma 4.2 and in the elliptic estimates of Proposition 4.3. The text asserts that zero is not an eigenvalue for g,q derived from the conductivity problem, but Proposition 4.3 is stated for arbitrary q∈D_N with ‖q‖_{H^s(Ω)}≤M, and no invertibility argument is given for this whole class. Since the smallness of ‖q‖_{L∞} is not guaranteed by D_N, a zero eigenfunction cannot be excluded. In addition, the equation for v in Lemma 4.2 has the wrong sign: since −Δ_{g0}w = −Δw + P(x,∂)w, the right-hand side should be −P(x,∂)u, not P(x,∂)u; the sign error does not affect the subsequent bound, but the missing invertibility argument is load-bearing for the multi-dimensional theorem.","section":"§4, Lemma 4.2 and proof of Proposition 4.3"},{"comment":"The numerical experiments do not test the admissible set of Theorem 2.1. In Example 5.1 the Gaussian parameters satisfy b∈(30,50) with Fourier cutoff R=1.8; substituting into (2.6) gives ̲N of order 10–100, whereas Theorem 2.1 and Remark 3.4 require N to be small, in fact exponentially small in R. In Example 5.2 the test and training perturbations are discontinuous characteristic functions, so δ∉C^2(Ω) and Δδ∉L^2, violating the hypotheses of (2.4); the Gaussian smoothing described in §2.3 is not applied in the experiments. The observed trends, such as better stability for smaller b or smaller support, may be suggestive, but they are not a validation of the theorem. The paper should either generate training data lying in E_{M,N} or explicitly state that the numerical section is illustrative rather than a test of the theory.","section":"§5, Examples 5.1 and 5.2"},{"comment":"The link between Theorem 2.1 and the robustness of the deep Calderón method is not a theorem. Theorem 2.1 concerns the map Λγ↦δ, whereas the deep Calderón method composes a truncated Fourier inversion with a trained U-net; the paper does not show that a U-net trained on data from E_{M,N} has a controlled Lipschitz constant or is kernel aware in the precise sense of [20]. The argument in §2.2 relies on an unquantified universal-approximation assumption and on the phrase 'well trained', and no condition on the training distribution or on the network's Lipschitz constant is derived from the stability estimate. Accordingly, the abstract and Section 6 statements that the stability theory 'relates to' or provides 'theoretical underpinnings' for the deep Calderón method should be substantially qualified.","section":"§2.1–§2.3, Section 6"}],"minor_comments":[{"comment":"The name 'Caldeón' appears to be a typo; it should read 'Calderón' throughout the title and abstract.","section":"Title and abstract"},{"comment":"The admissible set E_{M,N} is defined for δ∈C^2(Ω) with supp(δ)⋐Ω, but the norm ‖χ(D)Δδ‖_{L2(R^2)} requires an extension of δ to R^2; the paper should state explicitly that δ is extended by zero outside Ω.","section":"§2.2, Eq. (2.4)"},{"comment":"The theorem says C is independent of δ, but the proof gives a constant that depends on the chosen N and on M, A, and Ω; the statement should clarify that C may depend on the admissible-set parameters and on the a priori data but not on the particular perturbation.","section":"§2.2, Theorem 2.1"},{"comment":"The potential q is written using Δ_g; since the fixed metric in the Schrödinger problem is g0, the formula should be rewritten in terms of Δ_{g0} (or a precise convention stated) so that q is a potential on a fixed metric.","section":"§4, Eq. (4.4)"},{"comment":"The admissible set D_N is stated for q∈L∞(Ω), but the proof uses q∈H^s(R^n); the paper should specify the compact extension by zero and the relation between the Sobolev norms on Ω and R^n.","section":"§4, Proposition 4.3"},{"comment":"The symbol ̲N is used both as the lower bound and as a variant of the parameter N, which is confusing; a distinct notation such as N_0 or N_min would be clearer.","section":"§2.3, Fig. 1 and Eqs. (2.6), (2.7)"},{"comment":"The constants C0 and C1 in the proof of Theorem 2.1 are not explicitly defined; for the claimed explicit exponential dependence in Remark 3.4, the paper should state how they depend on the cutoff R, on M, and on the a priori bounds for A and Ω.","section":"§2.2, Remark 3.4 and §3"},{"comment":"The symbol χ is used both for the Fourier cutoff χ(D) and as the characteristic function of the ball {x:|x−x0|<β}; the notation should be disambiguated.","section":"§5, Example 5.2"}],"recommendation":"major_revision","confidential_remarks":"The multi-dimensional result appears to be vacuous in its present form for standard domains because the admissible set D_N contains only q=0 when N is below the threshold |Ω|^{-1/2}, and the numerical section does not test the two-dimensional admissible set. These are the main obstacles. If the authors restrict the formal claims to the two-dimensional theorem and re-frame the deep-learning discussion as a heuristic guideline, the paper could be acceptable after a major revision. The title typo 'Caldeón' should also be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The 2D stability estimate in Theorem 2.1 is a real result, self-contained and worth reading closely. The thing advertised in the title—the connection to the deep Calderón method—is not actually tested; the numerical data sit outside the admissible set on which the theorem applies.\n\nWhat is new: the admissible set E_{M,N} is a low-frequency dominance condition, ||δ||_∞ ≤ N ||χ(D)Δδ||_2, and it gives a conditional Lipschitz stability estimate for anisotropic conductivities in a conformal class. That is a meaningful addition to a literature that mostly has logarithmic stability or finite-dimensional restrictions. The 2D proof is elementary and the dependence of the constants on N is explicit, which is nice. The guideline that good stability requires small N (and thus small b for Gaussians) is useful.\n\nWhere it is soft. First, the numerical experiments do not respect the hypotheses. Example 5.1 uses Gaussians with b ∈ (30,50); the paper's own formula (2.6) gives N ~ O(b), so the required smallness condition C1 N^2 < 1/2 is off by orders of magnitude. Example 5.2 uses discontinuous characteristic functions, which are not C^2 and for which the condition (2.4) is not even defined. The authors note that sampling from E_{M,N} is not clear, but then proceed with data that plainly violate it, and the claimed 'subsets of E_{M,N}' is inaccurate. Second, the Gaussian calculation in (2.6) has a factor error: Δδ contributes |ξ|^4 to the integrand, not |ξ|^2. The qualitative behavior is the same, but the numbers are wrong. Third, the multi-dimensional proof asserts non-resonance of the Schrödinger operator rather than proving it, and the admissible condition is placed on q rather than δ, so the theorem is one step removed from the conductivity itself. Fourth, the step from Lipschitz stability of the inverse map to stability of a trained CNN is heuristic; the paper acknowledges this, but the experiments do not narrow the gap.\n\nBottom line: the 2D theorem deserves a serious referee and the paper should be sent for review, but the application and numerical sections need major work. I would not cite the paper yet.","headline":"A worthwhile 2D stability theorem for anisotropic EIT, but the numerical application to the deep Calderón method does not test the theorem's admissible set and should be reconsidered.","tokens_in":26349,"tokens_out":6956,"would_cite":false,"duration_ms":59388,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","65N21"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that electrical impedance tomography with anisotropies is Lipschitz stable for a restricted but infinite-dimensional class of conductivity perturbations dominated by low frequencies, and uses this to explain when…","keywords":["electrical impedance tomography","anisotropic conductivity","Lipschitz stability","deep Calderón method","kernel awareness","Dirichlet-to-Neumann map","conditional stability","inverse problems"],"falsifier":"Search numerically over smooth, compactly supported perturbations in $E_{M,N}$ with a fixed small $N$, compute the finite-element Dirichlet-to-Neumann discrepancy for each, and minimize the ratio $\\|\\delta\\|_{L^\\infty(\\Omega)}/\\|\\Lambda_\\gamma-\\Lambda_{\\gamma_0}\\|_*$; if a sequence inside $E_{M,N}$ makes this ratio tend to infinity, or if two perturbations in the class have arbitrarily small boundary discrepancy but nonnegligible difference, the Lipschitz estimate is false. Verifying that the ratio stays bounded for such a family would support the theorem quantitatively.","tokens_in":25428,"feed_emoji":"⚡","tokens_out":16947,"duration_ms":133573,"temperature":0.7,"pith_summary":"The paper proves a conditional Lipschitz stability estimate for anisotropic electrical impedance tomography: within an admissible class of smooth, compactly supported conformal perturbations whose amplitude is controlled by the $L^2$ norm of their low-frequency Laplacian, the size of the conductivity perturbation is bounded by a constant times the discrepancy between the boundary Dirichlet-to-Neumann maps, with the constant independent of the perturbation. The same conditional stability is established in higher dimensions through a conformal metric and a Schrödinger-equation reformulation. The authors then use this stability to analyze the deep Calderón method, a two-stage deep-learning reconstruction pipeline: if training data come from such a low-frequency-dominated class with a small stability constant, a well-trained network is Lipschitz stable and kernel-aware, and severe ill-posedness appears only outside the class. Numerical experiments with Gaussian bumps and smoothed disks illustrate the predicted trade-off: broad or small features reconstruct stably, while sharp, large, or out-of-distribution features degrade.","feed_headline":"Boundary data pin down low-frequency conductivity changes","feed_subtitle":"The deep Calderón method is provably stable when its training data obey one low-frequency amplitude bound.","key_machinery":"The load-bearing object is the admissible set $E_{M,N}=\\{\\delta\\in C^2(\\Omega): \\operatorname{supp}(\\delta)\\Subset\\Omega,\\ \\|\\delta\\|_{L^\\infty(\\Omega)}<M,\\ \\|\\delta\\|_{L^\\infty(\\Omega)}\\leq N\\|\\chi(D)\\Delta\\delta\\|_{L^2(\\mathbb{R}^2)}\\}$, with $\\chi(D)$ the Fourier multiplier by the characteristic function of the unit disc. This condition ensures that what the boundary data can see—the low-frequency Laplacian of the perturbation—controls the perturbation's full amplitude. The proof's workhorse is the boundary identity $\\int_{\\partial\\Omega} u_1(\\Lambda_\\gamma-\\Lambda_{\\gamma_0})u_2\\,dS=-\\frac{|k|^2}{2}\\widehat{\\delta}(-k)$ plus error terms, where $u_1,u_2$ are complex exponential harmonic functions tuned to wave vector $k$; Plancherel's theorem converts the resulting estimate into a bound on $\\|\\chi(D)\\Delta\\delta\\|_{L^2}$, and the condition $C_1N^2<1/2$ closes the argument. In dimensions $n\\geq3$, the machinery shifts to the conformal metric $g=(\\det\\gamma)^{1/(n-2)}\\gamma^{-1}$, the associated Schrödinger equation with potential $q$, complex geometric optics solutions, and an analogous low-frequency admissibility condition on $q$.","core_discovery":"On the paper's own terms, the central result is Theorem 2.1: for conductivities of the form $\\gamma=(1+\\delta)(I+A)$, where $A$ is a known elliptic anisotropy and $\\delta$ lies in the admissible set $E_{M,N}$, for $N>0$ sufficiently small there holds $\\|\\delta\\|_{L^\\infty(\\Omega)}\\leq C\\|\\Lambda_\\gamma-\\Lambda_{\\gamma_0}\\|_*$ with $C$ independent of $\\delta$. The admissible set consists of smooth, compactly supported perturbations satisfying $\\|\\delta\\|_{L^\\infty(\\Omega)}\\leq N\\|\\chi(D)\\Delta\\delta\\|_{L^2(\\mathbb{R}^2)}$, where $\\chi(D)$ is the Fourier multiplier cutting to the unit disc; the condition makes the amplitude of $\\delta$ visible through its low-frequency Laplacian. The proof reconstructs the low-frequency Fourier transform of $\\delta$ from boundary integrals of complex exponential harmonic functions in the manner of Calderón's original method, bounds the error terms arising from the known anisotropy and from the nonlinearity, and applies Plancherel's theorem. In dimensions $n\\geq 3$, the analogous statement is proved by transforming the conductivity equation to a Schrödinger equation on a conformally related metric, using complex geometric optics solutions, and placing the admissibility condition on the potential $q$. For the deep Calderón method, the stability estimate is interpreted as a sufficient condition on the training data: a well-trained postprocessing network is Lipschitz stable, and therefore kernel-aware, on the admissible class, with ill-posedness relegated to data outside it.","pith_inferences":["Beyond the paper: because the admissibility condition is an a priori smoothness and low-frequency constraint that boundary measurements alone do not reveal, the practical working rule is to train on low-frequency-dominated images and to distrust network outputs on sharper or more oscillatory inputs, even when those inputs resemble the training set.","Beyond the paper: the numerical contrast between sign-changing and sign-definite perturbations suggests a testable refinement of the stability theory—single-signed perturbations may admit a smaller stability constant, and computing the ratio $\\|\\delta\\|_{L^\\infty(\\Omega)}/\\|\\Lambda_\\gamma-\\Lambda_{\\gamma_0}\\|_*$ for the two families would indicate whether the sign restriction relaxes the admissibl","Beyond the paper: the paper's explanation for failed two-disk reconstructions (Fourier cancellation between nearby disks) can be tested directly by evaluating the required $N$ in the admissible-set condition for pairs of smoothed disks; if $N$ grows as the separation shrinks, the stability class itself predicts the resolution limit independently of the trained network.","Beyond the paper: one could compose the Lipschitz stability on $E_{M,N}$ with the Lipschitz constant of a trained postprocessor to obtain an explicit noise-robustness bound for the deep Calderón method, converting the qualitative kernel-awareness argument into a measurable error estimate that depends on the training distribution."],"forward_implications":["On the admissible class $E_{M,N}$ with $N$ sufficiently small, the conductivity perturbation is recovered from boundary measurements with error at most a constant times the measurement discrepancy, so inversion there is Lipschitz stable.","A well-trained deep Calderón postprocessor trained on such a class is kernel-aware and stable on that class; the severe ill-posedness of EIT only reappears when the network is evaluated on data outside the class.","For Gaussian perturbations, stability requires the width parameter $b$ to be small, meaning smooth broad features are stable while sharper or finer features are unstable, as the numerical experiments show.","For smoothed piecewise-constant features, smaller objects keep the stability parameter small more easily, and reconstructions degrade for larger or out-of-distribution objects.","The truncation radius $R$ in the initial Calderón reconstruction step should be kept small, because the stability constant grows exponentially with $R$, forcing $N$ to be exponentially small for large $R$."],"supporting_citations":[{"why":"supplies the original complex-exponential harmonic probe through which boundary integrals yield the low-frequency Fourier transform of the perturbation.","marker":"[14]"},{"why":"introduces the deep Calderón method whose robustness and training-data dependence the paper analyzes.","marker":"[16]"},{"why":"defines the kernel-awareness criterion for stable accurate networks, used to link Lipschitz stability to reliable deep reconstruction.","marker":"[20]"},{"why":"provides the conformal transformation formula relating the conductivity Dirichlet-to-Neumann map to a Schrödinger map with explicit potential, used in higher dimensions.","marker":"[24]"},{"why":"supplies an earlier Lipschitz stability result for conformal classes of anisotropic conductivities that the present admissible-set result extends.","marker":"[30]"},{"why":"supplies the elliptic existence, uniqueness, and regularity estimates used to bound the correction terms in the proof.","marker":"[33]"},{"why":"establishes equality of the conductivity and Laplace-Beltrami Dirichlet-to-Neumann maps, enabling the geometric reformulation in dimensions $n\\geq3$.","marker":"[48]"},{"why":"gives the classical logarithmic stability estimate for isotropic EIT that the paper's conditional Lipschitz estimate improves on a restricted class.","marker":"[51]"},{"why":"constructs the complex geometric optics solutions used for the multi-dimensional recovery of the potential.","marker":"[60]"}],"fun_headline_variants":["Anisotropy proof stabilizes deep EIT","Deep Calderón method gets provable stability","Low-frequency bound enables stable EIT","Stability theory underpins deep EIT imaging","Anisotropic EIT stability boosts deep reconstruction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on assuming the unknown perturbation $\\delta$ satisfies the a priori bound $\\|\\delta\\|_{L^\\infty(\\Omega)}\\leq N\\|\\chi(D)\\Delta\\delta\\|_{L^2(\\mathbb{R}^2)}$ with $N$ sufficiently small; this low-frequency-amplitude condition is strong, cannot be verified from boundary measurements alone, and the stability claim simply does not apply to perturbations outside it.","fun_headline_variants_meta":{"raw":{"variants":["Anisotropy proof stabilizes deep EIT","Deep Calderón method gets provable stability","Low-frequency bound enables stable EIT","Stability theory underpins deep EIT imaging","Anisotropic EIT stability boosts deep reconstruction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000361,"raw_usage":{"total_tokens":1968,"prompt_tokens":984,"completion_tokens":984,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":916}},"tokens_in":600,"tokens_out":984,"duration_ms":8216,"temperature":1.0,"reasoning_tokens":916,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:54:26.824210+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search numerically over smooth, compactly supported perturbations in $E_{M,N}$ with a fixed small $N$, compute the finite-element Dirichlet-to-Neumann discrepancy for each, and minimize the ratio $\\|\\delta\\|_{L^\\infty(\\Omega)}/\\|\\Lambda_\\gamma-\\Lambda_{\\gamma_0}\\|_*$; if a sequence inside $E_{M,N}$ makes this ratio tend to infinity, or if two perturbations in the class have arbitrarily small boundary discrepancy but nonnegligible difference, the Lipschitz estimate is false. Verifying that the ratio stays bounded for such a family would support the theorem quantitatively.","supporting_citations":[{"cited_title":"Calderón , On an inverse boundary value problem , in Seminar on Numerical Analysis and its Applica- tions to Continuum Physics (Rio de Janeiro, 1980), Soc","cited_arxiv_id":null,"evidence_quote":"supplies the original complex-exponential harmonic probe through which boundary integrals yield the low-frequency Fourier transform of the perturbation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the deep Calderón method whose robustness and training-data dependence the paper analyzes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the kernel-awareness criterion for stable accurate networks, used to link Lipschitz stability to reliable deep reconstruction."},{"cited_title":"Dos Santos Ferreira, C","cited_arxiv_id":null,"evidence_quote":"provides the conformal transformation formula relating the conductivity Dirichlet-to-Neumann map to a Schrödinger map with explicit potential, used in higher dimensions."},{"cited_title":"Gaburro and E","cited_arxiv_id":null,"evidence_quote":"supplies an earlier Lipschitz stability result for conformal classes of anisotropic conductivities that the present admissible-set result extends."},{"cited_title":"Gilbarg and N","cited_arxiv_id":null,"evidence_quote":"supplies the elliptic existence, uniqueness, and regularity estimates used to bound the correction terms in the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes equality of the conductivity and Laplace-Beltrami Dirichlet-to-Neumann maps, enabling the geometric reformulation in dimensions $n\\geq3$."},{"cited_title":"Liu , Stability estimates for the two-dimensional inverse conductivity problem , PhD thesis, University of Rochester, New York, 1997","cited_arxiv_id":null,"evidence_quote":"gives the classical logarithmic stability estimate for isotropic EIT that the paper's conditional Lipschitz estimate improves on a restricted class."},{"cited_title":"Sylvester and G","cited_arxiv_id":null,"evidence_quote":"constructs the complex geometric optics solutions used for the multi-dimensional recovery of the potential."}],"review_version":2}