{"id":"9f2b4f32-3d1a-40be-94d6-e02120aaee41","arxiv_id":"2501.00947","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The magnetic D-to-N ground state energy obeys λ = α̂ b^{1/2} − α̂² + (1/3) max κ + o(1) in 2D, with a 3D limit given by the boundary infimum of λ^DN(ϑ(x)) |B(x)|^{1/2}.","lead":"This paper proves that the lowest eigenvalue of the magnetic Dirichlet-to-Neumann operator on smooth bounded planar domains grows like a universal constant times the square root of the magnetic field strength, with the next-order correction governed by boundary curvature. The analysis extends to three dimensions and variable fields, yielding sharp asymptotics with leading order determined entirely by boundary data.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.6's 3D lower bound is asserted by analogy, not proved; the local half-space comparison with the exact angle-dependent coefficient is the missing load-bearing step.","rationale":"I read the paper in good faith and identify the same weakest assumption as the reader: the 3D lower bound in Proposition 6.9 is asserted by analogy rather than proved. The 2D results are supported by detailed upper and lower estimates, and the Robin-Laplacian comparison in Sections 4-5 is internally coherent. The 3D upper bound is a plausible quasimode construction and is not the bottleneck. The bottleneck is the lower bound: without a localized, angle-dependent half-space comparison with controlled errors, the lim-inf inequality (6.22) does not follow from the cited Lu-Pan Neumann bound alone. The text itself flags the incomplete construction of the half-space D-to-N operator in Section 6.5, which reinforces the concern. Because the missing step is a proof detail rather than a demonstrated contradiction, the appropriate verdict remains CONDITIONAL: the paper's central 3D theorem should be accepted only after the lower-bound argument is completed or replaced by a full proof. The reader's verdict already captures this, so no verdict change is needed.","tokens_in":30317,"tokens_out":6690,"duration_ms":66501,"concrete_test":"Write out the proof of Proposition 6.9 for a single boundary point p: for every epsilon>0 and every u supported in a small boundary neighborhood of p, derive the local lower bound ||(-i\\nabla-bA)u||_Omega^2 >= (lambda^DN(vartheta(p))|B(p)|^{1/2}-epsilon) b^{1/2} ||u||_{partial Omega}^2 + o(b^{1/2}), using the bulk Neumann bound (6.23), the gauge approximation Lemma 5.4 of [27], and the half-space variational definition (6.3). Then verify that a partition-of-unity sum of these local inequalities, with the same error bookkeeping as in Proposition 4.2, yields exactly (6.22). If this derivation cannot be completed, then Theorem 1.6 lacks a proof of its lower bound; if it can, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central 3D assertion, Theorem 1.6, needs both Proposition 6.10 (upper bound) and Proposition 6.9 (lower bound). Proposition 6.10 is a credible three-case quasimode construction, but Proposition 6.9 is given in four sentences: it says to follow the 2D proof, to replace the 2D Neumann lower bound by the Lu-Pan estimate (6.23), to use Lemma 5.4 from [27] for the gauge approximation, and to 'implement the constant magnetic field results obtained in the previous subsection'. No estimates are written down. In the 2D proof being adapted (Proposition 4.2), the lower bound requires: (i) a bulk lower bound on the magnetic Neumann energy; (ii) a boundary partition of unity at scale b^{-rho}; (iii) local gauge reduction to a constant field B(p_j); (iv) the half-plane D-to-N inequality giving the coefficient alpha|B(p_j)|^{1/2}; and (v) uniform control of all error terms. In 3D, the analogous chain must produce a local lower bound with coefficient lambda^DN(vartheta(p_j))|B(p_j)|^{1/2}, using the half-space model operator (6.2). None of the local comparison estimates, the error terms, or the uniformity in x is displayed. The gap is especially concrete when the infimum in Theorem 1.6 is attained at a boundary point with vartheta>0: the theorem's coefficient is lambda^DN(vartheta)|B|^{1/2}, not the simpler value alpha|B|^{1/2} established in Proposition 6.1. Section 6.5 further concedes that the half-space D-to-N operator is not constructed, and Remark 6.11 defers the technical verification. This does not disprove the theorem, but it is exactly the load-bearing part of the 3D matching result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the ground state energy of the magnetic Dirichlet-to-Neumann operator on bounded regular domains in dimensions 2 and 3 in the strong-field limit. In two dimensions it proves, for fields that are constant near the boundary, the two-term asymptotics λ^DN(bA,Ω) = α̂ b^{1/2} − α̂² + (1/3) max_{∂Ω} κ + o(1), and for variable non-vanishing boundary fields, the leading term α̂ (inf_{∂Ω}|B|)^{1/2} b^{1/2}. In three dimensions it states an analogue with coefficient λ^DN(ϑ(x))|B(x)|^{1/2}, depending on the angle between the magnetic field and the boundary normal. The paper also gives a Robin-Laplacian comparison yielding a splitting estimate for 2D domains with a unique non-degenerate curvature maximum, and a weak-field expansion λ^DN(bA,Ω) = b² |∂Ω|^{-1} ∫_Ω |A_Ω|² dx + o(b²).","tokens_in":30686,"tokens_out":23068,"duration_ms":229932,"significance":"If the main theorems are correct, the paper settles the Chakradhar–Gittins–Habib–Peyerimhoff conjecture and establishes a striking boundary-only leading-order dependence for the magnetic D-to-N eigenvalues. A clear strength is that the universal constant α̂ is not fitted: it is derived from an explicit harmonic-oscillator/Robin spectral problem (Proposition 2.1), and the 2D proof is detailed, with explicit quasimodes, Agmon-type concentration estimates, and a Robin-comparison argument. The 3D upper bound is plausible and the model half-space analysis in Proposition 6.1 is useful. However, the 3D lower bound is only sketched, and there are internal inconsistencies in a corollary and a remark that need to be corrected before the paper can be accepted.","major_comments":[{"comment":"The proof of the 3D lower bound, which is load-bearing for Theorem 1.6, is not actually carried out. The text only says to follow the 2D proof, to replace the 2D Neumann bound by the Lu–Pan estimate (6.23), to use Lemma 5.4 from [27], and to implement the constant-field results. None of the local half-space comparison estimates, the gauge-approximation error terms, the boundary partition at scale b^{-ρ}, or the uniformity in the boundary point p is displayed. In particular, when the infimum in Theorem 1.6 is attained at a point with ϑ(p)>0, the coefficient is λ^DN(ϑ(p))|B(p)|^{1/2}, which is not the simpler value α̂|B(p)|^{1/2} established in Proposition 6.1; the local comparison needed to obtain this angle-dependent coefficient is exactly the missing step. Remark 6.11 concedes that the half-space D-to-N operator is not constructed, further underlining that the local variational inequality is assumed rather than proved.","section":"§6.3, Proposition 6.9"},{"comment":"The displayed disk formula λ^DN(bA,D(0,R)) = (bR/2) I'_0(bR²/4)/I_0(bR²/4) grows linearly in b as b→∞, whereas Theorem 1.1 and Proposition 3.2 give λ^DN(bA,Ω) ∼ α̂ b^{1/2} for the same disk. The formula cannot be the lowest D-to-N eigenvalue of the disk; it appears to be the value for the n=0 (radial) boundary mode only. This internal inconsistency should be corrected by identifying the formula as a sector value or by giving the actual explicit spectral resolution.","section":"Remark 1.10"},{"comment":"The inequality direction in Corollary 1.3 appears to be reversed. Under Theorem 1.2, λ^DN(bA,Ω) ≈ α̂ b^{1/2} − α̂² + (1/3) max_{∂Ω} κ, while for the disk B of the same area the curvature term is (1/3) sqrt(π/|Ω|). Pankrashkin's inequality (cited as [31]) gives max_{∂Ω} κ ≥ sqrt(π/|Ω|), so the disk has the smaller or equal constant-order term and therefore λ^DN(bA,Ω) ≥ λ^DN(bA,B) for large b, not ≤ as stated.","section":"Corollary 1.3"},{"comment":"The proof of Theorem 5.2 uses Proposition 5.3, whose equality λ_j(bA,Ω) = −b^{1/2}γ_j(b) is conditional on the simplicity of μ_j(γ_j(b),b) for all j up to the relevant order. The proof does not establish this simplicity, and the asymptotic input (5.1) is quoted for fixed γ while the theorem requires the b-dependent value γ_j(b) to be inserted. The local uniformity in γ is asserted, but the simplicity condition needed to identify the j-th Robin zero with the j-th D-to-N eigenvalue should be proved or explicitly referenced from [7].","section":"§5, Theorem 5.2 and Proposition 5.3"}],"minor_comments":[{"comment":"The third moment identity contains a typo: the integrand should be (t−α̂)³|f∗(t)|² dt, not |f∗(t)|³ dt, which has the wrong homogeneity.","section":"Proposition 2.1(iii)"},{"comment":"The space C^∞_0(R^3_+) in the variational definition of λ^DN(ϑ) should be replaced by smooth functions compactly supported in the closure R^3_+ ∪ ∂R^3_+; otherwise the traces at x_1=0 vanish identically and the quotient is undefined.","section":"Eq. (6.3)"},{"comment":"The sign convention for the exterior-disk formula should be clarified: for R<0, equation (4.16) gives a different α̂² term than the exterior-disk formula (4.15) if R^{-1} is negative. The authors should state the curvature convention used for exterior boundaries.","section":"Eqs. (4.14)–(4.16)"},{"comment":"The definition of the trace space ̂H^{1/2}(R²) and the formal weak form (6.26) are not needed for the variational results, but the notation suggests a self-adjoint operator that is not constructed. A brief statement that only the variational ground-state energy is used would avoid confusion.","section":"Section 6.5"}],"recommendation":"major_revision","confidential_remarks":"The 3D part of the paper is not yet at the standard of a complete proof: Proposition 6.9 is a sketch, and Theorem 1.6 depends on it. The inconsistent Corollary 1.3 and Remark 1.10 should also be fixed before publication, as they currently assert false statements if read literally."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The 2D results are the real contribution and they look solid. Theorem 1.2's two-term expansion with the curvature correction is genuinely new outside the disk, and Theorem 1.4's variable-field leading term is a clean extension of the surface-superconductivity machinery to the D-to-N setting. The constant α̂ is derived from an explicit one-dimensional Robin/harmonic-oscillator problem and used as a benchmark, not fitted; I see no circularity. The concentration estimates and the Robin-Laplacian comparison are careful, and the weak-field expansion is a nice bonus.\n\nThe weak spot is exactly where the reader put it: Proposition 6.9, the 3D lower bound, is not proved in the text. Four sentences tell us to follow the 2D proof, replace a Neumann lower bound by the Lu-Pan bound, use Lemma 5.4 of [27], and 'implement' the constant-field results. But the load-bearing estimate — a local half-space comparison that yields the exact angle-dependent coefficient λ^DN(ϑ(x))|B(x)|^{1/2} — is never written. The 2D chain requires a bulk lower bound, a boundary partition of unity, a gauge reduction, a half-plane D-to-N inequality, and uniform error control; none of those steps is displayed for 3D. Section 6.5's admission that the half-space D-to-N operator is not constructed (Remark 6.11 defers the verification) compounds the issue. I would not bet against the theorem, but as written, the lower-bound half of Theorem 1.6 is a proof sketch, not a proof.\n\nThe 2D parts do not have this problem. Assumption 4.1 is a bit of a black box, but the examples cover the cases that matter and the assumption is standard in this literature. The paper is by people who know the material, and the citation pattern is appropriate, including their own prior work where it is genuinely prior.\n\nSo: this deserves peer review, and a serious referee should ask for a full write-up of the 3D lower bound before acceptance. The 2D theorems justify publication on their own; the 3D theorem should be either proved properly or stated as a conditional result. I'd bring it to reading group for the 2D methods, and I'd cite the 2D curvature result.","headline":"The 2D magnetic D-to-N asymptotics are solid and new; the 3D theorem is plausible but the lower bound is only sketched, so treat Theorem 1.6 as provisional.","tokens_in":31240,"tokens_out":3056,"would_cite":true,"duration_ms":28963,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J50","35P20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the strong-field limit of the lowest magnetic Dirichlet-to-Neumann eigenvalue is governed by boundary magnetic data, with an explicit curvature correction and no bulk contribution at leading order.","keywords":["magnetic Dirichlet-to-Neumann operator","eigenvalue asymptotics","strong magnetic field","boundary magnetic field","parabolic cylinder functions","Robin magnetic Laplacian","magnetic harmonic extension"],"falsifier":"Take the disk of radius $R$ and compute the exact D-to-N eigenvalue $\\lambda=(bR/2)I_0'(bR^2/4)/I_0(bR^2/4)$ for a range of large $b$; Theorem 1.2 predicts $\\lambda-\\hat{\\alpha}b^{1/2}+\\hat{\\alpha}^2-1/(3R)=o(1)$, so any deviation not tending to zero would show the two-term law fails. Alternatively, solve the three-dimensional variational problem on a cylinder whose boundary field angle is known and check that $b^{-1/2}\\lambda^{\\mathrm{DN}}$ converges to the boundary infimum in Theorem 1.6.","tokens_in":30075,"feed_emoji":"🧲","tokens_out":11256,"duration_ms":105726,"temperature":0.7,"pith_summary":"$\\lambda^{\\mathrm{DN}}(bA,\\Omega)$ denotes the lowest eigenvalue of the magnetic Dirichlet-to-Neumann operator. The paper establishes that as the field strength $b\\to+\\infty$, this eigenvalue grows like $b^{1/2}$, with a leading constant fixed by the magnetic field on the boundary. In two dimensions and constant unit field, the two-term law is $\\lambda^{\\mathrm{DN}}(bA,\\Omega)=\\hat{\\alpha} b^{1/2}-\\hat{\\alpha}^2+\\frac{1}{3}\\max_{\\partial\\Omega}\\kappa+o(1)$, where $-\\alpha$ is the unique negative zero of the parabolic cylinder function $D_{1/2}$ and $\\hat{\\alpha}=\\alpha/\\sqrt{2}$. In three dimensions and variable fields, $b^{-1/2}\\lambda^{\\mathrm{DN}}\\to\\inf_{x\\in\\partial\\Omega}\\lambda^{\\mathrm{DN}}(\\vartheta(x))|B(x)|^{1/2}$, so interior values of the magnetic field do not enter the leading term. These results refine a recent conjecture and connect the D-to-N problem to magnetic Robin Laplacian asymptotics.","feed_headline":"Boundary magnetic data control strong-field eigenvalues","feed_subtitle":"Only boundary magnetic data set the leading D-to-N eigenvalue; a curvature correction follows.","key_machinery":"The argument runs through the variational quotient $\\inf_u\\|(-i\\nabla-bA)u\\|_\\Omega^2/\\|u\\|_{\\partial\\Omega}^2$. Near a boundary point, a gauge transformation flattens the potential to a constant-field approximation, and parallel coordinates reduce the model to a half-plane with tangential field; the model constant $\\hat{\\alpha}$ is the bottom of a Robin harmonic oscillator, equivalently the zero of $\\Theta(\\gamma)$ at $\\gamma=-\\hat{\\alpha}$. Upper bounds are built from quasimodes localized in the normal direction using the profile $f_*$; lower bounds use the identity $\\mathrm{Re}\\int_\\Omega(-i\\nabla-A)u\\cdot(-i\\nabla-A)(w^2u) = \\int_\\Omega|(-i\\nabla-A)(wu)|^2-\\int_\\Omega|\\nabla w|^2|u|^2$, partitions of unity, and Agmon-type exponential decay of the magnetic harmonic extension. For the second term, the disk and exterior-disk models provide curvature comparison; for 3D, the half-space model $\\lambda^{\\mathrm{DN}}(\\vartheta)$ is minimized at $\\vartheta=0$, and the Lu-Pan Neumann lower bound supplies the localization step.","core_discovery":"The central claim is that the strong-field ground state of the magnetic D-to-N operator is a boundary-layer state. For a regular planar domain with constant unit field, Theorem 1.2 gives $\\lambda^{\\mathrm{DN}}(bA,\\Omega)=\\hat{\\alpha}b^{1/2}-\\hat{\\alpha}^2+\\frac{1}{3}\\max_{x\\in\\partial\\Omega}\\kappa_x+o(1)$, and the same two-term form holds for each fixed eigenvalue once the curvature maximum is isolated. For variable fields in two and three dimensions, the leading coefficient is $\\inf_{x\\in\\partial\\Omega}\\lambda^{\\mathrm{DN}}(\\vartheta(x))|B(x)|^{1/2}$, where $\\vartheta(x)$ is the angle between the magnetic vector field and the normal; the bulk of $B$ is invisible at leading order, in contrast to the Neumann magnetic Laplacian. The paper also proves a weak-field limit $\\lambda^{\\mathrm{DN}}(bA,\\Omega)=b^2|\\partial\\Omega|^{-1}\\int_\\Omega|A_\\Omega|^2dx+o(b^2)$ and a $b^{-1/4}$ eigenvalue splitting under a non-degenerate curvature maximum.","pith_inferences":["Beyond the stated theorems, the boundary-only leading term suggests that at strong fields the D-to-N operator acts as a boundary observable: bulk magnetic wells cost too much energy, so the ground state lives in a boundary layer and measurements of its energy mainly reveal boundary field magnitude and angle.","A natural follow-up is to make the half-space D-to-N operator self-adjoint; the paper deliberately avoids this by using the variational ground-state energy, so $\\lambda^{\\mathrm{DN}}(\\vartheta)$ would then be a spectral quantity rather than only an infimum.","One testable extension is to tune the magnetic field to vanish on part of the boundary; the paper's admissible-field condition forbids this, and a sharp answer would show whether the $b^{1/2}$ law breaks or acquires a new exponent."],"forward_implications":["In two dimensions with constant field, the low-lying eigenvalues all have expansion $\\lambda_j=\\hat{\\alpha}b^{1/2}-\\hat{\\alpha}^2+\\frac{1}{3}\\max_{\\partial\\Omega}\\kappa+(2j-1)c_*b^{-1/4}+o(b^{-1/4})$ when the curvature has a unique nondegenerate maximum.","If the magnetic field vanishes only on a finite set of smooth interior curves and stays nonzero on the boundary, the leading term is still $\\hat{\\alpha}(\\inf_{\\partial\\Omega}|B|)^{1/2}b^{1/2}$; interior zeros do not affect it.","In three dimensions, the leading constant is an infimum over the boundary of the half-space D-to-N energy $\\lambda^{\\mathrm{DN}}(\\vartheta(x))$ weighted by $|B(x)|^{1/2}$, and a boundary component homeomorphic to $S^2$ forces $\\vartheta=0$ somewhere, yielding the universal constant $\\hat{\\alpha}$ for constant field.","The D-to-N eigenvalues are zeros of Robin Laplacian eigenvalues via $\\lambda_j=-b^{1/2}\\gamma_j(b)$, so two-term and splitting asymptotics transfer between the two problems."],"supporting_citations":[{"why":"Serves as the conjecture that the D-to-N ground state energy tends to infinity with $b$, which the paper refines and proves.","marker":"[4]"},{"why":"Supplies the half-plane and disk analysis, the constant $\\hat{\\alpha}$, and the two-term expansion that this paper extends to general domains.","marker":"[17]"},{"why":"Defines $\\Theta(\\gamma)$ and its zero $\\gamma_0=-\\hat{\\alpha}$, the key one-dimensional model behind the boundary layer.","marker":"[18]"},{"why":"Provides the two-dimensional magnetic Neumann lower bounds and admissible-field conditions used in Assumption 4.1.","marker":"[13]"},{"why":"Provides the three-dimensional Neumann magnetic Laplacian lower bound invoked in the proof of Proposition 6.9.","marker":"[27]"},{"why":"Gives the Robin magnetic Laplacian spectral asymptotics with curvature and gap constants used for the 2D eigenvalue splitting.","marker":"[7]"},{"why":"Gives Robin Laplacian lower bounds with $\\Theta$ used in the Agmon-type exponential decay of the magnetic harmonic extension.","marker":"[19]"},{"why":"Supplies the three-dimensional Neumann upper-bound strategy adapted to construct the 3D D-to-N quasimodes.","marker":"[33]"}],"fun_headline_variants":["Boundary magnetic data set strong-field D-N eigenvalues","Magnetic D-N eigenvalues: boundary data rule at high field","High-field D-N spectra locked by boundary magnetic values","Strong-field D-N eigenvalues: boundary data win over bulk field"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the three-dimensional lower bound really does follow by repeating the two-dimensional localization proof with the Neumann lower bound of Lu and Pan in place of the two-dimensional bound; the paper invokes this adaptation rather than carrying it out, and Section 6.5 leaves the half-space D-to-N operator as a formal variational object.","fun_headline_variants_meta":{"raw":{"variants":["Boundary magnetic data set strong-field D-N eigenvalues","Magnetic D-N eigenvalues: boundary data rule at high field","High-field D-N spectra locked by boundary magnetic values","Strong-field D-N eigenvalues: boundary data win over bulk field"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000699,"raw_usage":{"total_tokens":3126,"prompt_tokens":885,"completion_tokens":2241,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":2174}},"tokens_in":501,"tokens_out":2241,"duration_ms":18245,"temperature":1.0,"reasoning_tokens":2174,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:39:18.882928+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the disk of radius $R$ and compute the exact D-to-N eigenvalue $\\lambda=(bR/2)I_0'(bR^2/4)/I_0(bR^2/4)$ for a range of large $b$; Theorem 1.2 predicts $\\lambda-\\hat{\\alpha}b^{1/2}+\\hat{\\alpha}^2-1/(3R)=o(1)$, so any deviation not tending to zero would show the two-term law fails. Alternatively, solve the three-dimensional variational problem on a cylinder whose boundary field angle is known and check that $b^{-1/2}\\lambda^{\\mathrm{DN}}$ converges to the boundary infimum in Theorem 1.6.","supporting_citations":[{"cited_title":"A note on the magnetic Steklov operator on functions","cited_arxiv_id":"2410.07462","evidence_quote":"Serves as the conjecture that the D-to-N ground state energy tends to infinity with $b$, which the paper refines and proves."},{"cited_title":"On the magnetic Dirichlet to Neumann operator on the disk -- strong diamagnetism and strong magnetic field limit--","cited_arxiv_id":"2411.15522","evidence_quote":"Supplies the half-plane and disk analysis, the constant $\\hat{\\alpha}$, and the two-term expansion that this paper extends to general domains."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines $\\Theta(\\gamma)$ and its zero $\\gamma_0=-\\hat{\\alpha}$, the key one-dimensional model behind the boundary layer."},{"cited_title":"Helffer and A","cited_arxiv_id":null,"evidence_quote":"Provides the two-dimensional magnetic Neumann lower bounds and admissible-field conditions used in Assumption 4.1."},{"cited_title":"Lu and X","cited_arxiv_id":null,"evidence_quote":"Provides the three-dimensional Neumann magnetic Laplacian lower bound invoked in the proof of Proposition 6.9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Robin magnetic Laplacian spectral asymptotics with curvature and gap constants used for the 2D eigenvalue splitting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives Robin Laplacian lower bounds with $\\Theta$ used in the Agmon-type exponential decay of the magnetic harmonic extension."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the three-dimensional Neumann upper-bound strategy adapted to construct the 3D D-to-N quasimodes."}],"review_version":1}