{"id":"d2dec5a5-e6ee-48f8-b282-37fdb916fe2e","arxiv_id":"2608.05547","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A new class of surface divergence-free kernel interpolants preserves the scalar kernel's Sobolev order with m=1 and adds a multiplier-preserving inverse Laplace-Beltrami variant.","lead":"This paper builds new matrix-valued kernels that interpolate tangent vector fields on spheres and other surfaces while automatically preserving the divergence-free condition. It shows a lower-order construction gains one full Sobolev order over the classical approach and adds a multiplier-preserving variant with error and stability guarantees.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved vector spherical harmonic addition formula (Lemma 2.1) is the keystone; a direct numerical check would settle it.","rationale":"The paper's central theoretical assertions are (i) the m=1 kernel has multipliers kappa_ell asymptotically equal to (1+lambda_ell)^{-sigma}, one Sobolev order better than the m=2 construction, and (ii) the inverse Laplace–Beltrami construction has multipliers exactly equal to the scalar Fourier coefficients. Both proofs reduce to expanding kernels of the form alpha(t)R + alpha'(t)Q in vector spherical harmonics, and the expansion coefficients are obtained by equating this expression with sum kappa_ell S_ell via Lemma 2.1. Hence the addition formula is the keystone of the entire Fourier analysis. The m=0 K0 variant, although prominent in the numerics, is explicitly unproved (Remark 3.3) and is not part of the central claim as stated by the authors; its failure for Wendland kernels is disclosed in Section 6.2 and does not threaten the m=1 theorem. The reader identified the same keystone, and I agree. The proposed numerical test is cheap and would settle whether the lemma actually lands. If Lemma 2.1 passes, the theory appears internally consistent, and the remaining caveats (K0 positive definiteness and imported polynomial-space estimates) justify keeping the conditional verdict rather than moving to full acceptance.","tokens_in":27665,"tokens_out":31670,"duration_ms":241107,"concrete_test":"Implement the direct sum S_ell(x,y) = sum_{k=1}^{2ell+1} y_{ell,k}(x) y_{ell,k}(y)^T with y_{ell,k} = (ell(ell+1))^{-1/2} x times grad Y_{ell,k} for a standard real orthonormal basis {Y_{ell,k}}, for ell=1,...,20 and random x,y in S^2. Compare to the closed form (2ell+1)/(4pi ell(ell+1)) [P''_ell(x·y) Q(x,y) + P'_ell(x·y) R(x,y)] from Lemma 2.1, using machine-precision Legendre derivatives. If the maximum relative Frobenius error exceeds 1e-10, the lemma is false and the central claim fails. Repeat for T_ell with z_{ell,k} = grad Y_{ell,k}/sqrt(ell(ell+1)). An independent symbolic derivation from the scalar addition theorem would provide full confirmation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Every multiplier identity in the paper—(3.5), (3.7), and the inverse Laplace–Beltrami construction of Theorem 3.4—is derived by invoking the vector spherical harmonic addition formulas of Lemma 2.1, which are cited to the authors' companion paper [39] and not proved here. If the formulas for S_ell or T_ell are algebraically wrong—wrong matrix term, wrong sign on Q, or an incorrect normalization—the claimed kappa_ell are wrong, and the central one-order gain (Corollary 3.2) and the multiplier-preserving property collapse. The manuscript also does not state the functional-analytic hypotheses under which the formulas are valid for the series manipulations in Theorem 3.1. Basic sanity checks (trace at x=y, ell=1 direct computation) pass, but that is not a proof. Because the entire Fourier analysis and all subsequent error estimates import this result, this is the single most load-bearing unverified step. The m=0 numerical variant, while prominent, is explicitly unproved in Remark 3.3 and does not affect the m=1 theorem, so it is secondary to the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a family of divergence-free tangential matrix-valued kernels on embedded surfaces, with the main analysis on the unit sphere. Starting from an isotropic ambient ansatz for curl-free kernels, the authors construct surface kernels of the form Kdiv = alpha(r)((n_x·n_y)I - n_y n_x^T) - beta(r)(n_x×(x-y))(n_y×(x-y))^T, and on S^2 reduce this to Kdiv = phi^(m-1)(t)R + phi^(m)(t)Q. They compute the vector spherical harmonic multipliers (Theorem 3.1), show that the m=1 variant preserves the scalar Sobolev order while the classical m=2 variant loses one order (Corollary 3.2), and introduce an inverse Laplace-Beltrami construction whose multipliers equal those of the underlying scalar kernel (Theorem 3.4). For interpolation at scattered nodes they prove a lower bound on the smallest eigenvalue of the interpolation matrix (Theorem 5.2), pointwise and fractional Sobolev error estimates (Theorems 5.8 and 5.11), and a Hilbert-scale superconvergence result (Theorem 5.13). Numerical experiments on S^2 and on three non-spherical surfaces illustrate convergence and stability, including a K0_div variant that is explicitly outside the proved theory.","tokens_in":27920,"tokens_out":17176,"duration_ms":140693,"significance":"If the results hold, the main contribution is substantial: for a scalar zonal kernel with Fourier coefficients decaying like l^{-2σ}, the m=1 construction gives divergence-free Fourier multipliers also decaying like l^{-2σ}, so the native space is H^σ_div, one Sobolev order smoother than the classical m=2 construction. The multiplier-preserving inverse Laplace-Beltrami construction is elegant and potentially useful. The paper also provides a clean Fourier-based stability proof and fractional-order Sobolev error estimates, including superconvergence for targets smoother than the native space. Strengths include the explicit multiplier computations, the direct proof strategy for Theorem 3.1, and the fact that the numerical rates for K1_div and K2_div match the stated exponents, with K2_div reproducing values from the earlier literature. The main caveats are the unproved vector addition formula imported from a companion preprint and the m=0 numerical variant, which the paper itself acknowledges lacks a positive-definiteness or native-space theorem.","major_comments":[{"comment":"The vector spherical harmonic addition formulas for S_l and T_l are cited from the companion preprint [39] and are not proved in this manuscript. These formulas are the keystone of the Fourier analysis: they are used in the proof of Theorem 3.1 to obtain the multiplier identity (3.5), in the m=2 comparison, in Corollary 3.2 via (3.7), and in Theorem 3.4. If a term, sign, or normalization in the formulas for Q, R, V, or W were incorrect, the central one-order gain and the multiplier-preserving property would collapse. Please include a proof, or at least a complete self-contained derivation in an appendix, and state explicitly the hypotheses under which the formulas hold for the series manipulations performed in Theorem 3.1.","section":"Section 2.1, Lemma 2.1"},{"comment":"The manuscript advertises lower-order variants, including m=0, but Remark 3.3 explicitly states that no general positive-definiteness or native-space theorem is established for m=0. Nevertheless, the text after (3.3) says that these cases still produce positive definite divergence-free kernels, and Section 6 presents K0_div convergence rates (Tables 1 and 2) and eigenvalue fits (Figure 4, with sigma=11/2) as if they were consequences of the theory. Since Theorem 5.13 and Corollary 5.5 require Assumption 5.1, which is unproved for K0_div, the K0 experiments are outside the paper's theorems. Either supply the missing theorem using the conditions from [38] or explicitly label the m=0 numerical results as heuristic and outside the proven framework.","section":"Section 3.1 and Remark 3.3"},{"comment":"The convergence rates reported for K0_div are presented as 'consistent with the Hilbert-scale superconvergence estimate established in Theorem 5.13', but Theorem 5.13 presupposes the multiplier condition (2.7) and positive multipliers, which are not established for K0_div. The same applies to the smallest-eigenvalue exponents sigma=11/2, 9/2, 7/2 in Figure 4; only the K1 and K2 cases are covered by the theorems in this paper. The text should distinguish clearly between proved rates and rates inferred from numerical fits, especially because the abstract emphasizes lower-order variants.","section":"Section 6, Tables 1 and 2 and Figure 4"}],"minor_comments":[{"comment":"The paragraph on the red blood cell surface says the interpolant is 'constructed using the proposed multiplier-preserving kernel K0_div', but K0_div was defined in Section 6 as the kernel with beta_0(r)=phi(r), whereas the multiplier-preserving construction is the one in Theorem 3.4 with K = psi'(t)Q + psi(t)R. Please correct the terminology or the kernel definition.","section":"Section 6.3, RBC paragraph"},{"comment":"The sentence 'As we show below, these cases still produce positive definite divergence-free kernels' is too strong given Remark 3.3, which disclaims a theorem for m=0. Please rephrase to state that the m=1 and m=2 cases are proved and that the m=0 case is formal and only supported numerically.","section":"Section 3.1, sentence after (3.3)"},{"comment":"The proof invokes [16, Theorem 4.1] and asserts that it is stated for any real tau>1. Since the fractional-order Sobolev estimates in Theorem 5.11 depend on this lemma, please quote the precise statement of that theorem or provide a proof, so that the reader can verify the fractional regularity range.","section":"Section 5.3, Lemma 5.10"},{"comment":"The manuscript contains several OCR-like artifacts: 'Fourier ana lysis' in the paragraph before Lemma 2.1, 'greaterorsimilar' in the stability discussion of Section 6.2, and irregular spacing in the title and abstract. Please clean these formatting issues before publication.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper relies on two companion preprints by the same group at load-bearing points: Lemma 2.1 from [39] and the m=0 conditions from [38]. The editors may wish to verify that these preprints are publicly and stably available, or require the authors to make the manuscript self-contained on these points. The central m=1 claim appears sound and the numerical rates for K1 and K2 match the theory, but the headline K0 gains are not backed by the theorems and should be either proved or explicitly labelled heuristic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the m=1 construction is the real news. The paper builds divergence-free tangential kernels on S2 from scalar zonal kernels, gives explicit Fourier multipliers, and Corollary 3.2 shows m=1 gets κℓ ≍ (1+λℓ)^−σ instead of the classical m=2 decay ℓ^(−2σ+2), a genuine one-order Sobolev gain. The inverse Laplace–Beltrami construction, with vector multipliers exactly equal to the scalar Legendre coefficients, is a nice trick. The stability and error analysis is standard for the area but competently executed, and the numerical rates match the stated exponents, including the superconvergence predictions.\n\nWhat is good: the construction cleanly separates geometric constraint enforcement from the scalar generator, the m=1 theorem is proved, and the error estimates cover fractional Sobolev spaces and Hilbert-scale superconvergence. The tables and figures are consistent with the theory, and the negative-eigenvalue behavior for some Wendland-based K0 kernels is honestly reported and attributed to a known dimensional positive-definiteness issue.\n\nThe soft spots, in order of size. First, Lemma 2.1 is load-bearing. Every multiplier identity — (3.5), (3.7), and Theorem 3.4 — uses the vector spherical harmonic addition formulas, and the paper simply cites them to a companion manuscript [39]. No proof appears here, and there is no statement of the functional-analytic hypotheses under which the formulas justify the distributional series manipulations in Theorem 3.1. The trace and ℓ=1 checks pass, but that is not a proof. If the formula for Sℓ or Tℓ has a wrong sign or normalization, the central one-order gain collapses. This should be the primary referee request: prove Lemma 2.1 in an appendix, or replace the citation with a fully stated standard reference.\n\nSecond, the m=0 K0 variant headlines several numerical gains, but Remark 3.3 explicitly says there is no positive-definiteness or native-space theorem for it. The experiments are therefore partly outrunning the theory. The behavior is plausible and the failures are consistent with the dimensionality caveat, but treating the K0 rates as established results is premature until that gap is closed.\n\nMinor: no code or data are shipped, which makes precise reproducibility harder, though the node sets and kernels are described in enough detail that a motivated reader could reproduce the experiments.\n\nThis paper is for researchers doing meshfree vector-field interpolation on the sphere and related surfaces. It deserves a serious referee and should go to peer review. My recommendation: send it out, but make the resolution of Lemma 2.1 and the status of K0 explicit conditions for acceptance.","headline":"The m=1 one-order gain and the multiplier-preserving inverse Laplace–Beltrami kernel are the real new results, but every multiplier identity rests on an unproved vector spherical harmonic addition formula quoted from a companion paper.","tokens_in":28442,"tokens_out":2138,"would_cite":false,"duration_ms":21718,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A05","41A25","43A90","65D12"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs divergence-free tangential kernels on the unit sphere and proves that a lower-order isotropic form preserves the full Sobolev regularity of the underlying scalar kernel, gaining one Sobolev order over the classical…","keywords":["divergence-free","vector fields","tangential vector fields","matrix-valued kernels","zonal kernels","multiplier","superconvergence","interpolation on the sphere"],"falsifier":"For a zonal kernel with known Fourier coefficients, e.g., the Gaussian $\\varphi(t)=\\exp(-\\varepsilon^2(2-2t))$ restricted to $S^2$, numerically project the kernel $K_{\\rm div}(\\cdot,y)e$ onto vector spherical harmonics for several degrees $\\ell$ and compare with the closed form in Corollary 3.2; any mismatch at a single degree would refute the main multiplier identity. Alternatively, for a smooth divergence-free target field and a scalar kernel with native space order $\\sigma$, measure the interpolation error in $L^\\infty$ as $h_X$ goes to zero; if the $m=1$ kernel converges at order $h_X^{\\sigma-1}$ or worse instead of $h_X^{\\sigma}$, the claimed one-order gain is false.","tokens_in":27453,"feed_emoji":"🌀","tokens_out":13528,"duration_ms":93930,"temperature":0.7,"pith_summary":"This paper develops a family of matrix-valued kernels for interpolating tangential, divergence-free vector fields on the unit sphere. The central claim is that the classical potential-based construction, which applies surface differential operators to a scalar kernel, unnecessarily loses one Sobolev order of regularity: the resulting vector multipliers decay like $\\ell^{-2\\sigma+2}$ when the scalar zonal kernel has coefficients of order $(1+\\ell(\\ell+1))^{-\\sigma}$. The paper shows that a lower-order isotropic construction, obtained by separating the geometric enforcement of the divergence-free constraint from the choice of scalar generator, yields multipliers of order $\\ell^{-2\\sigma}$ for the same scalar kernel, so the native space is the full $H^\\sigma_{\\rm div}(S^2)$ instead of $H^{\\sigma-1}_{\\rm div}(S^2)$. It further introduces an inverse Laplace--Beltrami construction whose vector multipliers equal the scalar multipliers exactly. If correct, these kernels enforce tangency and zero surface divergence at no cost in regularity, and the paper's stability and error estimates — including superconvergence for extra-smooth targets — make them directly usable for scattered-data approximation on spheres and other embedded surfaces.","feed_headline":"A lower-order kernel preserves full smoothness for sphere fields","feed_subtitle":"Divergence-free interpolation on the sphere reaches the scalar kernel's full regularity without extra derivatives.","key_machinery":"The load-bearing identity is the vector spherical harmonic addition formula of Lemma 2.1 (taken from the authors' earlier paper [39]), which expresses the subspace projection kernels $S_\\ell(x,y)=\\sum_{k=1}^{2\\ell+1}y_{\\ell,k}(x)y_{\\ell,k}(y)^\\top$ and $T_\\ell(x,y)$ as linear combinations of $P_\\ell''(t)$, $P_\\ell'(t)$ and the geometric matrices $Q$ and $R$ defined above. This identity converts the geometric kernel (3.3) into a spectral expansion, so that positivity, native-space equivalence, stability, and convergence all become statements about the scalar Fourier difference $\\widehat\\varphi(\\ell-1)-\\widehat\\varphi(\\ell+1)$. The proof then applies two further quantitative tools: a filtered vector-kernel localization estimate (Lemma 5.3) bounding the kernel of a spectral cut-off in terms of $L^2(1+L\\theta)^{-\\nu}$, and a spherical packing bound (Lemma 5.4) that converts separation into a decay estimate for off-diagonal sums. Together these give the small-eigenvalue bound $\\lambda_{\\min}(A_{K_{\\rm div},X})\\ge C L^2\\min_{1\\le\\ell\\le L}\\kappa_\\ell$ with $L\\sim q_X^{-1}$, from which the stability and all error estimates follow.","core_discovery":"The paper's central discovery is that the order of differentiation in the surface divergence-free kernel construction is not fixed by the constraint. On $S^2$, for every integer $m\\ge 1$, the matrix kernel $K_{\\rm div}(x,y)=\\varphi^{(m-1)}(x\\cdot y)R(x,y)+\\varphi^{(m)}(x\\cdot y)Q(x,y)$ is tangential and divergence-free, where $R(x,y)=tI-yx^\\top$, $Q(x,y)=-(x\\times y)(x\\times y)^\\top$, and $t=x\\cdot y$. For $m=1$, the vector spherical harmonic multipliers are $\\kappa_\\ell = \\frac{\\lambda_\\ell}{2\\ell+1}(\\widehat\\varphi(\\ell-1)-\\widehat\\varphi(\\ell+1))$; when $\\widehat\\varphi(\\ell)\\asymp(1+\\lambda_\\ell)^{-\\sigma}$ with $\\lambda_\\ell=\\ell(\\ell+1)$, this gives $\\kappa_\\ell\\asymp(1+\\lambda_\\ell)^{-\\sigma}$, i.e., the scalar kernel's Sobolev order is preserved. The classical $m=2$ potential-based kernel instead gives $\\kappa_\\ell\\asymp\\ell^{-2\\sigma+2}$, a loss of one order. The paper also proves that the inverse Laplace--Beltrami kernel $K_{\\rm div}(x,y)=\\psi(t)R(x,y)+\\psi'(t)Q(x,y)$ with $\\psi(t)=\\frac{1}{1-t^2}\\int_{-1}^t(c_\\varphi-\\varphi(s))\\,ds$ has multipliers $\\kappa_\\ell=\\widehat\\varphi(\\ell)$, matching the scalar kernel exactly.","pith_inferences":["The $m=0$ variant, although lacking a general positive-definiteness theorem here, converges fastest numerically (about $O(h^{11})$ on smooth fields with the Matern kernel). One testable conjecture the paper leaves open is that for scalar kernels whose radial derivatives satisfy stronger positivity conditions, the $m=0$ construction achieves a native space of order $\\sigma+1/2$ or $\\sigma+1$; a dir","The inverse Laplace--Beltrami construction effectively transplants scalar Fourier multipliers onto the divergence-free subspace. This suggests a general recipe for other two-point homogeneous spaces where vector addition formulas exist; the paper does not pursue it, but the same cancellation of the $\\lambda_\\ell$ factor should work there.","The slower convergence on the bumpy sphere suggests that strong curvature variation breaks the clean $H^\\sigma_{\\rm div}$ native-space equivalence. A plausible extension, which the paper lists as future work, is a curvature-aware normalization of the kernel; one could test this by computing the interpolation error on a family of surfaces interpolating between the sphere and increasingly oscillator","The observed smallest eigenvalues follow the exponents $2\\sigma-2$ predicted for $m=1$ and $m=0$ variants, so the Fourier-localization proof likely captures the true spectral behavior. A matching upper bound for $\\lambda_{\\min}$ remains open and would round out the stability picture."],"forward_implications":["For any scalar zonal kernel with $\\hat\\varphi(\\ell)\\asymp(1+\\lambda_\\ell)^{-\\sigma}$ and $\\sigma>1$, the $m=1$ kernel is positive definite on tangent data and its native space is norm-equivalent to $H^\\sigma_{\\rm div}(S^2)$, so all the scalar kernel's smoothness is available for divergence-free interpolation.","The interpolation matrix in local tangent frames satisfies $\\lambda_{\\min}(A_{K,X})\\ge C q_X^{2\\sigma-2}$ and $\\operatorname{cond}_2(A_{K,X})\\le C q_X^{-2\\sigma}$, so stability degrades exactly as it would for a scalar kernel of order $\\sigma$, not $\\sigma-1$.","For targets $f\\in H^\\tau_{\\rm div}$ with $1<\\tau\\le\\sigma$, the interpolant satisfies $\\|f-I_X f\\|_{H^s}\\le C\\rho_X^{\\sigma-\\tau}h_X^{\\tau-s}\\|f\\|_{H^\\tau}$ for $0\\le s\\le\\tau$, and targets in the native space converge at the full $\\sigma$ rate in $L^\\infty$.","Extra-smooth targets produce superconvergence: for $g\\in F^{1+\\vartheta}_{\\rm div}$, $\\|g-I_Xg\\|_{N_K}\\le C h_X^{\\vartheta\\sigma}\\|g\\|_{F^{1+\\vartheta}_{\\rm div}}$ and $\\|g-I_Xg\\|_{H^s}\\le C h_X^{(1+\\vartheta)\\sigma-s}$ for $0\\le s\\le\\sigma$; smooth test fields on $S^2$ show the predicted high rates, for example roughly $O(h^{11})$ for the $m=0$ variant with the Matern 7/2 kernel.","Because the construction uses the normal cross-product and tangent projection, the same kernel formula applies to any smooth oriented embedded surface; the numerical examples on a torus, a red blood cell surface, and a bumpy sphere confirm tangency and divergence-free reconstruction outside the sphere."],"supporting_citations":[{"why":"Supplies the vector spherical harmonic addition formulas in Lemma 2.1 that are the foundation of the multiplier analysis.","marker":"[39]"},{"why":"Defines the classical m=2 surface divergence-free RBF construction and its error estimates, which the paper's m=1 construction is designed to improve.","marker":"[16]"},{"why":"Provides the baseline Sobolev-order loss for the potential-based kernel and the smooth and limited-regularity test fields used in the numerical comparisons.","marker":"[17]"},{"why":"Gives the operator-based surface divergence-free kernel via cross-product and tangent projection, which the isotropic representation generalizes.","marker":"[31]"},{"why":"Provides the Hilbert-scale projection principle used in Lemma 5.12 to derive superconvergence.","marker":"[21]"},{"why":"Supplies the fractional-order sampling inequality used in Lemma 5.9 for Sobolev error estimates with noninteger orders.","marker":"[2]"},{"why":"Introduces the isotropic matrix-kernel representation and the formal m=0 construction, and describes the stronger positive-definiteness requirement in higher dimension.","marker":"[38]"},{"why":"Provides the standard filtered kernel localization estimates and Bernstein inequalities used in the stability proof in Lemma 5.3.","marker":"[5]"}],"fun_headline_variants":["Divergence-free kernels match scalar smoothness on sphere","Lower-order sphere kernel preserves full regularity","No extra derivatives for divergence-free sphere fields","Sphere vector fields keep scalar kernel's Sobolev order","Divergence-free interpolation reaches scalar kernel smoothness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the vector spherical harmonic addition formulas quoted from the authors' earlier paper [39] are correct and valid under the convergence conditions used here; the manuscript does not prove them, and every multiplier identity and error estimate rests on them.","fun_headline_variants_meta":{"raw":{"variants":["Divergence-free kernels match scalar smoothness on sphere","Lower-order sphere kernel preserves full regularity","No extra derivatives for divergence-free sphere fields","Sphere vector fields keep scalar kernel's Sobolev order","Divergence-free interpolation reaches scalar kernel smoothness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000283,"raw_usage":{"total_tokens":1723,"prompt_tokens":1045,"completion_tokens":678,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":605}},"tokens_in":661,"tokens_out":678,"duration_ms":6951,"temperature":1.0,"reasoning_tokens":605,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T11:07:18.813388+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a zonal kernel with known Fourier coefficients, e.g., the Gaussian $\\varphi(t)=\\exp(-\\varepsilon^2(2-2t))$ restricted to $S^2$, numerically project the kernel $K_{\\rm div}(\\cdot,y)e$ onto vector spherical harmonics for several degrees $\\ell$ and compare with the closed form in Corollary 3.2; any mismatch at a single degree would refute the main multiplier identity. Alternatively, for a smooth divergence-free target field and a scalar kernel with native space order $\\sigma$, measure the interpolation error in $L^\\infty$ as $h_X$ goes to zero; if the $m=1$ kernel converges at order $h_X^{\\sigma-1}$ or worse instead of $h_X^{\\sigma}$, the claimed one-order gain is false.","supporting_citations":[{"cited_title":"Vector field multiplier operators and matrix-valued kernel quasi-interpolation","cited_arxiv_id":"2605.05610","evidence_quote":"Supplies the vector spherical harmonic addition formulas in Lemma 2.1 that are the foundation of the multiplier analysis."},{"cited_title":"Fuselier, F.J","cited_arxiv_id":null,"evidence_quote":"Defines the classical m=2 surface divergence-free RBF construction and its error estimates, which the paper's m=1 construction is designed to improve."},{"cited_title":"Fuselier and G.B","cited_arxiv_id":null,"evidence_quote":"Provides the baseline Sobolev-order loss for the potential-based kernel and the smooth and limited-regularity test fields used in the numerical comparisons."},{"cited_title":"Narcowich, J.D","cited_arxiv_id":null,"evidence_quote":"Gives the operator-based surface divergence-free kernel via cross-product and tangent projection, which the isotropic representation generalizes."},{"cited_title":"Arcang´ eli, M.C","cited_arxiv_id":null,"evidence_quote":"Supplies the fractional-order sampling inequality used in Lemma 5.9 for Sobolev error estimates with noninteger orders."},{"cited_title":"Error estimates for vector field interpolation based on generalized matrix-valued kernels","cited_arxiv_id":"2608.04313","evidence_quote":"Introduces the isotropic matrix-kernel representation and the formal m=0 construction, and describes the stronger positive-definiteness requirement in higher dimension."},{"cited_title":"Dai and Y","cited_arxiv_id":null,"evidence_quote":"Provides the standard filtered kernel localization estimates and Bernstein inequalities used in the stability proof in Lemma 5.3."}],"review_version":1}