{"id":"dbe21ef3-a627-4ffd-be4a-59509fdea004","arxiv_id":"2607.25211","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Steklov eigenvalues of a manifold with many small critically-sized holes converge to weighted Laplace–Beltrami eigenvalues at the optimal rate, with the next-order term given by an indefinite Coulomb energy mediated by the reduced Green function.","lead":"This paper proves sharp convergence rates and a second-order asymptotic expansion for Steklov eigenvalues on manifolds perforated by many small holes, showing the leading correction is an indefinite Coulomb-type interaction energy between the holes. It is a quantitative bridge between spectral geometry and the calculus of variations of Coulomb gases.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's 'optimal' rate lacks a matching lower bound; the sharpness example concerns the auxiliary cell function, not the eigenvalue gap.","rationale":"The reader's stated weakest assumption — the per-cell neutrality condition (2.8) — is actually an explicit hypothesis of the theorem, and Remark 2.1 explains how to satisfy it exactly by adjusting β. A failure of exact mass balance would change the problem, but that is not a hidden assumption or an internal inconsistency; it is part of the setup. I therefore do not see the neutrality condition as the most load-bearing concern.\n\nThe reader's principal reason for CONDITIONAL — the absence of a demonstrated lower bound for the eigenvalue gap — is the real soft spot. Theorem 1.2 asserts 'optimal' convergence rates, but the only sharpness example in the paper concerns the L∞ norm of the global cell function Φ^ε, not the spectrum. The expansion of Theorem 1.3 identifies the leading correction, but no argument is given that this leading term is generically of size ω_d(ε). Because the Coulomb-type energy is indefinite and involves the eigenfunction U, it could vanish or be smaller for specific configurations; without a lower-bound example, the optimality claim is unsupported.\n\nThe constructive proof of the upper bound and the expansion appears coherent: the global cell function estimates, the J1/J2 decomposition, and the corrector equation with the reduced Green function fit together. The missing lower bound is fixable — one can likely use the positive-definite self-energy of each mass-neutral cell against the singular part of Gλ, summed over cells where |U|≥c>0, to obtain a matching lower bound. Therefore the appropriate verdict is unchanged: CONDITIONAL, pending a lower-bound example or a non-vanishing-energy argument.","tokens_in":83389,"tokens_out":27921,"duration_ms":256213,"concrete_test":"On the flat torus T^2 with β=1, take the first nonzero eigenvalue λ=4π², U(x)=cos(2πx_1), and S_ε=εZ²∩[0,1)² with mass-balanced radii satisfying (2.8). Numerically compute σ^ε_1−λ_1 (or directly the leading double-integral λ²∬GλUU dμ^ε dμ^ε) for ε=2^{-n}, n=3,...,8, and check whether |σ^ε_1−λ_1|/ω_2(ε) stays bounded below by a positive constant. Alternatively, analytically bound the diagonal self-energy per cell: for cells with |U(p)|≥c>0, the self-interaction of μ^ε against the singular part of Gλ is positive and O(ω_d(ε)); summing over the positive-density set of such cells would yield a matching lower bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper claims optimal convergence rate |λ_k − σ^ε_k| ≤ Cω_d(ε) in Theorem 1.2, and the abstract/title advertise 'optimal convergence rates'. However, no lower bound for the eigenvalue gap is proved. The only explicit sharpness result (Prop. 3.9, §3.4) shows that the L∞ norm of the auxiliary cell function Φ^ε can be of order ε when the point set is not mass-centered; it does not address the eigenvalue gap. Theorem 1.3 provides an expansion, but it does not establish that the leading Coulomb-type term λ²∬GλUU dμ^ε dμ^ε is bounded away from zero in ω_d(ε) units. Since the Coulomb functional is indefinite and U appears quadratically, it could in principle vanish (or be of smaller order) for symmetric configurations or for eigenfunctions that vanish at the hole centers, making |σ^ε−λ| much smaller than ω_d(ε). Thus the 'optimality' claim is an overstatement. The upper bound and expansion may still be correct, but the sharpness is unproven—this is the load-bearing gap in the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Steklov eigenvalues of a closed manifold perforated by many small geodesic balls, with each hole radius chosen so that its boundary area exactly balances the weighted volume of the corresponding Voronoi cell. The main results are (i) an upper bound of order ω_d(ε) for the difference between the k-th Steklov eigenvalue and the k-th weighted Laplace–Beltrami eigenvalue, together with an H^1 convergence rate of order √ω_d(ε) for harmonically extended eigenfunctions (Theorem 1.2), and (ii) a second-order expansion of the eigenvalue gap whose leading term is an indefinite Coulomb-type energy of the discrepancy measure μ^ε = β dV − dA|∂Ω^ε, mediated by the reduced Green function of Δ+λβ (Theorem 1.3). The proof introduces a global cell function Φ^ε, obtains sharp norm estimates for it (Theorem 3.1), and uses a corrector equation with a Green-function representation in Section 5.","tokens_in":83608,"tokens_out":17093,"duration_ms":156280,"significance":"If the results are correct, the second-order expansion is an original and valuable quantitative link between spectral homogenization on manifolds and Coulomb-gas/interaction-energy theory. The proof is largely self-contained, the estimates are explicit, and the derivation is parameter-free: the hole radii are determined by the mass-balance equation (2.8) and no eigenvalue data are fitted. The introduction of a global cell function, rather than per-cell functions, is a conceptually useful tool. However, the advertised optimality of the convergence rate is not actually proved, and the central theorem is unreadable as printed because of severe textual corruption.","major_comments":[{"comment":"The manuscript repeatedly calls the bound (1.8) ‘optimal’ and the abstract/title advertise ‘optimal convergence rates’, but no matching lower bound for |σ^ε−λ| is proved. The only explicit sharpness statement, Proposition 3.9, concerns the L∞ norm of the auxiliary cell function Φ^ε, not the eigenvalue gap. Theorem 1.3 gives an expansion, but the leading Coulomb-type term is indefinite and depends on the configuration and on U; it may vanish for symmetric configurations or for eigenfunctions with zeros at the hole centers, and the remaining volume term (for d≥3) could in principle cancel it. Since no argument shows that the leading term is bounded below by c ω_d(ε), the claim ‘optimal’ is an overstatement. The authors should either prove a lower bound under a precise nondegeneracy hypothesis or rephrase the claim as a sharp upper bound.","section":"§1.1, Theorem 1.2; §3.4, Prop. 3.9"},{"comment":"The displayed formulas of Theorem 1.3 are heavily corrupted by long strings of non-mathematical symbols (e.g. ‘⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨...’). As printed, these are not well-formed assertions, making it impossible to verify the exact coefficients, the scales at which each term appears, and the claimed error orders. Since this theorem is the central quantitative result, the manuscript is not in publishable form. The authors must replace the corrupted displays with clean, parseable formulas and re-check the ordering of retained terms versus the error O(ω_d(ε)^{3/2}) in dimensions d≥4.","section":"§1.1, Theorem 1.3, Eqs. (1.12)–(1.14)"}],"minor_comments":[{"comment":"The phrase ‘The rate ω_d(ε) is optimal’ is not supported by the preceding statements; see major comment.","section":"§1.1, after Theorem 1.2"},{"comment":"The proof refers to Figure 1, which is not included in the provided text; the figure is important for the construction of the perturbed point set and should be supplied.","section":"§3.4, Proposition 3.9"},{"comment":"The author name appears as ‘Ragha Vendra Venkatraman’ in the running header; this should be corrected.","section":"Header/running title"},{"comment":"There are numerous stray symbol sequences and OCR-like artifacts throughout the text (not only in Theorem 1.3), e.g. in Section 5 and the appendix. A full proofread of the LaTeX source is needed.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The provided manuscript text is not in a clean compilable form: the central theorem's formulas are corrupted, and several other passages contain similar artifacts. I have treated the surrounding prose as the intended statement, but the editor should verify the original source file. The technical core appears coherent, but the optimality claim needs to be either proved (with a nontrivial lower bound for the eigenvalue gap) or properly qualified. I recommend major revision rather than rejection, because the proof structure and the second-order expansion are plausible and valuable if the issues above are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. The paper does prove a genuinely new second-order expansion: σ^ε − λ is expressed, at leading order, as an indefinite Coulomb-type energy of the discrepancy measure β dV − dA|∂Ω_ε, mediated by the reduced Green function of Δ+λβ. That is a real advance over the qualitative convergence in [21,23], and it connects spectral geometry to Coulomb-gas variational problems in a concrete way. The proof is also parameter-free — hole radii are fixed by the mass-balance condition, nothing is fitted to a target eigenvalue.\n\nThe global cell function Φ^ε is a good idea. Replacing cell-by-cell local problems with a single global problem is the right move for disordered point sets. The estimates in Theorem 3.1, including the L∞ sharpness example for Φ^ε, look solid. The reduction in (4.3) to the two terms J1 and J2, the bounds in Lemmas 4.3 and 4.4, and the corrector machinery in Section 5 are all coherent. The representation formula (Corollary 5.7) is clean. I couldn't check every displayed estimate line-by-line because the extract is corrupted by formatting garbage, but the lemmas are all present and the structure is credible.\n\nThe main soft spot is the word 'optimal.' Theorem 1.2 gives the upper bound |σ^ε − λ| ≤ C ω_d(ε), and Theorem 1.3 gives an expansion, but no lower bound on the gap is proved. Proposition 3.9 shows the L∞ norm of the cell function can be order ε in 2D; that concerns an auxiliary function, not the eigenvalue gap. Since the Coulomb functional is indefinite, the leading term could vanish for symmetric configurations or eigenfunctions that vanish on the holes, which would make the gap smaller than ω_d(ε). So the sharpness claim is unproven. That is not a flaw in the proof of the upper bound or the expansion — it is an inflated claim in the abstract and theorem statement. The authors should either prove a lower bound or soften 'optimal' to 'sharp rate in the worst case' or 'the expected rate.'\n\nWho should read it: spectral geometers, people in quantitative homogenization, and anyone working with Coulomb/Riesz gases on manifolds. It deserves a serious referee. My recommendation: send it out, and tell the referee to push on the lower bound and to request a clean version for verification.\n\nReading group? Yes, this would spark a good discussion about what 'optimal rate' should mean when the leading term can vanish.","headline":"Genuine new expansion for Steklov spectra on perforated manifolds; the upper bounds are sharp in scale, but the 'optimal' claim lacks a lower-bound proof.","tokens_in":84148,"tokens_out":3296,"would_cite":true,"duration_ms":34042,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15","58J50","35B27","35C20","35J08"],"pacs":[],"model":"deepseek-v4-flash","headline":"Perforating a closed manifold with many small, mass-balanced holes makes its Steklov spectrum converge to the weighted Laplace–Beltrami spectrum at an optimal rate, with a next-order correction governed by an indefinite Coulomb-type interac","keywords":["Steklov eigenvalues","Laplace-Beltrami eigenvalues","perforated manifolds","quantitative homogenization","Coulomb-type interaction energy","reduced Green function","optimal convergence rates","mass balance condition"],"falsifier":"On a flat 2-torus with a maximally ε-separated set of mass-balanced holes, compute the first nonzero Steklov eigenvalue and compare σ^ε−λ with the Coulomb integral λ²∬ G_λ U U dμ^ε dμ^ε; if the difference does not track that integral at order ε²|log ε|, or if changing the hole configuration changes the sign of the leading correction in a way the indefinite kernel cannot reproduce, the expansion is falsified.","tokens_in":83205,"feed_emoji":"⚡","tokens_out":5236,"duration_ms":52290,"temperature":0.7,"pith_summary":"The paper establishes sharp quantitative convergence of Steklov spectra to weighted Laplace–Beltrami spectra when a closed manifold is perforated by many tiny geodesic balls. The holes are sized so that each hole's boundary area equals the weighted volume of its surrounding Voronoi cell, making each hole neutral and suppressing monopole effects. Under this balance, eigenvalues differ by at most ω_d(ε)=ε^{d/(d−1)} (with a log factor in dimension 2), eigenfunctions by √ω_d, and a second-order expansion shows the leading correction is an indefinite Coulomb-type energy of the discrepancy between bulk and surface measures, mediated by the reduced Green function of Δ+λβ. The result turns a spectral-geometric homogenization statement into an interaction-energy statement, linking it to systems of point charges.","feed_headline":"Perforating a manifold shifts its spectrum by a Coulomb interaction","feed_subtitle":"Sharp rates show Steklov eigenvalues track weighted Laplace-Beltrami ones, with a signed Coulomb-type correction.","key_machinery":"The key object is a global cell function Φ^ε satisfying ΔΦ^ε = β away from the hole boundaries and a jump of normal derivative equal to 1 across ∂Ω^ε. This function quantifies the signed discrepancy measure μ^ε = βdV − dA and converts the eigenvalue difference σ^ε−λ into two controlled terms via integration by parts. Together with the reduced Green function G_λ — the kernel of the inverse of Δ+λβ on the orthogonal complement of the λ-eigenspace, with Coulomb-type singularity near the diagonal — it turns the spectral correction into a double-integral interaction energy. The per-cell neutrality condition μ^ε(V_p)=0 for every Voronoi cell suppresses monopole contributions and is what makes the","core_discovery":"The central claim is that, for a simple weighted Laplace–Beltrami eigenvalue λ with eigenfunction U, the corresponding Steklov eigenvalue σ^ε on the critically perforated manifold admits an expansion whose leading term, of order ω_d(ε), is governed by the indefinite Coulomb-type energy λ² ∬_{M×M} G_λ(x,y) U(x)U(y) dμ^ε(x)dμ^ε(y), where dμ^ε = βdV − dA on the hole boundaries and G_λ is the reduced Green function of Δ+λβ. A companion bound shows |σ^ε−λ| ≤ C ω_d(ε), with eigenfunctions converging at rate √ω_d in H¹, uniformly across eigenvalue clusters. In dimensions two and three, two correction scales are identified explicitly. The perforated manifold behaves, to leading order, like a system","pith_inferences":["If the per-cell neutrality condition were only approximately satisfied, a monopole term would enter at a larger scale, so the sharp rate ω_d(ε) is itself a signature of exact mass balance rather than a generic homogenization phenomenon.","Because the Coulomb-type interaction is indefinite, any variational limit built from these energies will not be coercive; quantitative Gamma-convergence would require a signed or conditional formulation.","The global cell-function technique should adapt to the Euclidean 'dynamical eigenvalue' setting mentioned in the paper, yielding a companion next-order expansion there.","A numerical experiment with prescribed hole configurations on a flat torus could test the predicted dependence of the eigenvalue shift on the reduced Green function and on the signed measure μ^ε, including the sign changes of the leading correction."],"forward_implications":["Steklov eigenvalues on mass-balanced perforated manifolds converge to weighted Laplace–Beltrami eigenvalues at the optimal rate ω_d(ε), and harmonically extended eigenfunctions converge at rate √ω_d in H¹.","The next-order correction is an indefinite Coulomb-type interaction energy, so the perforated manifold is, to leading order, a discrete system of neutral charges coupled to a background charge through the reduced Green function.","In dimensions two and three, two distinct correction scales appear, allowing finer predictions than the qualitative convergence results that preceded this work.","The expansion is carried out for simple eigenvalues and adapted to spectral clusters with multiplicity, giving a uniform statement across degenerate eigenvalues.","The result bridges spectral geometry of perforated manifolds and the variational theory of Coulomb-type interaction energies, opening the door to quantitative variational analysis of shape-optimization limits."],"fun_headline_variants":["Holes punch a Coulomb hole in spectra","Steklov eigenvalues feel a signed Coulomb interaction","Perforations shift spectra with a signed Coulomb term","Critical holes induce an indefinite Coulomb energy","Perforated manifolds shift spectra via indefinite Coulomb energy"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the per-cell neutrality condition (1.1)/(2.8): each hole's surface area must exactly equal the weighted volume of its Voronoi cell, so that μ^ε(V_p)=0; if this mass balance fails, a monopole contribution enters at a larger scale and both the convergence rate and the form of the leading correction change.","fun_headline_variants_meta":{"raw":{"variants":["Holes punch a Coulomb hole in spectra","Steklov eigenvalues feel a signed Coulomb interaction","Perforations shift spectra with a signed Coulomb term","Critical holes induce an indefinite Coulomb energy","Perforated manifolds shift spectra via indefinite Coulomb energy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000536,"raw_usage":{"total_tokens":2403,"prompt_tokens":727,"completion_tokens":1676,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":471,"completion_tokens_details":{"reasoning_tokens":1605}},"tokens_in":471,"tokens_out":1676,"duration_ms":11222,"temperature":1.0,"reasoning_tokens":1605,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T03:05:23.790489+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a flat 2-torus with a maximally ε-separated set of mass-balanced holes, compute the first nonzero Steklov eigenvalue and compare σ^ε−λ with the Coulomb integral λ²∬ G_λ U U dμ^ε dμ^ε; if the difference does not track that integral at order ε²|log ε|, or if changing the hole configuration changes the sign of the leading correction in a way the indefinite kernel cannot reproduce, the expansion is falsified.","supporting_citations":[],"review_version":1}