{"id":"62d4c1ee-1a57-4854-84f8-07f49815eae5","arxiv_id":"2607.19008","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The reciprocal-sum deficit for the first d nonzero Neumann eigenvalues controls eigenvalue displacements, spectral splitting, and Fraenkel asymmetry with sharp, optimal exponents.","lead":"This paper proves sharp quantitative stability for a recently established spectral isoperimetric inequality: if a domain's first d Neumann eigenvalues nearly achieve the ball's reciprocal-sum value, the domain must be nearly round and its eigenvalues nearly match those of the ball. The estimates turn a sharp inequality into a tool for detecting near-roundness from low-frequency spectral data.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"All theorems depend on the unpublished He–Li–Tang reciprocal-sum inequality [9] and its asserted Lipschitz extension, which is only sketched; if that base inequality fails on bounded Lipschitz domains, every stability result fails at the stated level of generality.","rationale":"I read the paper in good faith. The internal development is largely coherent: the quantitative matrix lemma (Lemma 2.2) is proved, the trace manipulations in Theorems 1.4 and 1.6 check out, the mass-transfer argument in Theorem 1.5 is valid, and the sharpness constructions use standard analytic perturbation theory with admissible spherical-harmonic perturbations. The only internal defect I found is the displayed equality in the lower-bound proof of μ1 in Section 3: the expression after 'Therefore' is not identically equal to 1/μ1; it is a valid upper bound after minor rearrangement. This does not change the qualitative conclusion, only the constant. The truly load-bearing assumption is the base reciprocal-sum inequality [9] for bounded Lipschitz domains. The paper itself flags this by asserting the extension without proof. Since Lemma 2.1 is elementary and true, the missing piece is the Lipschitz-regularity step and the equality characterization in [9]. If that fails, all main theorems collapse at their stated level of generality. This is exactly the reader's weakest_assumption, so I agree. The appropriate verdict remains CONDITIONAL: conditional on a complete verification of [9] in the Lipschitz category.","tokens_in":16824,"tokens_out":35878,"duration_ms":275356,"concrete_test":"Obtain or independently reconstruct the full proof of Theorem 1.1 from [9] for bounded Lipschitz domains, and isolate every step that could require boundary regularity: the Weinberger translation via Brouwer, the matrix inequalities (2.7)-(2.8), the use of Hersch's variational principle, and the equality characterization. If the proof goes through verbatim for Lipschitz Ω, the concern is resolved; if it requires a smoothing/approximation argument, supply that argument explicitly. As a computational falsification probe, evaluate D(Ω) for a cube and a rectangular box in d=2,3 with volume |B|; a negative value would disprove the base inequality, while a positive value only provides supporting evidence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is conditional on Theorem 1.1, imported from the unpublished preprint [9]. Every main result (Theorems 1.2, 1.4, 1.5, 1.6, 1.8, 1.9, Proposition 1.10) is stated for bounded Lipschitz domains and uses D(Ω) ≥ 0, with equality only for balls, as its starting point. Section 1.1 asserts that the smooth-domain proof in [9] 'remains valid, with the same proof' for bounded Lipschitz domains, but no proof of this extension is given. Section 2 only sketches Theorem 1.1, citing [9, Lemma 2.2] and deferring Bessel monotonicity to '[an] analysis of the relevant Bessel functions' and '[a]nother computation'. The quantitative Lemma 2.2 is proved in full, and Lemma 2.1 itself is true (it follows from Cauchy–Schwarz), so the weak point is not the linear algebra but the Lipschitz-regularity step and the equality characterization. If the HLT proof needs C^1 boundaries for the transplantation/Hersch argument or for the equality case, then D is not known to be a nonnegative deficit on the stated domain class, and the large-deficit cases in Theorems 1.2 and 1.4 do not repair this. A secondary internal issue: the displayed equality after (3.5) for 1/μ1 is not an identity; the claimed right-hand side is only an upper bound. This is repairable (one can use 1/μ1 ≤ 1/λ + (d-1)C4 D^{1/2}/λ² after D ≤ 1) and does not threaten the existence of γ_d, but it should be corrected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves quantitative stability estimates for the normalized reciprocal-sum deficit D(Ω) of the first d nonzero Neumann eigenvalues, assuming the reciprocal-sum isoperimetric inequality recently proved by He, Li, and Tang. The main results are: quadratic control of the individual normalized eigenvalue displacements and of the Fraenkel asymmetry by D (Theorems 1.2 and 1.5), linear control of the displacement of the sum of the cluster (Theorem 1.4), and bounds on the first spectral gap and the full cluster width in terms of the Szegő–Weinberger deficit (Theorems 1.6 and 1.8). Theorem 1.9 asserts that all stated exponents are optimal via first-order volume-preserving perturbations of the ball, and Proposition 1.10 gives an improved two-dimensional constraint on the joint spectral image of the first two nonzero eigenvalues.","tokens_in":17152,"tokens_out":23622,"duration_ms":189005,"significance":"If the underlying reciprocal-sum inequality is available, this is a valuable and coherent contribution: it gives the first quantitative stability for the Ashbaugh–Benguria reciprocal-sum inequality in arbitrary dimension, with explicit constants and optimal exponents. The matrix method is transparent, the quantitative refinement in Lemma 2.2 is the right tool, and the trace cancellation leading to the linear sum control is elegant. The sharpness constructions are explicit and the paper is unusually honest about its constants. The principal caveat is that every theorem is conditional on Theorem 1.1, which is imported from an unpublished preprint and is asserted, rather than proved, to extend to bounded Lipschitz domains.","major_comments":[{"comment":"The entire paper is built on Theorem 1.1, which is imported from the unpublished preprint [9] and asserted to extend from smooth domains to bounded Lipschitz domains 'with the same proof'. No proof of this extension is given; Section 2 only sketches the argument and defers the Bessel monotonicity and the equality characterization to '[an] analysis of the relevant Bessel functions' and '[a]nother computation'. Since Theorems 1.2, 1.4, 1.5, 1.6, 1.8, 1.9, and Proposition 1.10 all state bounded Lipschitz domains and all use D ≥ 0 (with the equality characterization) as their starting point, a failure of the claimed Lipschitz extension would invalidate the results at the stated level of generality. The authors should either prove the extension, including the equality case, or restate the main results for the class of domains covered by [9].","section":"1.1, Theorem 1.1; Section 2"},{"comment":"The displayed identity d/(λ+D) - (d-1)/(λ+C4D^{1/2}) = 1/(λ+D) + (d-1)C4D^{1/2}/(λ(λ+C4D^{1/2})) is not an identity: after the common denominator, the left-hand side has numerator λ + dC4D^{1/2} - (d-1)D, while the right-hand side has numerator λ² + dλC4D^{1/2} + (d-1)C4D^{3/2}. The subsequent bound 1/μ1 ≤ 1/λ + C5D^{1/2} can still be justified because the intended right-hand side is an upper bound once D ≤ 1, so the error is repairable; nevertheless, the line as written is false and must be corrected.","section":"Section 3, after Eq. (3.5)"}],"minor_comments":[{"comment":"In the statement of Lemma 2.2, 'Theorem 2.1' should read 'Lemma 2.1'; in the proof of Lemma 2.4, 'Theorem 2.2' should read 'Lemma 2.2'.","section":"Section 2"},{"comment":"The text and the Figure 1.1 caption refer to 'Theorem 1.10' where the statement is Proposition 1.10; the proof of Section 7 also ends with 'This proves Theorem 1.10'.","section":"Section 7 and Figure 1.1"},{"comment":"The proof of Lemma 4.1 begins with 'Proof of Theorem 4.1'; this should be 'Proof of Lemma 4.1'.","section":"Section 4"},{"comment":"There are typographical artifacts in the header ('ST ABILITY', 'EIGENV ALUES') that should be cleaned in the final version.","section":"Title and abstract"},{"comment":"The paper should state explicitly that 'domain' means connected open set; the use of the Szegő–Weinberger bound x ≤ λ in Section 7 and parts of the sharpness discussion relies on connectedness if disconnected sets are allowed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the dependence on the unpublished preprint [9] and the asserted Lipschitz-domain extension. If the editor can confirm the status of [9] or the authors can supply the extension as an appendix, the manuscript is likely acceptable after fixing the local algebraic error. The internal argument is otherwise coherent, with tracked constants and credible sharpness constructions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know this about arXiv:2607.19008: the quantitative results are real and well executed, but the whole paper is conditional on an unpublished preprint [9] by He, Li, and Tang, and the authors only assert that the base inequality extends to bounded Lipschitz domains 'with the same proof' without giving that proof. If [9] is correct and the extension is routine, this is a strong stability paper; if not, the theorems fail at the stated level of generality.\n\nWhat is genuinely new: Lemma 2.2, the quantitative matrix lemma, is proved in full and yields D ≥ c3 Σ z_i^2, giving quadratic control of the individual eigenvalue displacements and the Fraenkel asymmetry, plus linear control of the cluster sum via a trace cancellation. The constants are tracked explicitly. The sharpness arguments (Theorem 1.9) use standard first-order perturbation theory and they work: the chosen deformations give D ~ t^2 while the relevant quantities are ~ t or ~ t^2, so the exponents are optimal. The two-dimensional spectral image refinement (Proposition 1.10) is a modest but real improvement over the Bucur–Henrot plus reciprocal-sum region. The internal logic is coherent; the large-deficit cases are covered by universal bounds, and the geometric stability argument retains a positive-semidefinite remainder that is handled sensibly.\n\nThe soft spots are two. First, the dependence on [9]: Theorem 1.1 is imported, and the paper's sketch defers the Bessel monotonicity to '[a]nother computation'. For bounded Lipschitz domains, the equality characterization via the radial monotonicity of G is plausible, but it is not written out. The authors should either include the proof of the Lipschitz extension or restrict the theorems to smooth domains. Second, the displayed equality after (3.5) is not an identity; the right-hand side is an upper bound. This is a minor typo and does not affect the existence of γ_d, but it should be corrected.\n\nOverall, the paper is for spectral geometers working on quantitative stability and shape optimization. It deserves a serious referee, but the referee should have access to [9] and should check the Lipschitz-extension claim. My recommendation: send it to peer review, with a request that the authors address the dependency on [9] and fix the equality typo.\n\nBest,\n\n[Your name]","headline":"A well-executed quantitative stability paper whose load-bearing wall is an unpublished preprint; referee it, but make the authors pin down the Lipschitz extension.","tokens_in":17701,"tokens_out":10351,"would_cite":true,"duration_ms":84662,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15","49R05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A small reciprocal-sum deficit forces a domain to be nearly a ball in spectrum and shape, at optimal rates.","keywords":["Neumann eigenvalues","reciprocal-sum isoperimetric inequality","quantitative stability","bounded Lipschitz domains","eigenvalue cluster","isoperimetric deficit","volume asymmetry"],"falsifier":"For the explicit family of volume-preserving perturbations of the unit ball with radial displacement $t(\\theta_1^2-\\theta_d^2)$, the paper's optimality analysis predicts $D(\\Omega_t)=O(t^2)$ with the first eigenvalue split of order $t$, and for perturbations orthogonal to all spherical harmonics of degree at most two it predicts $D(\\Omega_t) \\asymp A(\\Omega_t)^2 \\asymp t^2$ while the cluster sum moves by order $t^2$. Computing these quantities for this family and finding any different asymptotic order would refute the claimed exponents; finding a bounded Lipschitz domain with $D(\\Omega)\\to 0$ but $A(\\Omega)$ bounded away from zero would refute the geometric stability statement.","tokens_in":16585,"feed_emoji":"🔵","tokens_out":10112,"duration_ms":84552,"temperature":0.7,"pith_summary":"The paper proves that, for bounded Lipschitz domains in any dimension $d \\ge 2$, the recently established reciprocal-sum isoperimetric inequality for the first $d$ nonzero Neumann eigenvalues is quantitatively stable. If the normalized reciprocal-sum deficit $D(\\Omega)$ is small, then every eigenvalue in the first cluster is close to the corresponding eigenvalue of the equal-volume ball, and the deficit controls the squared displacement of each eigenvalue, the displacement of the cluster sum linearly, and the square of the volume-asymmetry from a ball. It also shows that the classical first-eigenvalue deficit controls the gap between the first two nonzero eigenvalues and the full width of the cluster. Nearly spherical deformations show every exponent in these controls is optimal. If the paper is right, a domain whose reciprocal sum nearly attains the ball's value must be nearly a ball in both spectrum and shape, at explicit rates.","feed_headline":"A tiny reciprocal-sum gap makes a domain mimic the ball","feed_subtitle":"Quadratic and linear bounds show a small reciprocal-sum deficit controls the whole first eigenvalue cluster.","key_machinery":"The machinery is a quantitative version of a linear-algebraic comparison lemma. With the mass matrix $A$ and energy matrix $B$ obtained from transplanted ball eigenfunctions, the comparison matrices are $M = aI + bZ$ and $N = \\lambda a I - bZ$, where $Z$ is the trace-free angular-imbalance matrix, $a$ and $b$ are explicit radial constants, and $\\lambda$ is the ball's first nonzero eigenvalue. The refined lemma shows that $\\operatorname{Tr}(B^{-1}A) - d/\\lambda$ is bounded below by a multiple of $\\|Z\\|_{HS}^2$; a variational comparison then converts this into the deficit bound. The trace condition $\\operatorname{Tr} Z = 0$ is essential, because it makes the linear term in the sum of the Ritz values vanish, leaving the linear estimate for the cluster sum. A separate positive-semidefinite remainder in the energy comparison, estimated by a mass-transfer argument, produces the geometric asymmetry bound.","core_discovery":"The central discovery is that the whole first eigenvalue cluster is governed by a single symmetric trace-free matrix $Z = \\int_\\Omega \\theta\\theta^T dx - \\int_B \\theta\\theta^T dx$, where $\\theta = x/|x|$, which measures the angular imbalance of the domain relative to the ball. Transplanting the ball's first eigenspace to the domain and comparing the resulting mass and energy matrices gives the quantitative matrix bound $D(\\Omega) \\ge c_3 \\|Z\\|_{HS}^2$. This yields the quadratic stability inequalities $D(\\Omega) \\ge \\gamma_d \\max_i(\\tilde\\mu_i(\\Omega)-\\tilde\\mu_i(B))^2$ and $D(\\Omega) \\ge \\kappa_d A(\\Omega)^2$, where $A$ is the minimal relative volume of the symmetric difference with an equal-volume ball, together with the linear control $D(\\Omega) \\ge \\tau_d |\\sum_i \\tilde\\mu_i(\\Omega) - \\sum_i \\tilde\\mu_i(B)|$. The trace-free structure of $Z$ is what makes the linear control possible: first-order terms cancel in the sum. The paper proves the gap estimates from the classical first-eigenvalue deficit, derives an improved two-dimensional constraint on the joint image of the first two normalized eigenvalues, and constructs first-order volume-preserving perturbations of the ball to show that all exponents are optimal.","pith_inferences":["The same matrix comparison should transfer to other transplantation-based eigenvalue inequalities, whenever a trace-free imbalance matrix can be defined, giving analogous stability rates for higher eigenvalues or other boundary conditions.","The optimality examples suggest that equality in the stability inequalities is approached only by nearly spherical domains, so the ball is quantitatively isolated as the unique minimizer of the reciprocal sum under a volume constraint.","A testable extension is to compute the explicit constants $\\gamma_d,\\tau_d,\\kappa_d$ for $d=2,3$ and compare the predicted envelopes with numerical spectra of explicit families of domains, which would also indicate how sharp the constants in $O(\\sqrt{\\varepsilon})$ are."],"forward_implications":["A bounded Lipschitz domain whose reciprocal-sum deficit is below $\\varepsilon$ has every one of its first $d$ normalized Neumann eigenvalues within $O(\\sqrt{\\varepsilon})$ of the ball's first eigenvalue.","The same deficit bounds the splitting $\\tilde\\mu_d(\\Omega)-\\tilde\\mu_1(\\Omega)$ by $O(\\sqrt{\\varepsilon})$, so the ball's $d$-fold eigenvalue can only split at a rate no faster than the square root of the deficit.","The displacement of the sum of the first $d$ eigenvalues is controlled linearly: $|\\sum_i \\tilde\\mu_i(\\Omega) - \\sum_i \\tilde\\mu_i(B)| = O(D(\\Omega))$, giving a Neumann-cluster analogue of known cluster stability for the fixed-boundary spectrum.","The classical first-eigenvalue deficit bounds the first spectral gap and the full cluster width linearly, so the ball maximizes certain convex combinations of low eigenvalues over an explicit range.","In dimension two, the improved bilinear constraint excludes part of the previously admissible $(\\tilde\\mu_1,\\tilde\\mu_2)$ region, and equality holds only for disks."],"supporting_citations":[{"why":"proves the base reciprocal-sum isoperimetric inequality for smooth domains and supplies the linear-algebraic lemma that the present paper quantifies; the stability theorems assume this inequality extends to bounded Lipschitz domains.","marker":"[9]"},{"why":"supplies the transplantation construction of trial functions from the ball's first eigenspace and the classical first-eigenvalue isoperimetric inequality used to define the deficits.","marker":"[18]"},{"why":"provides the sharp upper bound for the second non-trivial Neumann eigenvalue in dimension two, used for the universal bounds and equality cases in the planar region.","marker":"[6]"},{"why":"provides the universal upper bound for higher Neumann eigenvalues in dimension at least three, used in the large-deficit cases.","marker":"[13]"},{"why":"supplies the nearly-spherical asymmetry estimate used to show that the exponents in the geometric and cluster-sum estimates are optimal.","marker":"[4]"}],"fun_headline_variants":["Reciprocal-sum gap pins domain to ball with optimal bounds","One matrix governs eigencluster and shape stability","Small deficit, big control: eigenvalue cluster and geometry","Quadratic stability: ball-like spectrum and shape from one gap","Optimal exponents for reciprocal-sum isoperimetric stability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reciprocal-sum inequality used as the starting point was proved only for smooth domains, and the paper assumes that it extends to all bounded Lipschitz domains with the same proof; if that extension fails, the stated theorems are false at their claimed level of generality.","fun_headline_variants_meta":{"raw":{"variants":["Reciprocal-sum gap pins domain to ball with optimal bounds","One matrix governs eigencluster and shape stability","Small deficit, big control: eigenvalue cluster and geometry","Quadratic stability: ball-like spectrum and shape from one gap","Optimal exponents for reciprocal-sum isoperimetric stability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1641,"prompt_tokens":949,"completion_tokens":692,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":614}},"tokens_in":565,"tokens_out":692,"duration_ms":7197,"temperature":1.0,"reasoning_tokens":614,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:34:47.550337+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the explicit family of volume-preserving perturbations of the unit ball with radial displacement $t(\\theta_1^2-\\theta_d^2)$, the paper's optimality analysis predicts $D(\\Omega_t)=O(t^2)$ with the first eigenvalue split of order $t$, and for perturbations orthogonal to all spherical harmonics of degree at most two it predicts $D(\\Omega_t) \\asymp A(\\Omega_t)^2 \\asymp t^2$ while the cluster sum moves by order $t^2$. Computing these quantities for this family and finding any different asymptotic order would refute the claimed exponents; finding a bounded Lipschitz domain with $D(\\Omega)\\to 0$ but $A(\\Omega)$ bounded away from zero would refute the geometric stability statement.","supporting_citations":[{"cited_title":"Weinberger, An isoperimetric inequality for the N -dimensional free membrane problem , J","cited_arxiv_id":null,"evidence_quote":"supplies the transplantation construction of trial functions from the ball's first eigenspace and the classical first-eigenvalue isoperimetric inequality used to define the deficits."},{"cited_title":"222 (2019), no","cited_arxiv_id":null,"evidence_quote":"provides the sharp upper bound for the second non-trivial Neumann eigenvalue in dimension two, used for the universal bounds and equality cases in the planar region."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the universal upper bound for higher Neumann eigenvalues in dimension at least three, used in the large-deficit cases."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the nearly-spherical asymmetry estimate used to show that the exponents in the geometric and cluster-sum estimates are optimal."}],"review_version":1}