{"id":"51d42a96-7e31-40e1-82b0-9f71f99a3f57","arxiv_id":"2507.04307","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every bounded Lipschitz domain, Pólya's eigenvalue bound holds up to factor 1+ε for all eigenvalues above an explicit threshold, and exact Pólya bounds are proved for new irregular domain classes.","lead":"This paper proves that on any bounded Lipschitz domain, all sufficiently large Dirichlet eigenvalues satisfy Pólya's conjecture up to a factor of (1+ε), with an explicit threshold Λ(ε,Ω). It also finds new classes of domains, including triangles and strip-tiling domains with many holes, where Pólya's conjecture holds exactly.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof gap: Section 4 Step 2 assumes the complement R\\Ω of a minimal rectangle is a Lipschitz domain; an explicit Lipschitz Ω tangent to a face gives R\\Ω with a cusp, so Theorem 1.5/1.3 lacks a proved premise.","rationale":"The reader identifies the same load-bearing assumption: Section 4, Step 2 asserts without proof that R\\Ω is a Lipschitz domain satisfying the boundary-layer estimate (4.2) with the original constant CLip(Ω). My independent check confirms the concern and makes it sharper: there exists a bounded Lipschitz Ω whose minimal rectangle R has complement R\\Ω with a cusp, so the asserted Lipschitz regularity is not merely unproved but false. The example is Ω={0<x<1, (x−1/2)^2<y<1} inside R=(0,1)^2; Ω is Lipschitz and R is its minimal admissible rectangle, while R\\Ω={0<y<(x−1/2)^2} fails the cone condition at (1/2,0). The paper gives no substitute argument for the lower bound on N_{R\\Ω}, and (4.2) is used directly to prove the upper bound for eigenvalues in Theorem 1.5 and hence the epsilon-loss Pólya statement in Theorem 1.3. The gap is repairable in principle if (4.2) can be proved directly for the complement of a Lipschitz domain in a rectangle, perhaps with a different constant, but as written the proof of the central claim is incomplete. I therefore keep the conditional verdict: the paper should not be accepted until the complement-domain estimate is either proved or replaced. The strip-tiling and admissible-class results in Sections 2–3 appear independent of this Step 2 issue and are not affected by this objection.","tokens_in":34695,"tokens_out":26647,"duration_ms":314891,"concrete_test":"Analytical check: for the example above, compute the ε-boundary layer of Ω1 near the cusp and compare with ε|∂Ω1|. If |{x∈Ω1:dist(x,∂Ω1)<ε}|/(ε|∂Ω1|) is unbounded as ε→0, then (4.2) fails and Step 2 collapses. If it is bounded, the counterexample still disproves the 'obviously Lipschitz' assertion, so the authors must either prove (4.2) directly for general Lipschitz Ω or replace CLip(Ω) by a separately controlled constant for R\\Ω.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The weakest point is the unproved claim in Section 4, Step 2 (before (4.2)) that Ω1=R\\Ω is 'obviously' a Lipschitz domain. This is false. Take R=(0,1)^2 and Ω={(x,y):0<x<1, (x−1/2)^2<y<1}. Ω is bounded Lipschitz: its boundary consists of the graph of the C^1 function f(x)=(x−1/2)^2 (with |f'|≤1), the top side, and the two vertical sides; R is its minimal admissible rectangle since Ω touches all four sides and has width 1. But Ω1={0<y<(x−1/2)^2} has a cusp at (1/2,0): the arcs y=0 and y=f(x) are tangent there, and no cone of positive aperture from (1/2,0) lies in Ω1, so Ω1 is not a Lipschitz domain. The subsequent bound (4.2) with CLip(Ω1)≤CLip(Ω) is therefore unsupported. Since Theorem 1.3's explicit Λ(ε,Ω) is derived from Theorem 1.5's Step 2, the proof of the central claim has a real gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a quantitative path toward Pólya's conjecture for Dirichlet eigenvalues on bounded Lipschitz domains. Its central result, Theorem 1.3, states that for every ε∈(0,1) and every bounded Lipschitz domain Ω⊂R^n, all eigenvalues λ_k(Ω) above an explicit threshold Λ(ε,Ω) satisfy the ε-loss Pólya bound k≤(1+ε)|Ω|ω(n)(2π)^{-n}λ_k(Ω)^{n/2}. This is derived from Theorem 1.5, a two-sided quantitative remainder estimate for the Weyl counting function with explicit constants, obtained without Neumann eigenvalues. The paper also proves refined eigenvalue estimates on strip-tiling domains and product domains, and exhibits classes of domains with irregular boundaries on which the full Pólya conjecture holds. The arguments use elementary one-dimensional eigenvalue estimates, a Laptev-type product argument, Whitney decompositions, and explicit counting estimates on cubes.","tokens_in":34977,"tokens_out":9052,"duration_ms":101826,"significance":"If the main results are correct, the paper makes a substantial contribution to a longstanding problem: it reduces the ε-loss form of Pólya's conjecture for all Lipschitz domains to a finite verification below an explicit eigenvalue threshold, and it provides new classes of non-tiling, non-product domains where the full conjecture holds. Strengths of the paper include the explicit nature of all constants, the elementary and checkable one-dimensional estimates in Section 2, the derivation of improved estimates for strip-tiling domains and triangles, and the clear statement of the computational reduction in Theorem 1.3. The paper does not rely on unproved numerical computations or fitted parameters, and it gives credit to the prior works it builds on. However, one load-bearing geometric assertion in the proof of Theorem 1.5 is not proved and is in fact false as stated, which prevents the paper from being accepted in its present form.","major_comments":[{"comment":"The sentence 'Obviously, Ω1=R\\Ω is also a Lipschitz domain' is not correct. For example, take Ω={(x,y)∈(0,1)^2 : (x−1/2)^2<y<1}. This is a bounded Lipschitz domain, and R=(0,1)^2 is its minimal admissible rectangle, but Ω1={0<y<(x−1/2)^2} has a cusp at (1/2,0), where the two boundary arcs y=0 and y=(x−1/2)^2 are tangent; Ω1 therefore does not satisfy the cone condition at that point. The subsequent estimate (4.2) is asserted without proof, and it is exactly what is needed to apply Step 1 to Ω1 and to obtain the upper bound (1.9). Since Theorem 1.3 depends on (1.9), this is a load-bearing gap. I am not claiming that (4.2) is necessarily false for Lipschitz Ω; a direct boundary-layer estimate for R\\Ω may well be true. But the manuscript must supply a proof, or replace the auxiliary domain by one for which the estimate is proved.","section":"Section 4, Step 2 of the proof of Theorem 1.5 (before Eq. (4.2))"},{"comment":"The proof of Corollary 1.6 is only a sketch: the displayed boundary-layer estimate |{x∈R\\Ω: dist(x,∂Ω)<ε}| ≤ |∂R\\∂Ω| ε is justified by the phrase 'convexity of R and Ω would imply' and a reference to Figure 5. This estimate is used for the convex case of Theorem 1.3, so it is not a cosmetic detail. The proof should be written out in full, specifying how the Whitney decomposition with respect to ∂Ω gives the claimed constant, or the corollary should be restated with a proof.","section":"Corollary 1.6, proof in Section 4"}],"minor_comments":[{"comment":"The word 'Obviously' should be removed, and the notation '∂R∩∂Ω1' clarified: it is used as if ∂R and ∂Ω1 have a well-defined common boundary portion, which needs explanation once the boundary-layer estimate is properly proved.","section":"Section 4, Step 2"},{"comment":"In the proof of Lemma 3.7, several inequalities are asserted with '>' without displaying the algebraic verification, especially in the estimate for n≥4; a short calculation or an appendix would improve readability and checkability.","section":"Lemma 3.7"},{"comment":"The proof shows that the function f(Λ) is monotone decreasing, and existence of Λ(ε,Ω) follows; it would be helpful to state explicitly in the theorem that Λ(ε,Ω) is the unique solution of (1.7) and (1.8), respectively.","section":"Theorem 1.3"}],"recommendation":"major_revision","confidential_remarks":"The gap in Section 4, Step 2 is a genuine missing proof, not a wording issue: the auxiliary domain R\\Ω is not Lipschitz in general, as the cusp example shows. The authors should either prove the boundary-layer estimate (4.2) directly for complements of Lipschitz domains in their minimal rectangles, or restructure the argument so that the auxiliary domain has the needed regularity. The rest of the paper contains many explicit, checkable estimates and appears sound in its main strategy, so the result is plausibly repairable within the scope of a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has two genuinely valuable pieces and one load-bearing proof gap. The valuable pieces are the refined eigenvalue estimates on product domains and strip-tiling domains (Theorems 1.10, 2.3), and the construction of domains that satisfy Pólya's conjecture exactly by removing admissible cubes from strip-tiling domains (Theorem 1.15). These arguments are explicit, the constants are written out, and the strategy is new: use a better-than-Pólya estimate on a larger tiling domain and subtract a quantitative lower bound for the removed cubes. That part deserves serious attention.\n\nThe problem is Section 4, Step 2 of Theorem 1.5. The authors write that R\\Omega, the complement of the minimal admissible rectangle, is 'obviously' a Lipschitz domain. That is false. Take R=(0,1)^2 and Omega = {0<x<1, (x-1/2)^2 < y < 1}. This is a bounded Lipschitz domain whose minimal admissible rectangle is R itself, but R\\Omega near (1/2,0) has a cusp: the two boundary arcs are tangent there, so no cone condition holds. The boundary-layer estimate (4.2) is therefore not justified by the argument given. Since Theorem 1.3 uses Theorem 1.5 directly, the epsilon-loss Pólya statement for general Lipschitz domains lacks a proved premise. The gap may be repairable—for this example (4.2) still holds with a slightly different constant—but the proof as written does not cover it, and the same issue can arise whenever Omega touches a face of the minimal rectangle.\n\nCorollary 1.6 is also only sketched, with the Whitney decomposition for the convex complement not fully specified. That is a minor issue by comparison.\n\nThe rest of the mathematics looks coherent and mostly checkable. I did not see circularity: the proofs use Pólya's theorem for tiling domains, Laptev's product argument, and Seeley's asymptotics, which are independent. No constants appear to be fitted.\n\nWho should read this? Spectral geometers working on Pólya's conjecture and Riesz means. The theorem on exact Pólya classes for irregular domains is a real advance even if Theorem 1.3 needs repair. I would send this to review: the core idea is important, and the gap is clearly isolated. The referee should ask for a proof or a corrected statement of the Lipschitz property of the complement, or a direct proof of (4.2).","headline":"Strong new results on exact Pólya classes for irregular domains, but the headline epsilon-loss theorem for all Lipschitz domains rests on a false 'obvious' Lipschitz claim about the rectangle complement.","tokens_in":35489,"tokens_out":4986,"would_cite":true,"duration_ms":54696,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15","35P20","42B37"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for every bounded Lipschitz domain and any ε>0, all sufficiently large Dirichlet eigenvalues satisfy the Pólya inequality up to a factor (1+ε), with an explicit threshold.","keywords":["Dirichlet eigenvalues","Weyl law","Pólya conjecture","Lipschitz domains","eigenvalue counting function","Riesz means","strip-tiling domains","quantitative remainder"],"falsifier":"For a concrete Lipschitz domain that touches the boundary of its minimal admissible rectangle (for example, a rectangle with a thin slit attached to one face), compute the measure of {x ∈ R\\Ω : dist(x, ∂(R\\Ω)) < ε} for small ε and check whether it is bounded by C_Lip(Ω) ε |∂(R\\Ω)| with the constant from (1.6); if the bound fails, the upper-bound estimate (1.9) and Theorem 1.3 are not established by this proof.","tokens_in":34495,"feed_emoji":"📐","tokens_out":5919,"duration_ms":56464,"temperature":0.7,"pith_summary":"The paper establishes that Pólya's conjecture holds up to an arbitrarily small relative loss for all large Dirichlet eigenvalues on every bounded Lipschitz domain in any dimension n ≥ 2. For each ε ∈ (0,1) it gives an explicit threshold Λ(ε,Ω) such that every eigenvalue λ_k(Ω) above it satisfies k ≤ (1+ε) times the Weyl term (|Ω|ω(n)/(2π)^n)$λ_k^{{n/2}}$. This reduces the ε-loss version of Pólya's conjecture to checking finitely many small eigenvalues, a computational problem. The paper also constructs, in all dimensions, a class of possibly irregular domains (strip-tiling domains with admissible cubes removed) that satisfy Pólya's conjecture exactly, and it shows that triangles enjoy an even stronger bound than Pólya predicted.","feed_headline":"Pólya's conjecture proven with ε-loss on every Lipschitz domain","feed_subtitle":"An explicit threshold makes checking the conjecture for large eigenvalues a finite computation.","key_machinery":"The argument rests on two main components. First, refined Riesz-means estimates for one-dimensional intervals (Lemma 2.1 and Lemma 2.2) feed into a product-domain estimate (Theorem 2.3), giving counting-function upper bounds with an explicit negative lower-order term C_1(n−1)|Ω₂|√λ. Second, the minimal admissible rectangle R = $R^{{n−1}}$ × I_min, the smallest rectangle of width equal to the domain's width that contains Ω, enables a proof that avoids Neumann eigenvalues: monotonicity gives N_Ω(λ) ≤ N_R(λ) − N_{R\\Ω}(λ), and the lower bound for N_{R\\Ω} comes from a Whitney decomposition of R\\Ω together with explicit lower bounds for the counting function on cubes (Lemmas 3.4 and 3.7).","core_discovery":"The central discovery is a quantitative two-sided estimate for the Dirichlet eigenvalue counting function on every bounded Lipschitz domain, with all constants explicit, obtained without using Neumann eigenvalues. The upper bound states that for each k, k is at most the Weyl term plus a lower-order boundary term controlled by C_Lip(Ω)|∂(R\\Ω)| times an explicit expression involving λ_k; the lower bound is the analogous Weyl term minus a boundary term. As a corollary, for any ε > 0 there is an explicit Λ(ε,Ω) such that the ε-loss Pólya inequality holds for all eigenvalues above Λ. The proof combines refined Riesz-means estimates on product domains with a Whitney decomposition of the domain and of the complement of its minimal enclosing rectangle, together with monotonicity of Dirichlet eigenvalues.","pith_inferences":["If the explicit threshold Λ(ε,Ω) can be computed for a given domain, Pólya's conjecture for that domain could in principle be settled by a finite numerical verification of the finitely many eigenvalues below the threshold.","The main obstruction to the full conjecture appears to be the lower bound on counting functions of complements R\\Ω; improving the Whitney-type decomposition or using a better covering may yield the full conjecture, not just the ε-loss version.","The explicit constants in the remainder estimate could be tested against numerical eigenvalue computations on standard domains such as rectangles and balls to gauge how sharp the boundary-layer coefficient is.","The construction with strip-tiling domains and removable cubes might extend to removing more general admissible sets whose counting-function lower bound is known, beyond dyadic cubes."],"forward_implications":["On every bounded Lipschitz domain, the ε-loss version of Pólya's conjecture holds for all eigenvalues above an explicit threshold, so verifying the full conjecture reduces to checking finitely many small eigenvalues.","The new remainder estimate is uniform in λ with explicit constants, giving a quantitative answer to the question of a uniform remainder for Weyl's law without invoking Neumann eigenvalues.","Strip-tiling domains, and hence all triangles in the plane, satisfy an inequality stronger than Pólya's conjecture, with a negative lower-order term proportional to λ_k^{(n−1)/2}.","In all dimensions n ≥ 2, there exist domains with rather irregular shapes (strip-tiling domains with admissible cubes removed) that satisfy Pólya's conjecture exactly, not merely up to ε.","For convex domains the threshold Λ(ε,Ω) can be taken smaller, since the Lipschitz-layer constant reduces to 1."],"supporting_citations":[{"why":"Pólya's tiling proof, whose method is adapted to strip-tiling domains in Theorem 1.10.","marker":"[37]"},{"why":"Laptev's product-domain inequality that Theorem 2.3 extends with explicit constants.","marker":"[28]"},{"why":"Netrusov–Safarov's quantitative Weyl remainder on rough domains, a prior result this paper's proof avoids using Neumann eigenvalues.","marker":"[34]"},{"why":"Frank–Larson's uniform remainder estimate for Lipschitz domains, the recent benchmark this paper's explicit-constant estimate complements.","marker":"[11]"},{"why":"Berezin's semiclassical bound, which together with Li–Yau gives the best prior global Pólya-type inequality.","marker":"[2]"},{"why":"Seeley's sharp two-term asymptotic remainder, used in Theorem 1.9 to handle smooth holes.","marker":"[39]"},{"why":"Source for the Whitney decomposition (Proposition 4.1) that underlies the counting-function lower bounds in Section 4.","marker":"[18]"}],"fun_headline_variants":["ε-loss Pólya proves for large eigenvalues on Lipschitz","Explicit threshold makes ε-Pólya a computational check","Strip-tiling domains surpass Pólya's estimate","Quantitative Weyl remainder yields ε-Pólya without Neumann","Pólya's conjecture up to ε: all Lipschitz, explicit constant"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes without proof that the complement of the minimal enclosing rectangle, R\\Ω, is a Lipschitz domain whose boundary layer has measure at most C_Lip(Ω) ε |∂(R\\Ω)|; if this fails for a domain touching the rectangle's faces or with corners, the main upper bound and Theorem 1.3 would not follow.","fun_headline_variants_meta":{"raw":{"variants":["ε-loss Pólya proves for large eigenvalues on Lipschitz","Explicit threshold makes ε-Pólya a computational check","Strip-tiling domains surpass Pólya's estimate","Quantitative Weyl remainder yields ε-Pólya without Neumann","Pólya's conjecture up to ε: all Lipschitz, explicit constant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000404,"raw_usage":{"total_tokens":2087,"prompt_tokens":910,"completion_tokens":1177,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":1089}},"tokens_in":526,"tokens_out":1177,"duration_ms":11948,"temperature":1.0,"reasoning_tokens":1089,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:50:45.494881+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete Lipschitz domain that touches the boundary of its minimal admissible rectangle (for example, a rectangle with a thin slit attached to one face), compute the measure of {x ∈ R\\Ω : dist(x, ∂(R\\Ω)) < ε} for small ε and check whether it is bounded by C_Lip(Ω) ε |∂(R\\Ω)| with the constant from (1.6); if the bound fails, the upper-bound estimate (1.9) and Theorem 1.3 are not established by this proof.","supporting_citations":[{"cited_title":"London Math","cited_arxiv_id":null,"evidence_quote":"Pólya's tiling proof, whose method is adapted to strip-tiling domains in Theorem 1.10."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Laptev's product-domain inequality that Theorem 2.3 extends with explicit constants."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Netrusov–Safarov's quantitative Weyl remainder on rough domains, a prior result this paper's proof avoids using Neumann eigenvalues."},{"cited_title":"Math.241 (2025), 999-1079","cited_arxiv_id":null,"evidence_quote":"Frank–Larson's uniform remainder estimate for Lipschitz domains, the recent benchmark this paper's explicit-constant estimate complements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Berezin's semiclassical bound, which together with Li–Yau gives the best prior global Pólya-type inequality."},{"cited_title":"in Math.29 (1978), 244-269","cited_arxiv_id":null,"evidence_quote":"Seeley's sharp two-term asymptotic remainder, used in Theorem 1.9 to handle smooth holes."},{"cited_title":"Texts in Math., 249 Springer, New York, 2008, xvi+489 pp","cited_arxiv_id":null,"evidence_quote":"Source for the Whitney decomposition (Proposition 4.1) that underlies the counting-function lower bounds in Section 4."}],"review_version":1}