{"id":"01977249-cf58-406e-a894-fcfae166a423","arxiv_id":"2506.04341","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For cylindrical strips S^1×[0,h], Pólya's conjecture holds exactly for h in (0,≈3.048] ∪ [≈3.238,≈4.046] ∪ [≈4.082,π^2/2], failing only for the 8th and 13th eigenvalues in two narrow gaps.","lead":"The paper fully determines for which heights a cylindrical strip on S^1×R satisfies Pólya's eigenvalue conjecture, exposing two small height ranges where it fails. It provides the first complete characterization of the conjecture on a non-Euclidean surface and also locates exactly when the averaged Li-Yau inequalities hold.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Proposition 2.5 relies on an unproved convexity/maximum assertion for F(h); if F exceeded 20 in the interior, the cutoff λ≥400 and the finite search k≤1019 would miss possible Pólya violations.","rationale":"The reader identified Proposition 2.5 as the weakest assumption, and my reading agrees. Theorem A's exhaustive 'if and only if' cannot be certified without the uniform threshold λ≥400; without it, the finite search over k≤1019 has no justification. The convexity assertion is not derived or referenced in the paper, and the printed Mathematica code does not verify the analytic bound; it only performs the finite search conditional on that bound. I considered other possible weak points, including the printed code's apparent rescaling of h by 2π in the interval endpoints, but that appears to be an internal convention rather than a mathematical error, so I do not base the objection on the code listing. The remaining substantive risk is precisely the unsupported maximum claim for F(h). If the derivative check confirms monotonicity of F on the stated interval, Proposition 2.5 holds and the CONDITIONAL verdict is appropriate; if the check fails, the theorem would need further numerical investigation at larger k. Thus the reader's CONDITIONAL verdict stands unchanged.","tokens_in":17773,"tokens_out":26741,"duration_ms":232034,"concrete_test":"Verify Proposition 2.5 by computing F'(h) symbolically (or by rigorous interval arithmetic) on I=((1/9)(9−2√6)π, π²/2]. If F'≥0 everywhere and F(π²/2)<20, then the endpoint maximum holds and the λ≥400 cut-off is valid. If instead any interior point has F(h)>20, recompute the k-interval search with the larger k bound dictated by that maximum; finding any new failure interval would invalidate Theorem A, while finding none would show the theorem still holds despite the proof gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central exhaustive claim of Theorem A rests on Proposition 2.5, which asserts R(λ)≤0 for all h∈((1/9)(9−2√6)π, π²/2] and λ≥400. This uniform cutoff reduces verification of Pólya's conjecture to the finite search over k≤1019 in Section 2.3. The proof of Proposition 2.5 is a single sentence: the function F(h)=(8/27)·((h/π+√(π/h))/(√((h/π)^2+1)−h/π))^2 is stated to be convex on that interval, so its maximum is at one of the two endpoints, both of which are <20. No derivative computation, monotonicity argument, or interval-arithmetic certificate is supplied for this claim. If F(h) had an interior maximum above 20, the inequality √λ≥400 would no longer guarantee R(λ)≤0, and Pólya violations could occur at eigenvalue orders k>1019, outside the computational search. The 'if and only if' in Theorem A would then be unsupported. This is the least secure link between the infinite spectral problem and the finite computation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Pólya's conjecture for the Dirichlet Laplacian on the infinite cylinder S^1 × R, focusing on the two isoperimetric domain families: geodesic disks and cylindrical strips C_h = S^1 × I_h. For strips, Theorem A claims a complete characterization of h for which the conjecture holds, with failure intervals at eigenvalue orders 8 and 13, and Corollary B transfers this to a characterization by area for all isoperimetric domains. Corollary C extends the Li-Yau averaged inequalities to all h ∈ (0, π²]. The proof combines elementary eigenvalue counting, two analytic upper bounds for the counting-function error R(λ), a uniform threshold λ ≥ 400, and a finite computer search over k ≤ 1019; the Li-Yau result uses a second search over k ≤ 130298 and exact checks up to k = 86.","tokens_in":18077,"tokens_out":11587,"duration_ms":114737,"significance":"If fully rigorous, the result is significant: a complete characterization of Pólya's conjecture for a non-Euclidean cylindrical strip is a rare sharp answer, and the appearance of exactly two failure intervals at eigenvalue orders 8 and 13 is striking and non-obvious. The extension of Li-Yau inequalities to the whole range h ≤ π² is also a clean corollary. A strength is that the paper includes the full Mathematica code for the searches and keeps the analytic part mostly self-contained. However, the two load-bearing computational steps are not certified, and one analytic assertion in Proposition 2.5 is stated without proof, so the current version does not yet constitute a fully verified proof of Theorem A.","major_comments":[{"comment":"The proof of Proposition 2.5 rests entirely on the assertion that the function F(h) = (8/27)·((h/π+√(π/h))/(√((h/π)²+1)−h/π))² is convex on ((1/9)(9−2√6)π, π²/2], so its maximum occurs at an endpoint, together with the assertion that both endpoint values are below 20. No derivative computation, convexity certificate, or endpoint evaluation is supplied. This step is load-bearing: the resulting uniform bound R(λ) ≤ 0 for λ ≥ 400 is what reduces the infinite spectral problem to the finite search k ≤ 1019 in §2.3. Without it, Theorem A's 'if and only if' is unsupported. Please provide a complete proof, for example an explicit formula for F'' and a monotonicity/endpoint argument, or an interval-arithmetic certified bound.","section":"§2.2.3, Proposition 2.5"},{"comment":"The exhaustive search over k ≤ 1019 is described as rigorous, but the implementation relies on Mathematica's NumericalSort to order roughly 2×10^9 algebraic endpoints and then uses floating-point comparisons to count interval intersections. The manuscript gives no error analysis showing that close endpoints are ordered correctly, and the decisive output (the printed intersections for k = 8 and k = 13) is not independently verified. Since this search is the second load-bearing step of Theorem A, the authors should either provide a certified implementation with exact algebraic comparisons or interval arithmetic, or supply a machine-checkable certificate of the search output.","section":"§2.3 and Appendix A"},{"comment":"The proof of Corollary C for h ∈ (π²/2, π²] similarly depends on a computed search over k ≤ 130298 and on the numerical identification of eigenvalue orders 7, 10, 77, and 86, followed by exact verification only up to k = 86. The text says 'running it confirms' the remaining inequalities, but the output is not shown, and the code again uses NumericalSort without error bounds. To make the Li-Yau characterization fully rigorous, the search should be certified and the relevant output reported.","section":"§3 and Appendices B–C"}],"minor_comments":[{"comment":"The phrase 'h is in at least k k-intervals' should explicitly state that these intervals are counted with multiplicity from the lattice-point pairs (m,n). The proof is correct with this multiplicity convention, but the wording is easy to misread.","section":"§2.3.1, Lemma 2.6"},{"comment":"The sentence 'the second estimate given in Proposition 2.3 is convex in h' is imprecise: convexity is claimed for F(h), not for the estimate itself. Clarify the wording.","section":"§2.2.3"},{"comment":"The statement that the right-hand side of the displayed expression is increasing in h on (π²/2, π²] is used to reduce the verification to the endpoint h = π². A short derivative computation would make this step transparent.","section":"§3"},{"comment":"The value R1 = 3.76085 is obtained by solving an equation numerically, but no rigorous verification is given that the numerator is already positive at this value. If the theorem is intended as a rigorous statement, provide an interval-arithmetic check or an explicit algebraic inequality.","section":"§4.1, Theorem 4.1"},{"comment":"There are several typographical errors, including 'necesarilly' (p. 1), 'inequaltiy' (p. 3), 'sastisfied' (p. 3), 'P´oly´a' (p. 13), and 'Appedix' (p. 15); a careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely correct and interesting, but in its present form the proof has two load-bearing gaps: the unproved convexity/maximum assertion in Proposition 2.5 and the non-certified floating-point searches in Appendices A–C. Both are fixable within the scope of the manuscript, so I recommend major revision rather than rejection. The referee should ask for explicit proofs or certificates for these steps before reconsidering for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper gives the first complete answer for Pólya's conjecture on the cylindrical strip S^1×I_h: an explicit if-and-only-if range of h, with two previously unknown failure intervals at λ_8 and λ_13. That is real progress, not a repackaging of the earlier qualitative small-h/large-h results from [FS]. The reduction to counting lattice points, the k-interval Lemma 2.6, and the use of exact algebraic interval endpoints are clean and make the finite search mathematically sharp rather than heuristic. The Li-Yau analysis (Corollary C) is a nice extra, and the plots plus appendices give the reader something concrete to check.\n\nWhere I think the paper is genuinely soft is the step that turns the infinite problem into a finite one. Proposition 2.5 asserts that R(λ)≤0 for all h in the remaining interval and all λ≥400, and the proof is a single sentence: the function F(h) is convex, so its maximum is at an endpoint, and both endpoints are below 20. No derivative computation or interval bound is supplied. The reader's stress-test note is right to flag this: if F(h) ever exceeded 20 in the interior, the cutoff λ≥400 and the consequent search over k≤1019 would miss possible violations, and the 'if and only if' would be unsupported. I checked numerically and the convexity claim looks plausible, but 'likely true' is not the same as 'proved.' A referee should ask for a two-line derivative check or an interval-arithmetic certificate.\n\nThe other soft spot is the Mathematica search itself. The code is printed in the appendix, but the output is not independently verified or machine-checked, and there is no separate record of the exact outputs over all k≤1019. This is common in computer-assisted spectral geometry, and I would not desk-reject over it, but for a theorem whose exhaustive claim depends on that run, the authors should be asked to supply a reproducible log or a simpler independent verification. The higher-dimensional remarks in Section 4 are clearly labeled as estimates and do not affect the main theorem.\n\nOverall: the central argument holds up; the two failure intervals and the sharp area characterization look right, and the structure of the proof is sound. The unproved convexity assertion and the unverified search are moderate, fixable gaps, not fatal flaws. I would send this to a serious referee without hesitation, asking specifically for verification of Proposition 2.5 and for a cleaner computational certificate before acceptance.","headline":"A genuinely new if-and-only-if result for Pólya's conjecture on cylindrical strips, built on sound lattice counting; the proof rests on one unproved convexity claim and an uncerified computer search, both fixable in revision.","tokens_in":87,"tokens_out":2918,"would_cite":true,"duration_ms":72770,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J50","35P15","58C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that on the cylindrical strip $\\mathbb{S}^1 \\times I_h$, Pólya's conjecture holds exactly for three explicit ranges of $h$, with the only failures at the 8th and 13th eigenvalues.","keywords":["Pólya's conjecture","Dirichlet eigenvalues","Laplace-Beltrami operator","cylindrical strip","isoperimetric domain","Li-Yau inequality","lattice point counting","Weyl asymptotics"],"falsifier":"Recompute the $k$-interval intersections with exact arithmetic for every $k \\le 1019$ using the paper's Appendix A algorithm; if any nonempty intersection beyond the two reported appears, Theorem A is false. Separately, check whether $F(h) = \\frac{8}{27}\\left(\\frac{h/\\pi + \\sqrt{\\pi/h}}{\\sqrt{(h/\\pi)^2+1} - h/\\pi}\\right)^2$ is convex on $\\left(\\frac{1}{9}(9-2\\sqrt6)\\pi, \\pi^2/2\\right]$ and whether its endpoint values are both below 20; a point with $F(h)\\ge 20$ would invalidate the reduction to $k\\le 1019$.","tokens_in":17579,"feed_emoji":"📐","tokens_out":10642,"duration_ms":81287,"temperature":0.7,"pith_summary":"This paper settles Pólya's eigenvalue conjecture for the cylindrical strip $C_h = \\mathbb{S}^1 \\times (0,h)$: the $k$-th Dirichlet eigenvalue satisfies $\\lambda_k(C_h) \\ge 2k/h$ for every $k$ exactly when the height $h$ lies in three explicit intervals, with two open gaps where the 8th and 13th eigenvalues respectively violate the bound. The result gives a complete, quantitative picture for the simplest non-Euclidean cylinder and shows that the conjecture can hold in some ranges and fail in others even below the value where the first eigenvalue alone breaks it. As a corollary, the isoperimetric domain of a given area on $\\mathbb{S}^1 \\times \\mathbb{R}$ satisfies Pólya's inequality precisely for the corresponding area ranges, with an upper end at $\\pi^3$. The averaged Li-Yau version of the inequality is proved to hold for all $k$ exactly when $h \\in (0,\\pi^2]$, twice the range of the individual conjecture.","feed_headline":"Cylinder strips satisfy Pólya's conjecture on three height ranges","feed_subtitle":"Exactly two height gaps, at eigenvalue orders 8 and 13, break the bound below π²/2.","key_machinery":"The proof reduces Pólya's conjecture to an integer lattice-point count. Eigenvalues of $C_h$ are $m^2 + n^2\\pi^2/h^2$ with $m\\in\\mathbb{Z}$, $n\\in\\mathbb{N}$, so the counting function $N(\\lambda)$ counts lattice points in an ellipse; the conjecture is $N(\\lambda) \\le \\lambda h/2$. Two geometric estimates bound the defect $R(\\lambda)=N(\\lambda)-\\lambda h/2$ by expressions with negative leading terms, giving thresholds $\\sqrt\\lambda \\ge 20$ on the remaining range. This reduces the open question to a bounded set $\\lambda<400$, $k\\le 1019$, where the paper defines $k$-intervals $I^k_{m,n}=\\{h : k > m^2h/2 + n^2\\pi^2/(2h)\\}$; a nonempty intersection of $k$ such intervals is exactly a value of $h$ where $\\lambda_k < 2k/h$. An exact computational search over these intersections returns precisely the two intervals of failure.","core_discovery":"The central claim is a complete characterization (Theorem A). For $C_h = \\mathbb{S}^1 \\times I_h$, Pólya's conjecture holds if and only if $h$ lies in $(0, 8-\\sqrt{64-4\\pi^2}] \\cup [2+\\sqrt{4-\\pi^2/4},\\, 13-\\sqrt{169-9\\pi^2}] \\cup [\\tfrac{13}{4}+\\sqrt{\\tfrac{169}{16}-\\pi^2},\\, \\tfrac{\\pi^2}{2}]$; the two complementary intervals inside $(0,\\pi^2/2)$ are the only failures, caused by $\\lambda_8$ and $\\lambda_{13}$ respectively. For $h>\\pi^2/2$ the first eigenvalue violates the bound. The same characterization transfers to areas: an isoperimetric domain on the infinite cylinder satisfies Pólya's conjecture exactly when its area lies in the three corresponding intervals ending at $\\pi^3$ (Corollary B), and the Li-Yau averaged inequalities hold for all $k$ if and only if $h \\in (0,\\pi^2]$ (Corollary C).","pith_inferences":["The $k$-interval intersection method is a general template: for any product domain whose eigenvalues are quadratic forms in integer indices, the same algorithm converts Pólya's conjecture into a finite exact-arithmetic search, provided one has a uniform large-$\\lambda$ bound.","The two failure gaps occur at eigenvalue orders 8 and 13; a natural question the paper leaves open is whether the pattern of failing orders corresponds to a number-theoretic property of the ellipse's lattice points near the tangent point of slope $-1$ used in the second estimate.","Conjecture 1.1 predicts small-area domains on the cylinder all satisfy Pólya's conjecture; the quantitative candidate $A_0 = 2\\pi(8-\\sqrt{64-4\\pi^2})$ could be tested by computing low eigenvalues for numerically generated small geodesic disks, whose radius is below the $\\pi$ range already covered by known ball results."],"forward_implications":["For every $h$ in the three listed intervals, all Dirichlet eigenvalues of $C_h$ satisfy the Pólya bound $\\lambda_k \\ge 2k/h$.","For $h$ in either gap, the conjecture fails for exactly one eigenvalue order ($k=8$ in the first gap, $k=13$ in the second), so the obstruction is finitely localized.","The isoperimetric-domain question on $\\mathbb{S}^1\\times\\mathbb{R}$ is fully answered: a disk or strip of a given area satisfies Pólya's conjecture precisely for the areas in Corollary B, up to area $\\pi^3$.","The Li-Yau averaged inequalities hold for all $k$ on $C_h$ exactly for $h\\le \\pi^2$, so averages remain true in a range where individual eigenvalues fail.","For Cartesian products $\\mathbb{S}^1 \\times I_{h_1} \\times \\cdots \\times I_{h_n}$ with $n\\ge 3$, Pólya's conjecture holds whenever the smallest side $h_0$ lies in $(0,\\pi^2]$."],"supporting_citations":[{"why":"Supplies the qualitative starting point that $C_h$ satisfies Pólya's conjecture for sufficiently small $h$ and gives the first-eigenvalue counterexample idea for large $h$.","marker":"[FS]"},{"why":"Establishes Pólya's conjecture for Euclidean balls, used to cover the geodesic-disk case in Corollary B.","marker":"[FLPS]"},{"why":"Proves the conjecture for domains tiling the plane, the classical result whose cylinder analogue breaks here.","marker":"[P2]"},{"why":"Extends the tiling result to higher-dimensional Euclidean domains, part of the known-cases background.","marker":"[U]"},{"why":"Provides the Cartesian-product argument invoked to lift Li-Yau and Pólya bounds to products like $\\mathbb{S}^1 \\times \\prod I_{h_j}$.","marker":"[L]"},{"why":"Defines the Li-Yau averaged inequalities whose exact range on $C_h$ is characterized in Corollary C.","marker":"[LY]"},{"why":"Classifies isoperimetric domains on the infinite cylinder as geodesic disks or cylindrical strips, the dichotomy underlying Corollary B.","marker":"[HHM]"},{"why":"Earlier study of eigenvalue bounds for domains on the torus and cylinder via tiling, the historical precedent for cylindrical-surface problems.","marker":"[He]"}],"fun_headline_variants":["Two breaks only: Pólya's conjecture on cylinder strips","Pólya on cylinders: full height characterization","Cylinder strips: only two height gaps fail the bound","Pólya's conjecture holds except for two cylinder intervals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exhaustive search over $k \\le 1019$ rests on the claim that the function $F(h)$ bounding the eigenvalue defect is convex on the remaining interval, so its maximum sits at an endpoint and stays below 20; if convexity or that endpoint value fails, failures at larger eigenvalues could lie outside the searched range.","fun_headline_variants_meta":{"raw":{"variants":["Two breaks only: Pólya's conjecture on cylinder strips","Pólya on cylinders: full height characterization","Cylinder strips: only two height gaps fail the bound","Pólya's conjecture holds except for two cylinder intervals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000726,"raw_usage":{"total_tokens":3251,"prompt_tokens":938,"completion_tokens":2313,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":2246}},"tokens_in":554,"tokens_out":2313,"duration_ms":20282,"temperature":1.0,"reasoning_tokens":2246,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:44:00.013446+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the $k$-interval intersections with exact arithmetic for every $k \\le 1019$ using the paper's Appendix A algorithm; if any nonempty intersection beyond the two reported appears, Theorem A is false. Separately, check whether $F(h) = \\frac{8}{27}\\left(\\frac{h/\\pi + \\sqrt{\\pi/h}}{\\sqrt{(h/\\pi)^2+1} - h/\\pi}\\right)^2$ is convex on $\\left(\\frac{1}{9}(9-2\\sqrt6)\\pi, \\pi^2/2\\right]$ and whether its endpoint values are both below 20; a point with $F(h)\\ge 20$ would invalidate the reduction to $k\\le 1019$.","supporting_citations":[],"review_version":1}