P\'{o}lya's conjecture on $\mathbb{S}^1 \times \R$

Pedro Freitas; Rui Wang

REVIEW 3 major objections 5 minor 16 references

P\'{o}lya's conjecture on $\mathbb{S}^1 \times \R$

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.04341 v1 pith:GZDJGERB submitted 2025-06-04 math.SP

Pedro Freitas , Rui Wang This is my paper

classification math.SP MSC 58J5035P1558C40

keywords Pólya'sconjectureDirichleteigenvaluesLaplace-BeltramioperatorcylindricalstripisoperimetricdomainLi-YauinequalitylatticepointcountingWeylasymptotics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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 $k\le 1019$.

Watch

Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

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 $(0,\pi^2]$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

Where 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.

The 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.

Overall: 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[§2.2.3, Proposition 2.5] 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.
[§2.3 and Appendix A] 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.
[§3 and Appendices B–C] 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.

minor comments (5)

[§2.3.1, Lemma 2.6] 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.
[§2.2.3] 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.
[§3] 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.
[§4.1, Theorem 4.1] 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.
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem A is established by exact eigenvalue counting, a logical equivalence, and an exhaustive finite search independent of fitted parameters.

full rationale

The derivation is self-contained. The eigenvalues are explicit via equation (2.1), and Pólya's conjecture is restated exactly as R(λ) ≤ 0. The analytic bounds in Propositions 2.2–2.3 and the uniform threshold in Proposition 2.5 come from lattice-point geometry and do not assume the target conclusion. Lemma 2.6 is a direct logical equivalence between λ_k < 2k/h and membership in at least k k-intervals, so the subsequent computer search over k ≤ 1019 enumerates precisely all possible counterexamples in the remaining parameter range; there are no fitted constants, no data-dependent selection, and no parameter tuned to force the answer. The cited works [FS], [F], [FLP], and [FMS] are used only for context and motivation, such as earlier examples of tiling domains and prior product results, not as load-bearing premises for Theorem A. The weakest point is Proposition 2.5's asserted convexity and endpoint-maximum property of F(h), which is not proved in the paper; if that assertion were false, the cutoff λ ≥ 400 and the finite search could miss counterexamples. That is a rigor or correctness concern, not circularity, because the convexity claim is an independent analytic assertion rather than an equivalent reformulation of the conclusion.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The proof relies on the standard eigenvalue formula for the rectangle, the isoperimetric classification of domains on the cylinder, and an unproved convexity assertion in Proposition 2.5. No free parameters are fitted; h is the variable under study. No new entities are introduced.

assumptions (3)

standard math Eigenvalues of the Dirichlet Laplacian on S^1×I_h are λ=m^2+n^2π^2/h^2 with m∈Z, n∈N.
Section 2 states this as the starting point, derived from separation of variables on the rectangle with periodic boundary conditions.
domain assumption Isoperimetric domains on the infinite cylinder S^1×R are geodesic disks or cylindrical strips bounded by two circles.
Introduction cites [HHM] for this classification, and it underpins Corollary B.
ad hoc to paper The function F(h) defined from the second estimate in Proposition 2.3 is convex on the interval ((1/9)(9−2√6)π, π²/2], so its maximum lies at an endpoint and is below 20.
Section 2.2.3 asserts convexity without proof; this justifies the λ≥400 threshold that reduces the computer search to k≤1019.

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Cite this review

Pith. "Pith review of P\'{o}lya's conjecture on $\mathbb{S}^1 \times \R$." pith.science (2026). https://pith.science/paper/GZDJGERB

@misc{pith2026250604341,
  author       = {Pith},
  title        = {Pith review of: P\'olya's conjecture on $\mathbbS^1 \times \R$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZDJGERB}},
  note         = {Machine review of arXiv:2506.04341}
}

abstract

We study the area ranges where the two possible isoperimetric domains on the infinite cylinder $\mathbb{S}^{1}\times \R$, namely, geodesic disks and cylindrical strips of the form $\mathbb{S}^1\times [0,h]$, satisfy P\'{o}lya's conjecture. In the former case, we provide an upper bound on the maximum value of the radius for which the conjecture may hold, while in the latter we fully characterise the values of $h$ for which it does hold for these strips. As a consequence, we determine a necessary and sufficient condition for the isoperimetric domain on $\mathbb{S}^{1}\times \R$ corresponding to a given area to satisfy P\'{o}lya's conjecture. In the case of the cylindrical strip, we also provide a necessary and sufficient condition for the Li-Yau inequalities to hold.

Figures

Figures reproduced from arXiv: 2506.04341 by the authors.

**Figure 1.** Lattice points and ellipse for h = 3, λ = 50 lattice squares given by Q := [ k,l∈Z [k,k+1]×[l,l+1]⊂E [k, k + 1] × [l, l + 1]. By excluding lattice points lying on the y-axis, we can associate each lattice point in E to a lattice square in Q, thus obtaining N(λ) = |Q| + h π √ λ ≤ |E| + h π √ λ − (|E| − |Q|) = λ h 2 + h π √ λ − |E \ Q| and R(λ) = N(λ) − λ h 2 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** First R(λ) estimates 1 2 |E+ \ Q+| ≥ 1 2 ex(ℓn)ϵn + ℓ Xn−1 j=0 1 2 (ex(j) − ex(j + 1)) = 1 2 [ex(ℓn)ϵn + ex(0) − ex(ℓn)] = 1 2 h ex(ℓn)(ϵn − 1) + √ λ i [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Second R(λ) estimates. Ri , Sj and T triangles are shaded with vertical lines, horizontal lines, and dots, respectively [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Plots of (λk − 2k/h)/ √ k for different values of k; the two pairs of vertical lines denote the intervals where λ8 and λ13 fail the conjecture and which are shown in [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Plots of (λk − 2k/h)/ √ k showing the two intervals where P´olya’s conjecture is not satisfied. 3. Li-Yau inequalities The Li-Yau inequalities for S 1 × Ih are given by (1.1). Proceeding as in the case of the first eigenvalue for P´olya’s inequality, we now see that fo…

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Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

B\' e rard and G

P. B\' e rard and G. Besson, Spectres et groupes cristallographiques. II. Domaines spheriques, Ann. Inst. Fourier (Grenoble) 30 (1980), 237--248

work page 1980

[2] [2]

Luzzini, L

D.\ Buoso, P. Luzzini, L. Provenzano and J. Stubbe, Semiclassical estimates for eigenvalue means of Laplacians on spheres, J. Geom. Anal. 33 :280 (2023)

work page 2023

[3] [3]

Filonov, M

N. Filonov, M. Levitin, I. Polterovich and D. A. Sher, P\' o lya's conjecture for Euclidean balls, Invent. Math. 234 (2023), 129--169

work page 2023

[4] [4]

Freitas, Upper and lower bounds for the first Dirichlet eigenvalue of a triangle, Proc

P. Freitas, Upper and lower bounds for the first Dirichlet eigenvalue of a triangle, Proc. Amer. Math. Soc. 134 (2006), 2083--2089

work page 2006

[5] [5]

Freitas, J

P. Freitas, J. Lagac\' e and J. Payette, Optimal unions of scaled copies of domains and P\'olya's conjecture, Ark. Math. 59 (2021),11--51

work page 2021

[6] [6]

Freitas, J

P. Freitas, J. Mao and I. Salavessa, P\' o lya-type inequalities on spheres and hemispheres, Ann.\ Inst. Fourier (Grenoble) (to appear), DOI:10.5802/aif.3657

doi:10.5802/aif.3657

[7] [7]

Math.\ Phys.\ 64, 121503 (2023) DOI:10.1063/5.0161050

P.\ Freitas and I.\ Salavessa, Families of non-tiling domains satisfying P\' o lya's conjecture, J. Math.\ Phys.\ 64, 121503 (2023) DOI:10.1063/5.0161050

doi:10.1063/5.0161050 2023

[8] [8]

He and Z

X. He and Z. Wang, Pólya’s conjecture for thin products, arXiv:2402.12093v1

arXiv