ProofAtlas

Spectral geometry

Pólya's Conjecture for the Dirichlet Laplacian

Collaboration beta

For every bounded planar drum, should each Dirichlet eigenvalue stay above the area-scaled Weyl prediction?

λk(Ω)≥4πk|Ω|(k≥1)
Known results and sources
Two differently shaped drums show nodal patterns beside an eigenvalue ladder and an unresolved Weyl-scale comparison.
The card presents bounded planar domains, their vibration modes, and the open universal spectral comparison without showing a discrete proof route.

Research problem

Exact mathematical statement

For a bounded planar domain Ω\Omega, let λk(Ω)\lambda_k(\Omega) be its kk-th Dirichlet Laplacian eigenvalue. Pólya's Dirichlet conjecture states

λk(Ω)≥4πk|Ω|(k≥1).\lambda_k(\Omega)\ge \frac{4\pi k}{|\Omega|}\qquad(k\ge1).

Equivalently, its eigenvalue counting function should satisfy

NΩ(E)≤|Ω|E4π.N_\Omega(E)\le \frac{|\Omega|E}{4\pi}.

The governing source says the full problem is open, with no complete proof and no certified counterexample.

Problem infographic

Problem at a glance

A three-panel explainer defines planar Dirichlet eigenvalues, states Pólya's lower bound, and asks whether it holds for every bounded shape.
The plate states the exact spectral inequality and marks its universal planar-domain form as open, without depicting the packet's proposed reduction.

Current mathematical picture

Where work on Pólya's Conjecture for the Dirichlet Laplacian stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSeparate global estimates on consecutive energy terms

The source warns that this is structurally dangerous and requires preserving the signed combination 21E_20-20E_21. A cancellation-preserving boundary-component inequality, signed containment flow, or state certificate may still control the total high-endpoint sum.

Route status · Narrowed route
Main reductionOuter-grid continuum bridge

The source reports an outer-grid and cell-average implication from discrete positivity to the continuum counting inequality.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve universal positivity of beta_(19,1) for every finite occupancy set.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Pólya's Conjecture for the Dirichlet Laplacian in numbers

544retained lines of mathematical investigation544 in the current working snapshot
Argument development
465 · 85%
Explored or eliminated routes
20 · 4%
Computational analysis
24 · 4%
Open obligations
20 · 4%
Definitions and setup
15 · 3%
6selected mapped statements1routes investigated4open questions4contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byGeorge PólyaProposed1954Status checkedAug 15, 2026
Source recordedAug 15, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

11 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

11 selected steps

Scroll horizontally to explore the route

Working route overview for Pólya's Conjecture for the Dirichlet LaplacianA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Do all planar Dirichlet eigenvalues stay above the Weyl scale? — Depends on missing premiseDo all planar Dirichleteigenvalues stay above theWeyl…Current reduction — Depends on missing premiseCurrent reductionOuter-grid continuum bridge — Depends on missing premiseOuter-grid continuum bridgeClosing target — Depends on missing premiseClosing targetDegree-twenty interior cells — Depends on missing premiseDegree-twenty interior cellsWidth-three structured class — Depends on missing premiseWidth-three structured classSeparate global estimates on consecutive energy terms — stoppedSeparate global estimates onconsecutive energy termsProve universal positivity of beta_(19,1) for every finite occupancy set. — OpenProve universal positivityof beta_(19,1) for everyfinite…Build the exact width-three end-debt automaton at b=25 as the next finite proof-engineering target. — OpenBuild the exact width-threeend-debt automaton at b=25as…Establish all-degree discrete positivity or a direct uniform lower-edge theorem sufficient for the continuum passage. — OpenEstablish all-degreediscrete positivity or adirect…Planar Dirichlet eigenvalue bound — OpenPlanar Dirichlet eigenvaluebound
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeSeparate global estimates on consecutive energy terms

The source warns that this is structurally dangerous and requires preserving the signed combination 21E_20-20E_21. A cancellation-preserving boundary-component inequality, signed containment flow, or state certificate may still control the total high-endpoint sum.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Prove universal positivity of beta_(19,1) for every finite occupancy set.Suggested move: Build a cancellation-preserving boundary-component inequality or signed containment-flow certificate on the first genuinely two-dimensional cycle-rich cores.
Ready to work on
02
Build the exact width-three end-debt automaton at b=25 as the next finite proof-engineering target.Suggested move: Retain the pinching debt in the state, prove transition compatibility, and verify all terminal states without interpreting negative local debt as a counterexample.
Ready to work on
03
Planar Dirichlet eigenvalue bound

Prove the Pólya lower bound for every Dirichlet eigenvalue of every bounded planar domain.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Establish all-degree discrete positivity or a direct uniform lower-edge theorem sufficient for the continuum passage.Suggested move: State the quantifiers and uniform constants required by the outer-grid limit, then prove a bound that does not depend on a finite Bernstein cutoff.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

The Dirichlet Pólya inequality remains open for general bounded planar domains. It is proved for important special classes, including tiling domains, the disk and planar sectors, and more recently annuli.

[2][3]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedThe same authors proved the Dirichlet conjecture for planar annuli in a peer-reviewed JLMS article combining analytic and rigorous computer-assisted estimates; arXiv v2 is marked as the final published version.[3]
  2. Peer reviewedFilonov, Levitin, Polterovich, and Sher proved the conjecture for the disk, planar sectors, and Dirichlet balls in every dimension.[2]
  3. Historical sourcePólya proved the spectral bound for tiling domains and formulated the broader eigenvalue problem.[1]
3 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusPólya's Conjecture for the Dirichlet Laplacian
Solved special casedisks, sectors, and Euclidean balls

The Dirichlet inequality is proved for disks and balls, with planar sectors also covered; these do not exhaust bounded planar domains.

[2]
Solved special caseplanar annuli

The planar annulus family is a proved special case of the general domain conjecture.

[3]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • computation · not independently reproducedComputer-assisted short-interval checks for balls and annuli

    The primary special-case results include rigorous computer-assisted components; no code or certificate was rerun in this intake.

    [2][3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetFormal bounded planar domains, Dirichlet Laplacians, eigenvalue ordering, area, and eigenvalue counting functions.
  • Formalization targetA formal Weyl-law framework and proof that the eigenvalue and counting-function formulations align under the chosen conventions.
  • Formalization targetChecked analytic and interval-arithmetic libraries supporting the known disk, sector, ball, and annulus special cases.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

4 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction2 of 62
  • lemma3 of 63
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
  • retained route statementDo all planar Dirichlet eigenvalues stay above the Weyl scale?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementOuter-grid continuum bridgeintermediate
  • retained route statementDegree-twenty interior cellsintermediate
  • retained route statementWidth-three structured classintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureSeparate global estimates on consecutive energy termsreported failure
  • Research targetProve universal positivity of beta_(19,1) for every finite occupancy set.open
  • Research targetBuild the exact width-three end-debt automaton at b=25 as the next finite proof-engineering target.open
  • Research targetEstablish all-degree discrete positivity or a direct uniform lower-edge theorem sufficient for the continuum passage.open
  • Research targetPlanar Dirichlet eigenvalue boundopen
  • Research targetLast degree-twenty endpointsuperseded
  • Research targetAll-degree continuum passagesuperseded
  • Narrowed routeSeparate global estimates on consecutive energy termsThe source warns that this is structurally dangerous and requires preserving the signed combination 21E_20-20E_21. A cancellation-preserving boundary-component inequality, signed containment flow, or state certificate may still control the total high-endpoint sum.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve universal positivity of beta_(19,1) for every finite occupancy set.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointProve universal positivity of beta_(19,1) for every finite occupancy set.

Pólya's Conjecture for the Dirichlet Laplacian · ready to start

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Research contextPrepared context for any AI agent

For every bounded planar drum, should each Dirichlet eigenvalue stay above the area-scaled Weyl prediction?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    On the eigenvalues of vibrating membranesoriginal source · George Pólya · Proceedings of the London Mathematical Society · 1961 · DOI 10.1112/plms/s3-11.1.419 · accessed Aug 15, 2026
  2. 2
    Pólya’s conjecture for Euclidean ballspeer reviewed result · Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher · Inventiones mathematicae · 2023-06-05 · ARXIV 2203.07696 · DOI 10.1007/s00222-023-01198-1 · accessed Aug 15, 2026
  3. 3
    Pólya's conjecture for Dirichlet eigenvalues of annulipeer reviewed result · Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher · Journal of the London Mathematical Society · 2026-02-09 · ARXIV 2505.21737 · DOI 10.1112/jlms.70425 · accessed Aug 15, 2026

Important qualifications

  • This record concerns the Dirichlet inequality for general bounded planar domains, not the Neumann analogue or higher-dimensional variants.
  • Disk, sector, ball, and annulus theorems are recorded as special cases and are not generalized to arbitrary planar domains.
  • The general open status is supported by the peer-reviewed balls paper's statement that only the first two eigenvalues are known in full generality and by the restricted scope of later primary results.
  • No packet URL or attachment was fetched, executed, rendered, or treated as external evidence.

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