{ "schemaVersion": "1.6", "slug": "square-maximises-dirichlet-ratio-quadrilaterals", "title": "The square maximises the ratio of the first two Dirichlet eigenvalues among quadrilaterals", "shortTitle": "The square maximises λ₂/λ₁ among quadrilaterals", "url": "https://evidencepress.org/releases/square-maximises-dirichlet-ratio-quadrilaterals/", "oneLine": "A computer-assisted proof that among all quadrilaterals, convex or not, the square uniquely maximises the ratio λ₂/λ₁ of the first two Dirichlet eigenvalues, with value 5/2: the k = 4 case of the polygonal Payne–Pólya–Weinberger conjecture.", "abstract": "This unrefereed candidate proves that λ₂(Q)/λ₁(Q) ≤ 5/2 for the Dirichlet Laplacian on every simple quadrilateral Q, convex or not, with equality only for squares. This is the case k = 4 of the polygonal Payne–Pólya–Weinberger conjecture (Siudeja; Antunes–Freitas); the triangle case is due to Siudeja and to Arbon, Mannan, Psenka and Ragavan. The proof is computer-assisted. Two analytic lemmas remove near-triangles and thin shapes. A certified local step covers an explicit neighbourhood of the square through an inertia criterion for a 3×3 Rayleigh–Ritz pencil and a Kato–Temple bound expanded as Taylor models. Certified box covers of the remaining convex shapes bound λ₂ by Rayleigh–Ritz and λ₁ by a Lehmann–Goerisch inequality with an exactly divergence-matched flux, transported by a piecewise-affine map and a Piola transform; a band of elongated shapes is covered by one-dimensional bounds certified by Sturm oscillation. Non-convex quadrilaterals are handled by analytic lemmas (Dirichlet–Neumann bracketing, one-dimensional bounds with exponential tails) and certified covers in charts uniform in the arm lengths. All certificates are computed in ball or exact rational arithmetic and replayed by fail-closed verifiers; the frozen archive was replayed from a fresh extraction. The paper also identifies two gaps in an earlier local argument (Arbon 2022).", "datePublished": "2026-10-03", "dateModified": "2026-10-03", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23109314", "doiUrl": "https://doi.org/10.5281/zenodo.23109314", "conceptDoi": "10.5281/zenodo.23109313", "pdfUrl": "https://github.com/ipitchford/square-maximises-dirichlet-ratio-quadrilaterals/releases/download/v0.1.0-candidate/square-maximises-dirichlet-ratio-quadrilaterals-v0.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/23109314/files/square-maximises-dirichlet-ratio-quadrilaterals-v0.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23109314", "repoUrl": "https://github.com/ipitchford/square-maximises-dirichlet-ratio-quadrilaterals", "releaseUrl": "https://github.com/ipitchford/square-maximises-dirichlet-ratio-quadrilaterals/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/square-maximises-dirichlet-ratio-quadrilaterals/index.md", "bibtexUrl": "https://evidencepress.org/releases/square-maximises-dirichlet-ratio-quadrilaterals/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/square-maximises-dirichlet-ratio-quadrilaterals.mp3", "imageUrl": "https://evidencepress.org/assets/og/square-maximises-dirichlet-ratio-quadrilaterals.png?v=c14fbd6d4d", "coverArtUrl": "https://evidencepress.org/assets/art/square-maximises-dirichlet-ratio-quadrilaterals.svg?v=d904dd9651", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/square-maximises-dirichlet-ratio-quadrilaterals.mp3", "name": "Audio briefing", "description": "AI-generated communication, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/square-maximises-dirichlet-ratio-quadrilaterals.txt" }, { "type": "video", "url": "https://youtu.be/ujzVbAeCBqc", "name": "The Square Maximises λ₂/λ₁ Among Quadrilaterals", "description": "Evidence Press video overview of this release. This is a communication aid; it does not extend the release's claims or assurance status and is not additional research evidence." } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed candidate produced by AI systems. A computer-assisted proof whose certificates were replayed by the producer, including from a fresh extraction of the frozen archive. No independent rerun, formal verification, specialist or journal review; the dart certificates have no second implementation; novelty is established only within a bounded literature search." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23109314", "note": "Public GitHub prerelease v0.1.0-candidate and published Zenodo record (4 files each); every file matched the SHA-256 sums by provider checksum, and the paper, supplement and checksum list also by full download from both providers." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Exhaustive fail-closed replays of every certified cover; then a replay of all five components from a fresh extraction of the frozen archive components (local step, band, shell, transport cover, darts), with receipts.", "evidenceUrl": "https://github.com/ipitchford/square-maximises-dirichlet-ratio-quadrilaterals/blob/v0.1.0-candidate/REPLAY_RECEIPT.md" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated party has reported rerunning the archive." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Separately written checkers (coverage, exclusions, certificate samples, receipts) come from the same producer workflow: implementation diversity, not independent reproduction. The dart certificates have no second implementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof assistant; the verifiers, Arb/FLINT, Julia and Python are trusted." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Cross-vendor model reviews and an external referee report of unrecorded authorship do not establish specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No journal or venue review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "evidenceUrl": "https://github.com/ipitchford/square-maximises-dirichlet-ratio-quadrilaterals/blob/v0.1.0-candidate/ENVIRONMENT.txt", "note": "Versions pinned (requirements.txt; Julia Project and Manifest); the dart replay binds a runtime inventory of the Python installation. No container image; the full replays take hours and CI runs structural checks only." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "Claude (Anthropic) research agent under user direction", "OpenAI GPT-5.6 ('Sol') as adversarial reviewer", "Python with python-flint (FLINT/Arb), SymPy, NumPy, SciPy and scikit-fem; Julia with Arblib.jl", "pandoc and pdfTeX" ], "humanRole": "The user chose the problem from an agent-compiled list, set compute and scope constraints, supplied an external referee report and the publication authority; scholarly creator Anonymous. No human mathematical contribution is claimed.", "disclosure": "AI-produced research, computation and drafting. Separate AI agents wrote the separately written checkers, verified the bibliography, typeset the supplement and ran literature searches. A cross-vendor model (GPT-5.6) reviewed every part adversarially in several rounds. An external referee report of unrecorded authorship, supplied by the publisher, recommended major revisions; all items were addressed. None of these is specialist or journal peer review." }, "attributionBoundary": null, "problem": { "name": "Polygonal Payne–Pólya–Weinberger conjecture for quadrilaterals (Siudeja 2010, Conjecture 1.2; Antunes–Freitas 2008)", "url": "https://doi.org/10.1512/iumj.2010.59.3744" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Research, the cross-vendor adversarial reviews, the response to a publisher-supplied external referee report, the fresh-extraction replay and publication are recorded separately; the research phase before publication intake is left-censored in the work ledger.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Dirichlet Laplacian", "eigenvalue ratio", "Payne–Pólya–Weinberger inequality", "Ashbaugh–Benguria inequality", "quadrilaterals", "spectral geometry", "shape optimisation", "computer-assisted proof", "validated numerics", "Lehmann–Goerisch method" ], "keyResults": [ "Theorem 1: λ₂(Q)/λ₁(Q) ≤ 5/2 for every simple quadrilateral Q, convex or not, with equality if and only if Q is a square.", "Theorem 2 (local step): ξ(Q_p) < 5/2 for 0 < |p|∞ ≤ 1/20 in the Endo–Osting parametrisation, certified by 960 direction boxes with exact rational sign bounds.", "The convex case away from the square: 8 216 shell certificates, 31 426 transported certificates with 18 756 exclusion leaves, and 44 268 one-dimensional certificates for the band 1/10 < width/diameter ≤ 2/5.", "Theorem 5 (darts): complete analytic lemmas for the degenerate and unbounded parts of the non-convex shape space (Supplement S) and certified covers of 462 849 leaves in 14 logs.", "An exact calculation showing that the first-order rule of an earlier local argument (Arbon 2022, eq. 3.16) is invalid for the double eigenvalue: for the trapezoid direction the true first-order term is 0, the rule gives about −1.664." ], "reviews": [ { "schemaVersion": "1.0", "reviewId": "square-maximises-dirichlet-ratio-quadrilaterals-external-referee-2026-10-02", "reviewClass": "external-review", "status": "published", "subject": { "canonicalUrl": "https://evidencepress.org/releases/square-maximises-dirichlet-ratio-quadrilaterals/", "doi": null, "version": "draft of 2 October 2026", "releaseTag": null, "title": "The square maximises the ratio of the first two Dirichlet eigenvalues among quadrilaterals" }, "reviewer": { "name": "External referee (identity and affiliation not recorded); report supplied by the publisher", "independence": "external, unverified" }, "assessment": { "title": "Referee report on the draft of 2 October 2026", "date": "2026-10-02", "recommendation": "Major revisions" }, "limitationsAndNonClaims": [ "The report's text and audit bundle are not redistributed; the response summarises every item.", "Reviewer identity and expertise are not recorded; this record does not establish specialist or journal peer review." ], "assuranceImpact": { "changesVerificationStatus": false, "changesPeerReviewedFlag": false, "changesIndependentReproductionFlag": false, "changesFormalVerificationFlag": false, "explanation": "A publisher-supplied report of unrecorded authorship; it led to repairs and new evidence but does not change the assurance dimensions." }, "publication": { "publicUrl": "https://github.com/ipitchford/square-maximises-dirichlet-ratio-quadrilaterals/blob/v0.1.0-candidate/REFEREE_RESPONSE.md", "publicSummary": "Recommended major revisions: a frozen proof archive; the angular hypothesis of the thin-wedge lemma and complete non-convex proofs; published correction coefficients, remainders and sign bounds for the local step; a soundness specification of the certificate primitives; separation of replay, independent implementation and sampling; minor corrections. All items were addressed: Supplement S, Appendices A–C, published local-step records with a separately written check, a verified bibliography, an exact calculation for the earlier local argument, two defects of the producer's own checkers repaired, a portability patch of the dart verifier, and a fresh-extraction replay of the frozen archive." } }, { "schemaVersion": "1.0", "reviewId": "square-maximises-dirichlet-ratio-quadrilaterals-adversarial-model-reviews", "reviewClass": "research-assessment", "status": "published", "subject": { "canonicalUrl": "https://evidencepress.org/releases/square-maximises-dirichlet-ratio-quadrilaterals/", "doi": null, "version": "working versions of 30 September – 2 October 2026", "releaseTag": null, "title": "The square maximises the ratio of the first two Dirichlet eigenvalues among quadrilaterals" }, "reviewer": { "name": "OpenAI GPT-5.6 ('Sol'), prompted to refute; several rounds", "independence": "producer-coordinated" }, "assessment": { "title": "Cross-vendor adversarial reviews of the local step, the global covers, Lemma B′ and the darts", "date": "2026-10-02", "recommendation": "Defects found in code and harness were repaired and the affected logs re-certified or replayed; the final dart rounds found no defect in the certificates" }, "assuranceImpact": { "changesVerificationStatus": false, "changesPeerReviewedFlag": false, "changesIndependentReproductionFlag": false, "changesFormalVerificationFlag": false, "explanation": "Producer-coordinated model review does not establish external peer review, reproduction or formal proof." }, "publication": { "publicUrl": "https://github.com/ipitchford/square-maximises-dirichlet-ratio-quadrilaterals/blob/v0.1.0-candidate/paper/PAPER.md", "publicSummary": "Findings and repairs are recorded in Appendix C of the paper; review transcripts are not redistributed." } } ], "evidencePackage": "Paper (with soundness specification, dependency table and assurance record) and Supplement S; all final certificate logs, verifiers, pins and manifests; exhaustive fail-closed replays bound by SHA-256 to log records and verifier sources; a fresh-extraction replay of the frozen archive components; separately written checks of coverage and exclusions (convex covers, band), of 60 transported and 400 band certificates, of the local-step records and of 160 000 sampled darts; an exact receipt state machine with a mutation test; the response to an external referee report.", "openProblems": [ "An independent rerun of the archive by an unaffiliated party, and an independently written checker for the dart certificates.", "Formalise the certificate primitives (Lehmann–Goerisch with transported fluxes, the Sturm cell rule, the Taylor-model inertia criterion) in a proof assistant.", "Extend the method to regular k-gons for k ≥ 5, where the conjectured maximisers are the regular polygons.", "Turn the local certificate into a quantitative stability (deficit) estimate near the square." ], "mathObjects": [], "relatedWorks": [ { "citation": "Arbon, Mannan, Psenka and Ragavan (2022). A proof of the triangular Ashbaugh–Benguria–Payne–Pólya–Weinberger inequality. J. Spectr. Theory 12(2), 515–533. The case k = 3.", "url": "https://doi.org/10.4171/JST/409", "doi": "10.4171/JST/409" }, { "citation": "Siudeja (2010). Isoperimetric inequalities for eigenvalues of triangles. Indiana Univ. Math. J. 59(3), 1097–1120. Acute triangles; states the polygonal conjecture (Conjecture 1.2).", "url": "https://doi.org/10.1512/iumj.2010.59.3744", "doi": "10.1512/iumj.2010.59.3744" }, { "citation": "Arbon (2022). Global and local bounds on the fundamental ratio of triangles and quadrilaterals. arXiv:2207.05814. Local result near the square; two gaps identified in §1 of this paper.", "url": "https://arxiv.org/abs/2207.05814", "doi": "10.48550/arXiv.2207.05814" }, { "citation": "Endo and Osting (2026). Maximizing the fundamental Laplace–Neumann eigenvalue on quadrilaterals. arXiv:2609.02784. The Neumann analogue; source of the parametrisation and box-cover architecture.", "url": "https://arxiv.org/abs/2609.02784", "doi": "10.48550/arXiv.2609.02784" }, { "citation": "Ashbaugh and Benguria (1992). A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions. Ann. of Math. 135(3), 601–628. The disk is extremal among all domains.", "url": "https://doi.org/10.2307/2946578", "doi": "10.2307/2946578" }, { "citation": "Antunes and Freitas (2008). A numerical study of the spectral gap. J. Phys. A 41(5), 055201. 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