A Rigorous Computer-Assisted Lower Bound for the Greene Critical Threshold in the Standard Map at the Golden Frequency

A Rigorous Computer-Assisted Lower Bound for the Greene Critical Threshold in the Standard Map at the Golden Frequency | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Rigorous Computer-Assisted Lower Bound for the Greene Critical Threshold in the Standard Map at the Golden Frequency Seer Kai This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9004238/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract This work establishes a certified lower bound of 0.94248 for the Greene critical threshold in the standard map at the golden frequency, a parameter that governs the destruction of the last invariant circle in this canonical model of Hamiltonian dynamics. Our approach combines mathematical rigor with computational verifiability through three elements. First, we employ directional outward rounding in interval arithmetic and use it to certify strict inequalities. Second, we partition the phase space into sixty-four cells, each validated by an independent Krawczyk certificate that establishes local existence and uniqueness with uniform safety margins. Third, we provide a reproducible audit trail through environment pinning and content checksums, enabling third-party verification without re-running the original computation. All sixty-four cells exhibit identical validation margins---consistent with scale-invariance at the golden frequency. While classical numerical methods estimate the threshold near 0.9716, our certified bound helps bridge the gap between high-precision estimates and fully rigorous, computer-assisted verification. The methodology is portable and offers clear pathways to tighter bounds through higher precision and adaptive techniques. Standard map Greene critical threshold Golden mean Computer-assisted proof Interval arithmetic KAM theory Rigorous numerics Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 06 Mar, 2026 Editor assigned by journal 06 Mar, 2026 Submission checks completed at journal 03 Mar, 2026 First submitted to journal 01 Mar, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9004238","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":602535923,"identity":"bd1d4c8b-9ffc-4dee-bd46-3ab120f01940","order_by":0,"name":"Seer Kai","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA00lEQVRIiWNgGAWjYBACCQbGBhAtx8BDqhZjUrRAQGID0Vok2w83f/hQcS99O88ZA4aPe2oZ+Gc34NcizZPYJjnjTHHuzt4eA8YZz44zSNw5gF+LHENiGzNvW0LuhvM8Bsw8B44xGEgkENDC/7D5899/CekGRGuRlkhskGZsSEgwONsD0lJDWIvkjIdtkj3HEgx39hwrODjjwAEeiRsEtEicT3/84UdNgrw5T/LGBx8O1MnxzyCgBQ4MgPgAA8NhoqMHogUI6ojXMQpGwSgYBSMGAADLyUOntwaBOgAAAABJRU5ErkJggg==","orcid":"","institution":"Peking University","correspondingAuthor":true,"prefix":"","firstName":"Seer","middleName":"","lastName":"Kai","suffix":""}],"badges":[],"createdAt":"2026-03-01 23:38:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9004238/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9004238/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104406348,"identity":"bb803e30-f938-4e38-929b-a5e727da83ca","added_by":"auto","created_at":"2026-03-11 12:25:24","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1140086,"visible":true,"origin":"","legend":"","description":"","filename":"KAMGREENEFULLPAPER.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9004238/v1_covered_2279f23d-add8-47c7-b96e-39014a8a6881.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Rigorous Computer-Assisted Lower Bound for the Greene Critical Threshold in the Standard Map at the Golden Frequency","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"international-journal-of-applied-and-computational-mathematics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [International Journal of Applied and Computational Mathematics](https://link.springer.com/journal/40819)","snPcode":"40819","submissionUrl":"https://submission.nature.com/new-submission/40819/3","title":"International Journal of Applied and Computational Mathematics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Standard map, Greene critical threshold, Golden mean, Computer-assisted proof, Interval arithmetic, KAM theory, Rigorous numerics","lastPublishedDoi":"10.21203/rs.3.rs-9004238/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9004238/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This work establishes a certified lower bound of 0.94248 for the Greene critical threshold in the standard map at the golden frequency, a parameter that governs the destruction of the last invariant circle in this canonical model of Hamiltonian dynamics. 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