Number-Theoretic Derivation of Sliding Window Coverage Parameter W for Xi Sequences

Number-Theoretic Derivation of Sliding Window Coverage Parameter W for Xi Sequences | 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 Number-Theoretic Derivation of Sliding Window Coverage Parameter W for X i Sequences Tony Yuan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8508267/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The X i , sequence, as a linear derived sequence of odd primes, its sliding window coverage isan important characterization of the uniformity of prime distribution. Aiming at the key problem that the value of the core parameter-sequence set size W_in existing verifcation methodsacks number-theoretic theoretical support and relies on empirical grading, this paper focuses onthe number-theoretic analysis and strict formula derivation of W, constructing a W calculationsystem with both theoretical rigor and engineering practicality. Firstly, based on the PrimeNumber Theorem (PNT) and the asymptotic distribution theorem of prime gaps, the intrin.sic number-theoretic correlation between W and prime density, maximum prime gap is strictlydemonstrated, establishing a rigorous logical chain of "prime distribution characteristics -X i sequence coverage requirements - W constraints". Secondly, a conservative coeffcient is introduced through the analysis of prime gap fuctuations to complete the strict derivation of the coreformula for W; boundary constraints are added in combination with the prime distribution char-acteristics of small N and extremely large N scenarios to form a complete number-theoreticallyprovable calculation method, and the probability guarantee and engineering verifcation of cov.erage effectiveness are provided. Finally, a sliding window coverage verification framework for X i ; sequences based on this method is constructed, and the effectiveness of the method is verifedthrough experiments in multi-scale N scenarios. Theoretical analysis shows that the derived Wcan strictly cover the X i , sequence coverage requirements corresponding to any prime gap within [1, N] in engineering application scenarios, and the derivation process relies entirely on corenumber-theoretic theorems without empirical assumptions. Experimental results demonstratethat the proposed method can effectively ensure coverage effectiveness and has good engineeringpracticality. The core value of this paper lies in providing a strict number-theoretic theoreticalbasis and standardized calculation scheme for W setting, as well as a promotable theoreticalparadigm for parameter optimization of prime-derived sequences. Prime Number Theorem Prime Gap Xi Sequence Sliding Window Coverage Verification Number-Theoretic Parameter Optimization Sequence Interval Length W Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted 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. 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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-8508267","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":568826645,"identity":"c962ce93-596c-4417-965e-155bb4dad2b0","order_by":0,"name":"Tony Yuan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABD0lEQVRIie2PMUvDQBiGEyqJw4HriX8iUjgVjuSHuHwhci4KHTMUPBHaJe4VBH+B0MnVKx9kOpo10qX5B+nWSbx06pKro+A9w8F7vA8vn+c5HH8VyOnD5xsqtfnmJvqPytYm3bPW3J/TIF3MAtEp8qDiNxNhFDLE4wB3v1YlCZ9LChoHF2dFhB6p4vcpmpUxv+5dIUtBIcfg6rUc4Yiusg+dGqUU97JPoXesWyGeyuY4i1YZU0bxJdqVdILUUxAhgWXGquZXioii+sYoSsWsPrSi9fASNIfTooTFi8yA1WYFLLeE0+L8a5tTOAmfsN3IOGHVbbNux7xXMRzR/ZTumtBf7xi0+ymxlx0Oh+M/8gNW7G1QrscErQAAAABJRU5ErkJggg==","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Tony","middleName":"","lastName":"Yuan","suffix":""}],"badges":[],"createdAt":"2026-01-03 16:55:18","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-8508267/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8508267/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":99795198,"identity":"deff4cbd-65db-49a8-ba59-3a6ab23c82e7","added_by":"auto","created_at":"2026-01-08 13:37:19","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":794540,"visible":true,"origin":"","legend":"","description":"","filename":"NumberTheoreticDerivationofSlidingWindowCoverage.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8508267/v1_covered_c5ec27f2-b380-41ce-84ab-7705934436d7.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eNumber-Theoretic Derivation of Sliding Window Coverage Parameter \u003cem\u003eW\u003c/em\u003e for \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e \u003c/em\u003eSequences\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Beihang University","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Prime Number Theorem, Prime Gap, Xi Sequence, Sliding Window, Coverage Verification, Number-Theoretic Parameter Optimization, Sequence Interval Length W","lastPublishedDoi":"10.21203/rs.3.rs-8508267/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8508267/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe X\u003csub\u003ei\u003c/sub\u003e, sequence, as a linear derived sequence of odd primes, its sliding window coverage isan important characterization of the uniformity of prime distribution. Aiming at the key problem that the value of the core parameter-sequence set size W_in existing verifcation methodsacks number-theoretic theoretical support and relies on empirical grading, this paper focuses onthe number-theoretic analysis and strict formula derivation of W, constructing a W calculationsystem with both theoretical rigor and engineering practicality. Firstly, based on the PrimeNumber Theorem (PNT) and the asymptotic distribution theorem of prime gaps, the intrin.sic number-theoretic correlation between W and prime density, maximum prime gap is strictlydemonstrated, establishing a rigorous logical chain of \"prime distribution characteristics -X\u003csub\u003ei\u003c/sub\u003e sequence coverage requirements - W constraints\". Secondly, a conservative coeffcient is introduced through the analysis of prime gap fuctuations to complete the strict derivation of the coreformula for W; boundary constraints are added in combination with the prime distribution char-acteristics of small N and extremely large N scenarios to form a complete number-theoreticallyprovable calculation method, and the probability guarantee and engineering verifcation of cov.erage effectiveness are provided. Finally, a sliding window coverage verification framework for X\u003csub\u003ei\u003c/sub\u003e; sequences based on this method is constructed, and the effectiveness of the method is verifedthrough experiments in multi-scale N scenarios. Theoretical analysis shows that the derived Wcan strictly cover the X\u003csub\u003ei\u003c/sub\u003e, sequence coverage requirements corresponding to any prime gap within [1, N] in engineering application scenarios, and the derivation process relies entirely on corenumber-theoretic theorems without empirical assumptions. Experimental results demonstratethat the proposed method can effectively ensure coverage effectiveness and has good engineeringpracticality. The core value of this paper lies in providing a strict number-theoretic theoreticalbasis and standardized calculation scheme for W setting, as well as a promotable theoreticalparadigm for parameter optimization of prime-derived sequences.\u003c/p\u003e","manuscriptTitle":"Number-Theoretic Derivation of Sliding Window Coverage Parameter W for Xi Sequences","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-07 02:49:22","doi":"10.21203/rs.3.rs-8508267/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"07239a95-d991-408f-9a37-fb94c63c437c","owner":[],"postedDate":"January 7th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-01-07T02:49:22+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-07 02:49:22","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8508267","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8508267","identity":"rs-8508267","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}