Quantum-Consistent Adelic Integration and Structure of Egyptian Fractions

Quantum-Consistent Adelic Integration and Structure of Egyptian Fractions | 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 Quantum-Consistent Adelic Integration and Structure of Egyptian Fractions Julian Del Bel This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6115299/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 Through diligent application of adelic integration and quantum arithmetic, we demonstrate the Egyptian fraction system’s remarkable anticipation of both number-theoretic profundities and quantum measurement theory. Our findings reveal a quantum-arithmetic framework revealing profound structural patterns in Egyptian fractions. We demonstrate these ancient decompositions exhibit: · Adelic balance through multiplicative normalization · Prime entanglement with optimized logarithmic spread (σlogs ≈ 0.17) · Dyadic quantization in Eye of Horus fractions · Computational validation via modern thresholding (< 10 −12 ) Statistical analysis shows Egyptian σlogs values significantly below Erdős-Kac predictions (p < 0.001), evidencing non-random optimization. Our adelic unity framework connects these features through number-theoretic quantum analogs, revealing an ancient system anticipating modern mathematical principles. Full Text Additional Declarations No competing interests reported. Supplementary Files RecursiveFractalization.pdf EuclideanFramework2.pdf Diophantine.pdf 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. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-6115299","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":448538876,"identity":"9d4ffacc-6763-4549-843d-1ce2391607ad","order_by":0,"name":"Julian Del Bel","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAx0lEQVRIiWNgGAWjYBACxgbmhgMMDDaMDew8DAwPiNPCCNKSBtQK1JJArD1A4jAJWphnJDYe+LnjvGw/M+/BBwkMdvKE7ZiR2HCw98xt45nNfMkGCQzJhg3EaDnA23Y7ccNhHjOJBIYDjERpOfi37Vzi/sM85j+AWuyJ0nKYt+1A4gZmHjOg9w8kEtbS87DhsGxbsvGMw3zJEgkGyckEtRi2Jx/++LbNTra/vffghw8VdraEtaCqMCCkHggIR8MoGAWjYBSMAgDasENKDiRoVwAAAABJRU5ErkJggg==","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Julian","middleName":"Del","lastName":"Bel","suffix":""}],"badges":[],"createdAt":"2025-02-26 17:38:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6115299/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6115299/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":83545075,"identity":"9a60725b-f815-414e-a414-0a4a61b3d848","added_by":"auto","created_at":"2025-05-28 08:54:18","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":416679,"visible":true,"origin":"","legend":"","description":"","filename":"tecurve66.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6115299/v1_covered_35bc2095-e999-4250-8ae1-78af5c5b2b05.pdf"},{"id":81604281,"identity":"4a187fde-b3e4-4e9d-b418-cd111a46f312","added_by":"auto","created_at":"2025-04-29 05:11:38","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":958013,"visible":true,"origin":"","legend":"","description":"","filename":"RecursiveFractalization.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6115299/v1/4a18b53760ccf9ca58f93b33.pdf"},{"id":81604284,"identity":"69dcbe2a-ee57-47d5-8711-93cfe9ed0dae","added_by":"auto","created_at":"2025-04-29 05:11:38","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":352696,"visible":true,"origin":"","legend":"","description":"","filename":"EuclideanFramework2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6115299/v1/17ef5d87eb9365905f01846c.pdf"},{"id":81604282,"identity":"481bff79-4b4f-4051-b5bc-9273d0e08aa3","added_by":"auto","created_at":"2025-04-29 05:11:38","extension":"pdf","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":336409,"visible":true,"origin":"","legend":"","description":"","filename":"Diophantine.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6115299/v1/ed3226919d97d11e178c92d4.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Quantum-Consistent Adelic Integration and Structure of Egyptian Fractions","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"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":"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":"","lastPublishedDoi":"10.21203/rs.3.rs-6115299/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6115299/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThrough diligent application of adelic integration and quantum arithmetic, we demonstrate the Egyptian fraction system’s remarkable anticipation of both number-theoretic profundities and quantum measurement theory. Our findings reveal a quantum-arithmetic framework revealing profound structural patterns in Egyptian fractions. We demonstrate these ancient decompositions exhibit:\u003c/p\u003e\n\u003cp\u003e·\u0026nbsp; Adelic balance through multiplicative normalization\u003c/p\u003e\n\u003cp\u003e·\u0026nbsp; Prime entanglement with optimized logarithmic spread (σlogs ≈ 0.17)\u003c/p\u003e\n\u003cp\u003e·\u0026nbsp; Dyadic quantization in Eye of Horus fractions\u003c/p\u003e\n\u003cp\u003e·\u0026nbsp; Computational validation via modern thresholding (\u0026lt; 10\u003csup\u003e−12\u003c/sup\u003e)\u003c/p\u003e\n\u003cp\u003eStatistical analysis shows Egyptian \u003csub\u003eσlogs\u003c/sub\u003e values significantly below Erdős-Kac predictions (p \u0026lt; 0.001), evidencing non-random optimization. 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