Spectral Isomorphism between Renormalization Flow in Non-Autonomous Quadratic Maps and Riemann Zeros

Spectral Isomorphism between Renormalization Flow in Non-Autonomous Quadratic Maps and Riemann Zeros | 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 Article Spectral Isomorphism between Renormalization Flow in Non-Autonomous Quadratic Maps and Riemann Zeros Liang Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9024307/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract Finding a dynamical system that corresponds to the non-trivial zeros of the Riemann \(\text{ζ}\) -function remains a long-standing core challenge in mathematical physics. This paper proposes a discrete dynamical operator driven by non-autonomous logarithmic cooling evolution ( \({\text{U}}_{\text{n}}\text{∼1/}{\text{ln}}^{\text{2}}\text{n}\) ), achieving global topological isomorphism with the Riemann zero manifold for the first time under finite precision. At the microscopic level, research shows that at a critical numerical discrete resolution ( \(\text{ϵ=0.001916}\) ), the system can spontaneously emerge with a “perfect quantum lock-in” approaching near-zero error for the first 6 zeros. On a macroscopic scale, the “Negative Energy Hypothesis” reveals that the inherent particle-hole symmetry of the real-valued transfer operator inevitably excites paired conjugate eigenstates; this mirror projection results in an irreducible interference envelope of approximately 2.68%. By filtering out these negative-frequency states to break the conjugate symmetry, the system achieves absolute asymptotic convergence to the true Riemann values, with the deep-water mean relative error plunging to 0.0839% and the maximum relative deviation strictly capped at 0.2401%. Furthermore, benchmarking with actual test data reveals the physical essence of the unexplained measurement divergences in recent USTC experiments: the theory-predicted 2.68% macroscopic envelope perfectly bounds the experimental deviation distribution, while the \(\text{N}\text{=20}\) dynamical resonance anomaly precisely explains the massive measurement error bars encountered in the physical ion-trap simulation. This study not only provides a brand-new non-autonomous thermodynamic implementation path for the Hilbert-Pólya conjecture but also proves that the spectral dispersion and divergent anomalies observed in real quantum systems are, in fact, the topological destiny at the underlying level of nature. Physical sciences/Mathematics and computing Physical sciences/Physics Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 14 Apr, 2026 Reviewers agreed at journal 14 Apr, 2026 Reviewers agreed at journal 01 Apr, 2026 Reviewers agreed at journal 29 Mar, 2026 Reviewers invited by journal 27 Mar, 2026 Editor invited by journal 27 Mar, 2026 Editor assigned by journal 04 Mar, 2026 Submission checks completed at journal 04 Mar, 2026 First submitted to journal 03 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. 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-9024307","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":614161688,"identity":"5359eb18-add8-4d65-972f-95c1d4e7d4bc","order_by":0,"name":"Liang Wang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9UlEQVRIiWNgGAWjYLCCBwYgkvHhAyDJw0eUlgSwFmZjEMXDRpwWMMlsJgGiCGoxOH728IuEArs8eQdmtsqvOXYybAzMDx/dwKflTF6aRYJBcrHhAWa227LbkoEOYzM2zsGjxexAjplBggFz4sYG/mO3JbcxA7XwsEnj1XL+DUhLPVALM1ux5LZ6IrTcyDF+kGBwOHE+AzMb48dthwlrsb/xxgwYyMcTNzAzM0szbjvOw8ZMwC+S/TnGHz78qU6c397M+PHntmp7fvbmh4/xaQECNnB0GBwGxgwPiMWMXzlYyQcQKd8ATDI/CKseBaNgFIyCEQgAZaJEahMEgeAAAAAASUVORK5CYII=","orcid":"","institution":"Huazhong University of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Liang","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2026-03-04 00:23:09","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9024307/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9024307/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105800817,"identity":"6f6e3b3e-6e3a-48e3-908e-537c12080a8b","added_by":"auto","created_at":"2026-03-31 09:35:42","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1189593,"visible":true,"origin":"","legend":"","description":"","filename":"renmannzeroFixedv2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9024307/v1_covered_a1136c72-f4ab-4752-a0b8-7690b3aea3bc.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Spectral Isomorphism between Renormalization Flow in Non-Autonomous Quadratic Maps and Riemann Zeros","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":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-9024307/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9024307/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFinding a dynamical system that corresponds to the non-trivial zeros of the Riemann \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{\u0026zeta;}\\)\u003c/span\u003e\u003c/span\u003e-function remains a long-standing core challenge in mathematical physics. This paper proposes a discrete dynamical operator driven by non-autonomous logarithmic cooling evolution (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{U}}_{\\text{n}}\\text{\u0026sim;1/}{\\text{ln}}^{\\text{2}}\\text{n}\\)\u003c/span\u003e\u003c/span\u003e), achieving global topological isomorphism with the Riemann zero manifold for the first time under finite precision. At the microscopic level, research shows that at a critical numerical discrete resolution (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{ϵ=0.001916}\\)\u003c/span\u003e\u003c/span\u003e), the system can spontaneously emerge with a \u0026ldquo;perfect quantum lock-in\u0026rdquo; approaching near-zero error for the first 6 zeros. On a macroscopic scale, the \u0026ldquo;Negative Energy Hypothesis\u0026rdquo; reveals that the inherent particle-hole symmetry of the real-valued transfer operator inevitably excites paired conjugate eigenstates; this mirror projection results in an irreducible interference envelope of approximately 2.68%. By filtering out these negative-frequency states to break the conjugate symmetry, the system achieves absolute asymptotic convergence to the true Riemann values, with the deep-water mean relative error plunging to 0.0839% and the maximum relative deviation strictly capped at 0.2401%. Furthermore, benchmarking with actual test data reveals the physical essence of the unexplained measurement divergences in recent USTC experiments: the theory-predicted 2.68% macroscopic envelope perfectly bounds the experimental deviation distribution, while the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{N}\\text{=20}\\)\u003c/span\u003e\u003c/span\u003e dynamical resonance anomaly precisely explains the massive measurement error bars encountered in the physical ion-trap simulation. This study not only provides a brand-new non-autonomous thermodynamic implementation path for the Hilbert-P\u0026oacute;lya conjecture but also proves that the spectral dispersion and divergent anomalies observed in real quantum systems are, in fact, the topological destiny at the underlying level of nature.\u003c/p\u003e","manuscriptTitle":"Spectral Isomorphism between Renormalization Flow in Non-Autonomous Quadratic Maps and Riemann Zeros","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-31 09:35:26","doi":"10.21203/rs.3.rs-9024307/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"137909017797786905522047731675288990752","date":"2026-04-14T18:55:00+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"185348120518375041866951102569462526465","date":"2026-04-14T18:33:34+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"41880144678271229070638574596954366556","date":"2026-04-01T16:44:49+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"253428824621606994351089366632749028340","date":"2026-03-29T20:11:08+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-03-27T16:24:59+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-03-27T11:54:48+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-03-04T05:28:02+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-03-04T05:24:03+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2026-03-04T00:07:05+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"3373c7bb-6aab-4ead-9ac6-f3fc11e0d7fc","owner":[],"postedDate":"March 31st, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":65347494,"name":"Physical sciences/Mathematics and computing"},{"id":65347495,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-03-31T09:35:26+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-31 09:35:26","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9024307","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9024307","identity":"rs-9024307","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}