Size-Invariant Performance in Cognitive Damping Systems: Definitive Empirical Proof of Internal Temporal Coherence (τ-Invariance) and Absolute O(1) Scalability

Size-Invariant Performance in Cognitive Damping Systems: Definitive Empirical Proof of Internal Temporal Coherence (τ-Invariance) and Absolute O(1) Scalability | 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 Size-Invariant Performance in Cognitive Damping Systems: Definitive Empirical Proof of Internal Temporal Coherence (τ-Invariance) and Absolute O(1) Scalability Abdulsalam Al-Mayahi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7852011/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 We present a definitive empirical validation of the Union Dipole Theory (UDT) through implementation of the Union Dipole Computing Engine (UDCE), a meta-cognitive control system founded on Internal Temporal Coherence (τ-Invariance). Using a large-scale τ-Blocks architecture tested across system sizes B = 10 to 10^6 , we show that the global control efficacy remains strictly constant (mean 72.0637%, σ = 0.0004%), establishing the first reproducible evidence of O(1) performance scalability in a nonlinear dynamical network. Comparative benchmarking against a tuned PID controller reveals a 7.55-fold improvement under equal decision complexity. Analytical derivation confirms that the decoupling between external complexity O(B) and internal performance O(1) arises from a conservation of effective computational energy per τ-cycle. These results constitute the first operational framework for the hypothesised Union Dipole Particle (UDP) and provide experimental support for UDT’s claim that internal time—not external spacetime—governs the scalability of physical and computational processes. Computational Physics Internal Time (τ-Invariance) Union Dipole Theory (UDT) Union Dipole Computing Engine (UDCE) O(1) Scalability Nonlinear Dynamical Systems Meta-Cognitive Control Systems Cognitive Damping Temporal Coherence Size-Invariant Performance Computational Physics Full Text Additional Declarations The authors declare potential competing interests as follows: The author is an independent researcher and the sole developer of the theoretical framework presented in this preprint. There are no financial, institutional, or personal relationships that could influence the results or interpretations of this work. 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-7852011","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":529046139,"identity":"aeac4b01-ba7b-4867-999e-140aad53ced6","order_by":0,"name":"Abdulsalam Al-Mayahi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABF0lEQVRIie3QMUvDQBTA8RcOMj249ZWU9CtcCESFYr/KhUJd4iSIoGCgEJdCNxHs4EfQpXMhEJd+gMB1iEsnBafSIaiXul7RbCL3H4/78e4dgM32FyMQUAF0AdgCsDlhvyESmsuuBJTtCIpv8lP8fjwnWR8jv11u6G0LA37Dimp7Af7B9ME8ZFWcU5wNkdTpvDOTEN/l7kkwWULYLSvzmDKJKE4ZgiYeyk8pGEaek2lLC6PoNUTW19hTyVoTGAjGN57zsZ+IHXFzFCpxG+I8MnQ9J9WEp0YSrIqzwzh7xkCNwqPZaLdL1JkUFBKZV/HV+Kl8ry99Xw1fyte+/rFpvqbtVd8nbn7Y3vQIku2Iru0Um81m+699ASfwVmQEotAjAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0009-0007-4130-6451","institution":"University of Surrey (Retired)","correspondingAuthor":true,"prefix":"","firstName":"Abdulsalam","middleName":"","lastName":"Al-Mayahi","suffix":""}],"badges":[],"createdAt":"2025-10-13 19:49:02","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":true,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7852011/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7852011/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":93555278,"identity":"01e6e0f3-e4b5-4012-828c-bd4d3e855ed0","added_by":"auto","created_at":"2025-10-15 06:25:21","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2151030,"visible":true,"origin":"","legend":"","description":"","filename":"AlMayahi2025TauInvarianceO1ScalabilityUDCE2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7852011/v1_covered_a39e7e72-eb83-4728-b1c9-6d5497fc3f0c.pdf"}],"financialInterests":"The authors declare potential competing interests as follows: The author is an independent researcher and the sole developer of the theoretical framework presented in this preprint.\nThere are no financial, institutional, or personal relationships that could influence the results or interpretations of this work.\n","formattedTitle":"\u003cp\u003eSize-Invariant Performance in Cognitive Damping Systems: Definitive Empirical Proof of Internal Temporal Coherence (τ-Invariance) and Absolute O(1) Scalability\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Independent Researcher (United Kingdom)","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":"Internal Time (τ-Invariance), Union Dipole Theory (UDT), Union Dipole Computing Engine (UDCE), O(1) Scalability, Nonlinear Dynamical Systems, Meta-Cognitive Control Systems, Cognitive Damping, Temporal Coherence, Size-Invariant Performance, Computational Physics","lastPublishedDoi":"10.21203/rs.3.rs-7852011/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7852011/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cstrong\u003eWe present a definitive empirical validation of the Union Dipole Theory (UDT) through implementation of the Union Dipole Computing Engine (UDCE), a meta-cognitive control system founded on Internal Temporal Coherence (τ-Invariance). Using a large-scale τ-Blocks architecture tested across system sizes \u003c/strong\u003eB = 10\u003cstrong\u003e to \u003c/strong\u003e10^6\u003cstrong\u003e, we show that the global control efficacy remains strictly constant (mean 72.0637%, σ = 0.0004%), establishing the first reproducible evidence of O(1) performance scalability in a nonlinear dynamical network. Comparative benchmarking against a tuned PID controller reveals a 7.55-fold improvement under equal decision complexity. Analytical derivation confirms that the decoupling between external complexity O(B) and internal performance O(1) arises from a conservation of effective computational energy per τ-cycle. These results constitute the first operational framework for the hypothesised Union Dipole Particle (UDP) and provide experimental support for UDT’s claim that internal time—not external spacetime—governs the scalability of physical and computational processes.\u003c/strong\u003e\u003c/p\u003e","manuscriptTitle":"Size-Invariant Performance in Cognitive Damping Systems: Definitive Empirical Proof of Internal Temporal Coherence (τ-Invariance) and Absolute O(1) Scalability","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-15 06:16:25","doi":"10.21203/rs.3.rs-7852011/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":"3a1a596c-c970-4803-b838-468dc85c8d54","owner":[],"postedDate":"October 15th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":56231762,"name":"Computational Physics"}],"tags":[],"updatedAt":"2025-10-15T06:16:25+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-15 06:16:25","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7852011","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7852011","identity":"rs-7852011","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}