Emergence as a Universal Phase Transition: From Mathematical Inevitability to Predictive Control Across AI, Physical, and Social Systems

Emergence as a Universal Phase Transition: From Mathematical Inevitability to Predictive Control Across AI, Physical, and Social Systems | 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 Physical Sciences - Article Emergence as a Universal Phase Transition: From Mathematical Inevitability to Predictive Control Across AI, Physical, and Social Systems Yusen Lin This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7718584/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 Emergence—the phenomenon where complex behaviors arise from simple components—has re- mained a deeply puzzling concept across scientific disciplines. Here, we present a unified mathemat- ical framework that explains emergence as a predictable phase transition governed by the size of a system’s exploration space. We demonstrate that this framework not only explains the emergent capabilities of artificial intelligence but also provides a fundamental basis for the phase transitions of water and the evolution of social complexity in human societies. We posit that emergence is not a mystical phenomenon but a predictable consequence of a system’s exploration of a high-dimensional state space. By defining the relationship between the microscopic rules of a system and its macro- scopic properties through the lens of exploration capacity, we provide a model that explains why low-probability, ”miraculous” behaviors become reliable and regular in sufficiently large systems, such as large language models. The core of our thesis is the derivation of an inverse exponential relationship between the probability of an emergent behavior and the size of the exploration space, offering a quantitative basis for this previously qualitative concept. This revised framework introduces the concept of emergent areas as behavioral classes that undergo phase transitions based on system scale and dimensionality. Physical sciences/Mathematics and computing/Computational science Physical sciences/Mathematics and computing/Computer science Physical sciences/Mathematics and computing/Information technology Full Text Additional Declarations There is NO Competing Interest. 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. 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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-7718584","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Physical Sciences - Article","associatedPublications":[],"authors":[{"id":522430926,"identity":"ff8d2582-080b-4a0f-8f62-39a8a45df329","order_by":0,"name":"Yusen Lin","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAUlEQVRIie3PMUrEQBTG8W8IJM2A7cLC5goZAjYKe5UXAkmTxcLGYtEBIdusbpvOK+QIwmBsgvWAoIG0FrERQVxNUNBmyJaC8y8ew2N+xQNstj/YfBgNMANYsxsREkwSEAJO8GtPZhJcf5FI7kzE6uK2oROVXq0Ua/nyYQZP3ZTsVRnJnN9Fkmq1KOvYCXl1HIIniQaZiSgyIaO8JziqpplLkZzw/Z7cm295fOrJVqX+pvXesi2djRPNeyIVQceus8iJMEbEOhMFVakodetM3y9J5MMtlHyYz/fqoOuWB76/idlz8UL+nqcq3R0mRvLd+c/THQaNAeB0/IvNZrP93z4B7hZZvLXMuToAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0000-2569-5919","institution":"Safeware Technologies Inc","correspondingAuthor":true,"prefix":"","firstName":"Yusen","middleName":"","lastName":"Lin","suffix":""}],"badges":[],"createdAt":"2025-09-26 07:00:47","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7718584/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7718584/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":92722970,"identity":"36679d64-6151-482d-8d45-c2264e82b660","added_by":"auto","created_at":"2025-10-03 13:56:16","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":675260,"visible":true,"origin":"","legend":"Article File","description":"","filename":"EmergentbehaviorsV1.1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7718584/v1_covered_03e5c8bd-aabb-44da-88f3-d6873cb78fb4.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Emergence as a Universal Phase Transition: From Mathematical Inevitability to Predictive Control Across AI, Physical, and Social Systems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"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-7718584/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7718584/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Emergence—the phenomenon where complex behaviors arise from simple components—has re-\r\nmained a deeply puzzling concept across scientific disciplines. 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