Critical Behavior in Sequential Causal Growth: Evidence that Lorentzian Order is a Boundary Phenomenon

Critical Behavior in Sequential Causal Growth: Evidence that Lorentzian Order is a Boundary Phenomenon | 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 Critical Behavior in Sequential Causal Growth: Evidence that Lorentzian Order is a Boundary Phenomenon Dewan Sajid Islam This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8920948/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 investigate stochastic sequential growth processes generating finite partially ordered sets without geometric input. The model exhibits two macroscopic regimes: a sparse phase with vanishing comparability and a dense phase approaching total order. A narrow crossover region produces ordering statistics numerically close to those of 3+1 dimensional Lorentzian causal sets at finite size. However, finite-size scaling demonstrates this behavior does not persist asymptotically. The ordering fraction flows toward the sparse phase as system size increases, indicating the Lorentzian value is not a fixed point of the dynamics. We therefore identify relativistic causal statistics as a finite-size crossover rather than a stable emergent structure. This provides a negative constraint on discrete spacetime models: local Markovian growth rules of this class cannot generically reproduce continuum Lorentzian order. Full Text Additional Declarations No competing interests reported. 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-8920948","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":609827995,"identity":"e21c8999-6821-4451-b4ea-1b1d0eebe4c3","order_by":0,"name":"Dewan Sajid Islam","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+UlEQVRIiWNgGAWjYBACxgYeEJUAJg98qACSzMwNxGnhYWNgfDjjDEgLI34tDAwQLQxALczGnG0QY/BqYG7vPfa4sC1Nxl6+x0yacV5tNH87UMuPim24HdZzLt14ZlsO0GE8ZtKF247nzjjM2MDYc+Y2bi0zcsykedsqIFpmbjuW2wDUwszYRqwW3jnHcucTqQXsMGNj3oaa3A0EtfScMZPmOZfGw3MsrfDhjGMHcjcCtRzE5xfDdmBA8ZQl27M3H95w4ENNXe6884cPPvhRgUdLA8gqNhCTwwBIHAaLHsCpHgjkweQfEMH+AEjU4VM8CkbBKBgFIxQAABWDVdoJspgGAAAAAElFTkSuQmCC","orcid":"","institution":"Independent Researcher, Biam Model School \u0026 College","correspondingAuthor":true,"prefix":"","firstName":"Dewan","middleName":"Sajid","lastName":"Islam","suffix":""}],"badges":[],"createdAt":"2026-02-19 21:38:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8920948/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8920948/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109160206,"identity":"00566e97-3cd3-40fd-b7e1-3a93df555819","added_by":"auto","created_at":"2026-05-13 07:31:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":251987,"visible":true,"origin":"","legend":"","description":"","filename":"photon7.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8920948/v1_covered_f99def54-1e88-4374-a68b-ab077a800f0b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Critical Behavior in Sequential Causal Growth: Evidence that Lorentzian Order is a Boundary Phenomenon","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-8920948/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8920948/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We investigate stochastic sequential growth processes generating finite partially ordered sets without geometric input. 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