Emergent Geometry in Complex Systems from Relational Dynamics. I. The Classical Setting

Emergent Geometry in Complex Systems from Relational Dynamics. I. The Classical Setting | 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 Emergent Geometry in Complex Systems from Relational Dynamics. I. The Classical Setting Ernesto Estrada This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9147871/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 Geometry is typically treated as a primitive background structure within which physical processes unfold. In this work we investigate an alternative possibility: that geometry can arise as a structural consequence of relational dynamics. We study the framework of Graph Dynamical Geometrization (GDG), in which a system is specified by a relational structure encoded by a graph together with a dynamical law acting on that structure. The dynamics defines a symmetric operator whose spectral evolution generates a positive–definite kernel. From this kernel one obtains a squared Euclidean distance matrix that induces a canonical embedding of the relational system into a Euclidean space. Geometry therefore appears not as an assumed background but as a structure generated by the dynamics of relations. This construction motivates a philosophical interpretation that we call relational–dynamical structural realism, according to which the fundamental ontology consists of entities connected by relations and governed by a dynamical law, while geometric structure emerges from the spectral organization of that dynamics. We analyze the status of the induced geometry within several forms of structural realism. The GDG framework shows that invariant metric relations generated by relational dynamics can possess explanatory and predictive significance. More generally, the results suggest a structural principle: geometry may emerge whenever relational dynamics admits a spectral articulation. Structural realism Emergent geometry Relational ontology Geometry–dynamics relation Spectral methods Complex systems 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-9147871","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":618057039,"identity":"5e091875-2715-418a-ab59-5a700eb8cde2","order_by":0,"name":"Ernesto Estrada","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBElEQVRIiWNgGAWjYNACNgY5NuaDDSAmjwGITMCrnBmsxZiNLZFELYkNbFBlBoScxM9//tiHH2V26X1szM0ffjAcljFnYH784UENg5x8A3YtkjOSmWf2nEvObWNjbJPsYTjMY9nAZiaRcIzB2OAAdi0GN5iZGXjbmHPb5BvbgG48zGNw/w3InQyJG3A4zP78YWbGv2316WxsjM2fwVoO8DB/AGqpn4/DYQYMyczMvG2HE4BaGqShWhgkgFoSGHA4TOJGsjGzzLnjhhC/GKTD/CJhuAGHFv7+g48Z35RVy8u3sT/+8KPC2h4UYh9/1NjI4woxDHfCrSdK/SgYBaNgFIwC7AAAmkJNPcc3oG0AAAAASUVORK5CYII=","orcid":"","institution":"Institute for Cross-Disciplinary Physics and Complex Systems","correspondingAuthor":true,"prefix":"","firstName":"Ernesto","middleName":"","lastName":"Estrada","suffix":""}],"badges":[],"createdAt":"2026-03-17 10:54:05","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9147871/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9147871/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109428905,"identity":"6142facd-df39-4257-be51-1265044a328c","added_by":"auto","created_at":"2026-05-18 03:40:36","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2355559,"visible":true,"origin":"","legend":"","description":"","filename":"FODAfiles.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9147871/v1_covered_5fe2221a-1001-4530-adce-c042104a42c7.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Emergent Geometry in Complex Systems from Relational Dynamics. 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