The Geometry of Meaning: Perfect Spacetime Representations of Hierarchical Structures

The Geometry of Meaning: Perfect Spacetime Representations of Hierarchical Structures | 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 The Geometry of Meaning: Perfect Spacetime Representations of Hierarchical Structures Andres Anabalon, Hugo Garcés, Julio Oliva, José Cifuentes This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6597545/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 show that there is a fast algorithm that embeds hierarchical structures in three-dimensional Minkowski spacetime. The correlation of data ends up purely encoded in the causal structure. Our model relies solely on oriented token pairs--local hierarchical signals--with no access to global symbolic structure. We apply our method to the corpus of WordNet . We provide a perfect embedding of the mammal sub-tree including ambiguities (more than one hierarchy per node) in such a way that the hierarchical structures get completely codified in the geometry and exactly reproduce the ground-truth. We extend this to a perfect embedding of a maximal unambiguous subset of the WordNet with 82, 115 noun tokens and a single hierarchy per token. We introduce a novel retrieval mechanism in which causality, not distance, governs hierarchical access. Our results seem to indicate that all discrete data has a perfect geometrical representation that is three-dimensional. The resulting embeddings are nearly conformally invariant, indicating deep connections with general relativity and field theory. These results suggest that concepts, categories, and their interrelations, namely hierarchical meaning itself, is geometric. Physical sciences/Mathematics and computing/Computer science Physical sciences/Physics/Particle physics/Theoretical particle physics 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. 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-6597545","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Physical Sciences - Article","associatedPublications":[],"authors":[{"id":453827923,"identity":"5ec16c6b-075b-4dab-94b0-55e0f334e2b8","order_by":0,"name":"Andres Anabalon","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxklEQVRIiWNgGAWjYHACNgaGAgY5EOvAA+K1GDAYg7UkkKIlsQHEJEoL/+zmZw9+GNikzw87/BBoi52cbgMBLRJ3jpkb9hik5W68nWYA1JJsbHaAkDU3EswkeAwO526cnQDSciBxGyEt8jfSv0n+MTicbjg7/QNxWgxu5JhJA21JkJfOIdIWwxs55cYyBmmGG6RzCg4kGBDhF7kb6dsevqmwkZefnb75w4cKOznC3oe7EKzSgFjlICDfQIrqUTAKRsEoGFEAAADBRXx6okbHAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-2551-1937","institution":"Universidad de Concepcion","correspondingAuthor":true,"prefix":"","firstName":"Andres","middleName":"","lastName":"Anabalon","suffix":""},{"id":453827924,"identity":"8f384119-bc19-45b9-bb2a-617247b45ea4","order_by":1,"name":"Hugo Garcés","email":"","orcid":"https://orcid.org/0000-0002-7809-708X","institution":"Universidad de Concepcion","correspondingAuthor":false,"prefix":"","firstName":"Hugo","middleName":"","lastName":"Garcés","suffix":""},{"id":453827925,"identity":"1298f005-4b02-458b-ada3-b4ffee582a8a","order_by":2,"name":"Julio Oliva","email":"","orcid":"","institution":"Universidad de Concepcion","correspondingAuthor":false,"prefix":"","firstName":"Julio","middleName":"","lastName":"Oliva","suffix":""},{"id":453827926,"identity":"0b822161-72a2-4d91-ae1b-796cbf5a12bd","order_by":3,"name":"José Cifuentes","email":"","orcid":"https://orcid.org/0000-0002-5303-2247","institution":"Universidad de Concepcion","correspondingAuthor":false,"prefix":"","firstName":"José","middleName":"","lastName":"Cifuentes","suffix":""}],"badges":[],"createdAt":"2025-05-05 22:40:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6597545/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6597545/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":83603673,"identity":"1c36efa8-0ee8-4ab6-bc43-9d175f40e656","added_by":"auto","created_at":"2025-05-29 10:00:48","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":801180,"visible":true,"origin":"","legend":"Article File","description":"","filename":"Embeddings5.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6597545/v1_covered_1ad7bd5e-9ad5-4649-bc0e-fe545a8cd3ae.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"The Geometry of Meaning: Perfect Spacetime Representations of Hierarchical Structures","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-6597545/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6597545/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe show that there is a fast algorithm that embeds hierarchical structures in three-dimensional Minkowski spacetime. 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