The Parabolic Law for Fermionic Masses: Geometric Regularity in the Standard Model

The Universal Quadratic Law for Fermionic Masses From Charged Fermions to Neutrinos: A Unified, Predictive Framework Across the Standard Model | 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 The Universal Quadratic Law for Fermionic Masses From Charged Fermions to Neutrinos: A Unified, Predictive Framework Across the Standard Model Iván Moreno Gil This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7294140/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract We propose the Universal Quadratic Law for Fermionic Masses (UQLFM), a compact framework that reproduces the entire charged–fermion spectrum and yields falsifiable predictions across leptons, quarks, and neutrinos. In this model, the masses of each sector are organized by a parabola in a discrete family index, with the vertex position n 0 (g) determined by intrinsic attributes: an electric–charge sector tag and a geometric mean mass scale. No free fit parameters are required once these sector invariants are specified. This elevates the UQLFM from a numerical pattern to a physically–anchored law. The framework accounts for charged fermion masses with sub–percent accuracy, predicts neutrino absolute masses in the tens–of–meV range consistent with oscillation and cosmological bounds, and implies strong constraints: exact matter–antimatter mass symmetry, exclusion of sequential fourth generations, and structural instability of heavy fermions as excitations above a sub–vacuum state. These features render the UQLFM highly testable against upcoming precision experiments in particle physics and cosmology. High Energy and Particle Physics Fermion mass hierarchy Neutrino absolute masses Quadratic mass law Unified mass formula Standard Model extensions Generation structure Fundamental energy scales Beyond Standard Model (BSM) physics Geometric approaches in particle physics. Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions 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. 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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-7294140","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":504366212,"identity":"93212c99-4083-415b-b1d0-d69797fad0ce","order_by":0,"name":"Iván Moreno Gil","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVRIiWNgGAWjYBACNihtx8/A3AAXlSBGS7JkAyORWmCAccMBYrXwiR0+9uHjDhtm4+MH26Ru7rBhMLjdfPAGQ0UdbodJpyXPnHkmjc/sTGKbdO6ZNAaDO8eSLRjOHMajJceYmbftMLPZAZCWtsMMBjdyzCQY2w7g1/K37TDj5v6HMC353yQY/+FzGFALI1DLBgmELWwSjA3MeP3C2NuWlixx42GzdW5bGo/kjTRji4RjuP0iPzv5MMPPNhs7/v7kg7dz22zk+G4kP7zxoQa3wzAAD5hMIF7DKBgFo2AUjAIsAAD4HU/Xd9RsowAAAABJRU5ErkJggg==","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Iván","middleName":"Moreno","lastName":"Gil","suffix":""}],"badges":[],"createdAt":"2025-08-04 19:38:14","currentVersionCode":2,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7294140/v2","doiUrl":"https://doi.org/10.21203/rs.3.rs-7294140/v2","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":90684514,"identity":"3edbfd3c-28f4-4550-8f65-3f2b4c5f5532","added_by":"auto","created_at":"2025-09-05 16:48:09","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":852540,"visible":true,"origin":"","legend":"","description":"","filename":"UQLFM.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7294140/v2_covered_652809fb-856a-456a-8487-e6cd324b62ef.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"The Universal Quadratic Law for Fermionic Masses From Charged Fermions to Neutrinos: A Unified, Predictive Framework Across the Standard Model","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":"Fermion mass hierarchy; Neutrino absolute masses; Quadratic mass law; Unified mass formula; Standard Model extensions; Generation structure; Fundamental energy scales; Beyond Standard Model (BSM) physics; Geometric approaches in particle physics.","lastPublishedDoi":"10.21203/rs.3.rs-7294140/v2","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7294140/v2","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe propose the Universal Quadratic Law for Fermionic Masses (UQLFM), a compact framework that reproduces the entire charged–fermion spectrum and yields falsifiable predictions across leptons, quarks, and neutrinos. 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