Fundamental Constants from Grassmannian Geometry: Deriving α = 1/137 via Kaluza-Klein Reduction on Gr(3,16)

Fundamental Constants from Grassmannian Geometry: Deriving α = 1/137 via Kaluza-Klein Reduction on Gr(3,16) | 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 Fundamental Constants from Grassmannian Geometry: Deriving α = 1/137 via Kaluza-Klein Reduction on Gr(3,16) Amin Al Yaquob This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8511766/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 Abstract We derive fundamental constants of the Standard Model from the complex Grassmannian Gr(3, 16) = U(16)/[U(3) °ø U(13)]. With only two integers (k = 3, n = 13), we obtain: (1) the electroweak mixing angle sin2 θW = k/n = 3/13 = 0.2308, matching the measured value 0.2312 to 0.2%; (2) the hierarchy ln(MPlanck/vEW) = kn = 39, matching 38.4 to 1.5%; (3) generations = k = 3, exactly; (4) fermions per generation = N = 16, matching the SO(10) spinor exactly. We further derive the fine structure constant via Kaluza-Klein reduction: α(MZ)−1 = dim(u(N))/2 = N2/2 = 128 (0.04% error), and α(0)−1 = N2/2 + Tr(P)2 = 128 + 9 = 137 (0.03% error). The framework is falsifiable: FCC-ee will measure sin2 θW to 10−5 precision by ∼2040. All seven fundamental quantities emerge from one geometric structure with zero free parameters. PACS 12.10.-g · 11.25.Mj · 12.15.-y · 02.40.-k Grassmannian Fine structure constant Weinberg angle Gauge unification Electroweak hierarchy Kaluza-Klein Full Text Additional Declarations No competing interests reported. Supplementary Files GD313extensionnote.tex 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-8511766","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":569915333,"identity":"e5d41b3d-4298-4241-ba29-75cd86c6d877","order_by":0,"name":"Amin Al Yaquob","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAv0lEQVRIiWNgGAWjYFACxgaGDwwWYCYz0VoYZzBIkKQFqJKHJC380w63PbapkUjsb88xfFzAYCen20BAi8TtxHbjnGMSiTPOvDE2nsGQbGx2gJA1txPbpHPYJBIbbuSYSfMwHEjcRkiLPEiLxT+JxPlEazEAaWFsk0jcQLQWQ6AWyd4+CeONZ54VG88wIMIvcrfTn0n8+GYjO+948sbHBRV2coS9DwWODQwJIHcSqRwE7BnAWkbBKBgFo2AUYAEAl41AoRX7xukAAAAASUVORK5CYII=","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Amin","middleName":"Al","lastName":"Yaquob","suffix":""}],"badges":[],"createdAt":"2026-01-04 09:08:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8511766/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8511766/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100419280,"identity":"ff3c7a22-cb17-48fc-bcfb-a2693008237f","added_by":"auto","created_at":"2026-01-16 13:26:46","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":2689,"visible":true,"origin":"","legend":"","description":"","filename":"935ff3c5057c439bbb7e4c6188c89673.json","url":"https://assets-eu.researchsquare.com/files/rs-8511766/v1/c1164cd911ff5ff635e3b06c.json"},{"id":101881240,"identity":"eafc7eac-daf1-4b0e-b68d-33cce7d9a03a","added_by":"auto","created_at":"2026-02-04 15:11:07","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":319206,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptP6.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8511766/v1_covered_786cb665-aa0c-428a-921c-87eceecaf252.pdf"},{"id":100419308,"identity":"47376105-03e6-4c8d-844e-55e5fee98956","added_by":"auto","created_at":"2026-01-16 13:26:53","extension":"tex","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":9890,"visible":true,"origin":"","legend":"","description":"","filename":"GD313extensionnote.tex","url":"https://assets-eu.researchsquare.com/files/rs-8511766/v1/ec854b0e1c563ea6a8de2ad5.tex"}],"financialInterests":"No competing interests reported.","formattedTitle":"Fundamental Constants from Grassmannian Geometry: Deriving α = 1/137 via Kaluza-Klein Reduction on Gr(3,16)","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":"Grassmannian, Fine structure constant, Weinberg angle, Gauge unification, Electroweak hierarchy, Kaluza-Klein","lastPublishedDoi":"10.21203/rs.3.rs-8511766/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8511766/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAbstract We derive fundamental constants of the Standard Model from the\u0026nbsp;complex Grassmannian Gr(3, 16) = U(16)/[U(3) °ø U(13)]. With only two integers\u0026nbsp;(k = 3, n = 13), we obtain: (1) the electroweak mixing angle sin2 θW =\u0026nbsp;k/n = 3/13 = 0.2308, matching the measured value 0.2312 to 0.2%; (2) the\u0026nbsp;hierarchy ln(MPlanck/vEW) = kn = 39, matching 38.4 to 1.5%; (3) generations\u0026nbsp;= k = 3, exactly; (4) fermions per generation = N = 16, matching\u0026nbsp;the SO(10) spinor exactly. We further derive the fine structure constant via\u0026nbsp;Kaluza-Klein reduction: α(MZ)−1 = dim(u(N))/2 = N2/2 = 128 (0.04% error),\u0026nbsp;and α(0)−1 = N2/2 + Tr(P)2 = 128 + 9 = 137 (0.03% error). The\u0026nbsp;framework is falsifiable: FCC-ee will measure sin2 θW to 10−5 precision by ∼2040. All seven fundamental quantities emerge from one geometric structure with zero free parameters.\u003c/p\u003e\n\u003cp\u003ePACS 12.10.-g · 11.25.Mj · 12.15.-y · 02.40.-k\u003c/p\u003e","manuscriptTitle":"Fundamental Constants from Grassmannian Geometry: Deriving α = 1/137 via Kaluza-Klein Reduction on Gr(3,16)","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-16 13:10:17","doi":"10.21203/rs.3.rs-8511766/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"e55032f4-742f-44a0-adc5-7387e8e65c06","owner":[],"postedDate":"January 16th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-02-03T16:11:04+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-16 13:10:17","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8511766","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8511766","identity":"rs-8511766","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}