The Four-Layer Arithmetic Structure of the Fine-Structure Constant

The Four-Layer Arithmetic Structure of the Fine-Structure Constant | 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Article The Four-Layer Arithmetic Structure of the Fine-Structure Constant Jian Zhou This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9335558/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 propose a conjecture: the fine-structure constant satisfies a four-layer sum rule α (Λφ + τ ζ ′ K (0)) + τ = 1 , where Λφ ≡ 360/φ 2 − ln φ = 137.027 is a constant built from the golden ratio φ = (1 + √ 5)/2, τ ≡ 4 kB TCMB/ER = 6.905×10 −5 is the ratio of the cosmic microwave background thermal energy to the Rydberg energy, and ζ ′ K (0) = − ln φ/2 is the derivative of the Dedekind zeta function of Q(√ 5) at its trivial zero. The four layers are: (i) 360/φ 2 = 137.508, the golden-angle geometric ground state; (ii) − ln φ, the Dirichlet regulator; (iii) τ , the cosmic thermodynamic parameter; and (iv) τ ζ ′ K (0), a cross-term coupling arithmetic and thermodynamic structures. Each successive layer reduces the residual by an order of magnitude: 3400 ppm → 69 ppm → 0.11 ppm → 0.009 ppm. The fourth layer yields α −1 = 137.035 998, matching CODATA 2022 to 0.009 ppm—a 12-fold improvement over the three-layer formula. A refined variant employing the exact Bernoulli sum 1 − φ ln φ achieves α −1 = 137.035 999 16 (δ = +0.6 ppb), within the TCMB measurement uncertainty. Two independent mathematical derivations—the Bernoulli generating function evaluated at the Dirichlet regulator RK = ln φ and the Seeley–DeWitt heat kernel on the hyperbolic symmetric space H 2 of Q(√ 5)— converge to the same first-order expression for the fourth layer, providing structural evidence beyond numerical fitting. Crucially, |ζ ′ K (0)| = ln φ/2 introduces no new parameter: it is the class number formula invariant hK RK /wK , fully determined by φ. We show that 360 = (dK + 1)/ζK (−3), where ζK (−3) = 1/60 = 1/|A5| is a special value of the Dedekind zeta function of Q(√ 5)—establishing that the geometric ground state is a purely number-theoretic quantity, independent of the degree convention. The Fermi–Dirac self-referential fixed point 1/φ shares the same discriminant ∆ = 5, so both quantum statistics select Q(√ 5). We discuss structural correspondences with the Bost– Connes system, the McKay correspondence 2I ↔ E8, a quantitative look-elsewhere analysis, five independent consistency checks, and falsifiable predictions. 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-9335558","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":619641360,"identity":"e5447670-570c-42ea-b683-0b59006805e4","order_by":0,"name":"Jian Zhou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7UlEQVRIiWNgGAWjYBACxgYGZhABBgc+AAk2dlK0HJwB0sJM2CKEFmYeCJ+A+vbDjw1/7rBJbOA//PCwza9t8nxAIz58zMHjsJ404wTJM2mJDRJpBodz+24btjEzMEvO3IbPLznMBwzbDgO18DAczu25zQjUwsbMi09L/xvmA4kgLfxnGA5b9ty2J6xlRg5zwkGQFoYchsMMP24nEqHlmbFhY1uacRvQLwd7G24ntzEzNuP1i2F/8mPJn202sv38hx9/+PHntu389uaDHz7i09IAZbCB7WwDkw04FEOAPCr3D17Fo2AUjIJRMEIBACyvUfrG5uSkAAAAAElFTkSuQmCC","orcid":"","institution":"Independent Researcher","correspondingAuthor":true,"prefix":"","firstName":"Jian","middleName":"","lastName":"Zhou","suffix":""}],"badges":[],"createdAt":"2026-04-06 15:53:29","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9335558/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9335558/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106724527,"identity":"923cdba2-40a1-48c0-8278-c0f04e9fc780","added_by":"auto","created_at":"2026-04-12 18:28:29","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":331864,"visible":true,"origin":"","legend":"","description":"","filename":"ZhouTheFourLayerArithmeticStructureoftheFineStructureConstant2026v8.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9335558/v1_covered_6f3de91a-ff0b-4dfb-9e81-4cf8627f83f7.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The Four-Layer Arithmetic Structure of the Fine-Structure Constant","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-9335558/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9335558/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe propose a conjecture: the fine-structure constant satisfies a four-layer sum rule α (Λφ + τ ζ ′ K (0)) + τ = 1 , where Λφ ≡ 360/φ 2 − ln φ = 137.027 is a constant built from the golden ratio φ = (1 + √ 5)/2, τ ≡ 4 kB TCMB/ER = 6.905×10 −5 is the ratio of the cosmic microwave background thermal energy to the Rydberg energy, and ζ ′ K (0) = − ln φ/2 is the derivative of the Dedekind zeta function of Q(√ 5) at its trivial zero. The four layers are: (i) 360/φ 2 = 137.508, the golden-angle geometric ground state; (ii) − ln φ, the Dirichlet regulator; (iii) τ , the cosmic thermodynamic parameter; and (iv) τ ζ ′ K (0), a cross-term coupling arithmetic and thermodynamic structures. Each successive layer reduces the residual by an order of magnitude: 3400 ppm → 69 ppm → 0.11 ppm → 0.009 ppm. The fourth layer yields α −1 = 137.035 998, matching CODATA 2022 to 0.009 ppm—a 12-fold improvement over the three-layer formula. A refined variant employing the exact Bernoulli sum 1 − φ ln φ achieves α −1 = 137.035 999 16 (δ = +0.6 ppb), within the TCMB measurement uncertainty. Two independent mathematical derivations—the Bernoulli generating function evaluated at the Dirichlet regulator RK = ln φ and the Seeley–DeWitt heat kernel on the hyperbolic symmetric space H 2 of Q(√ 5)— converge to the same first-order expression for the fourth layer, providing structural evidence beyond numerical fitting. Crucially, |ζ ′ K (0)| = ln φ/2 introduces no new parameter: it is the class number formula invariant hK RK /wK , fully determined by φ. We show that 360 = (dK + 1)/ζK (−3), where ζK (−3) = 1/60 = 1/|A5| is a special value of the Dedekind zeta function of Q(√ 5)—establishing that the geometric ground state is a purely number-theoretic quantity, independent of the degree convention. The Fermi–Dirac self-referential fixed point 1/φ shares the same discriminant ∆ = 5, so both quantum statistics select Q(√ 5). We discuss structural correspondences with the Bost– Connes system, the McKay correspondence 2I ↔ E8, a quantitative look-elsewhere analysis, five independent consistency checks, and falsifiable predictions.\u003c/p\u003e","manuscriptTitle":"The Four-Layer Arithmetic Structure of the Fine-Structure Constant","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-09 02:45:44","doi":"10.21203/rs.3.rs-9335558/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":"a1d88104-6575-4204-9de7-17b58b98c34d","owner":[],"postedDate":"April 9th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-04-09T02:45:44+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-09 02:45:44","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9335558","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9335558","identity":"rs-9335558","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}