Universal Spectral Scaling and Hierarchical Organization in Atomic Spectra

Universal Spectral Scaling and Hierarchical Organization in Atomic Spectra | 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 Universal Spectral Scaling and Hierarchical Organization in Atomic Spectra Abdennour Abbas This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8629139/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 A model-independent analysis of more than a century of empirical atomic spectral data across the periodic table reveals that atomic spectra exhibit a self-similar, scale-invariant multifractal-like organization, consistent with a hierarchical network of effectively coupled oscillatory modes. Across elements, core spectral observables, including line positions, spectral edges, bandwidths, fine-structure splittings, and spectral families, collapse onto deterministic global scaling relations governed by a single empirical spectral scale, the maximum observed frequency \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\) . When additional atomic observables are indexed by this same spectral scale, including isotopic state structure and reported nuclear decay sequences and rates, they likewise collapse onto the same global relations. These observables follow simple, near-perfect power-law relations to \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\) through systematic mode coupling, splitting, and hierarchical branching, without invoking particles, discrete energy transitions, or quantum-number formalisms as primitive organizing concepts. The spectral phylogeny (hierarchical spectral lineage) reconstructed from mode trajectories and splitting events shows that atomic spectra form a generational cascade of dynamically selected configurations within a single hierarchical spectral system, rather than a collection of independent objects. Atomic identity spectrally emerges from the discrete fractional spacing of stable \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\) attractors, with spacing that is large at low \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\) and converges toward a universal minimum of approximately 2% of \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\) , below which distinct stable spectral configurations are not observed. This empirically derived organization reproduces and refines major chemical groupings, indicating that regularities commonly attributed to shell structure and orbital occupation can arise directly from invariant spectral organization. Analysis of hydrogen demonstrates that the same \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\) -governed hierarchy observed across the periodic table is already present internally within its spectrum, indicating a nested and recursively organized oscillatory structure. These results show that atomic spectra constitute a low-dimensional, self-similar oscillatory hierarchy governed by an empirically accessible spectral invariant, providing a model-independent basis for spectral regularities and elemental classification. Spectral scaling Atomic spectra Self-similarity Spectral phylogeny Coupled oscillators Maximum frequency (vmax) Full Text Additional Declarations Competing interest reported. The author is an Associate Professor of Nanotechnology at the University of Minnesota. This work was carried out independently and is not part of, or funded by the University. The author is also the founder and sole shareholder of Spectral Dynamics, Inc., a startup company formed on January 6, 2026 after completion of the research reported here, which was developed over several years. The company has filed two patent applications related to potential downstream applications of this work. The company had no role in the conception, execution, interpretation, or reporting of this work. 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. 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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-8629139","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":577438577,"identity":"d2775bdb-864c-4bdb-a2f7-88a63c6d0a16","order_by":0,"name":"Abdennour Abbas","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVRIiWNgGAWjYDACCR4gUQHjFTCDqQNAzINfyxkQC6TagFgtjG1oWvAC/tm9Bz/+nFeXx89+/tiHHwbWidvZTyce+PGHQYYflyV3ziVL8247XCzZk8w8s8cgPXFnT+6Gg71tDDySDTj03MgxkGbcdiBxww1mZgYeg8OJGw7kbjjM2ABkH8CuQ/5GjvHPn3PqEvcDtTD+AWk5/3bDYYY/uLUY3Mgxk+BtYE7cIMHMzAy25QbQFgY23FoM75xLs+Y5drhY4kyyMbOMQbrxhhtvQX6RwOkXudu9h2/+qAGGWPvBx4xvKqxlN5zP3fzhxx8be1whBgMJMIYj1GwJAhqQtNgTVDoKRsEoGAUjDgAAYaBgstjEIrkAAAAASUVORK5CYII=","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Abdennour","middleName":"","lastName":"Abbas","suffix":""}],"badges":[],"createdAt":"2026-01-18 04:23:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8629139/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8629139/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102745999,"identity":"63566670-df4a-42a6-8f53-6b7f4fcccdd6","added_by":"auto","created_at":"2026-02-16 08:55:08","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2160859,"visible":true,"origin":"","legend":"","description":"","filename":"Paper2UniversalSpectralScalinginAtomsFoundations011625v3.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8629139/v1_covered_0427d324-a48b-4322-84e5-2f8240798c01.pdf"}],"financialInterests":"Competing interest reported. The author is an Associate Professor of Nanotechnology at the University of Minnesota. This work was carried out independently and is not part of, or funded by the University. The author is also the founder and sole shareholder of Spectral Dynamics, Inc., a startup company formed on January 6, 2026 after completion of the research reported here, which was developed over several years. The company has filed two patent applications related to potential downstream applications of this work. 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Across elements, core spectral observables, including line positions, spectral edges, bandwidths, fine-structure splittings, and spectral families, collapse onto deterministic global scaling relations governed by a single empirical spectral scale, the maximum observed frequency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\nu\\:}_{\\text{m}\\text{a}\\text{x}}\\)\u003c/span\u003e\u003c/span\u003e. When additional atomic observables are indexed by this same spectral scale, including isotopic state structure and reported nuclear decay sequences and rates, they likewise collapse onto the same global relations. These observables follow simple, near-perfect power-law relations to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\nu\\:}_{\\text{m}\\text{a}\\text{x}}\\)\u003c/span\u003e\u003c/span\u003e through systematic mode coupling, splitting, and hierarchical branching, without invoking particles, discrete energy transitions, or quantum-number formalisms as primitive organizing concepts.\u003c/p\u003e \u003cp\u003eThe spectral phylogeny (hierarchical spectral lineage) reconstructed from mode trajectories and splitting events shows that atomic spectra form a generational cascade of dynamically selected configurations within a single hierarchical spectral system, rather than a collection of independent objects. Atomic identity spectrally emerges from the discrete fractional spacing of stable \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\nu\\:}_{\\text{m}\\text{a}\\text{x}}\\)\u003c/span\u003e\u003c/span\u003e attractors, with spacing that is large at low \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\nu\\:}_{\\text{m}\\text{a}\\text{x}}\\)\u003c/span\u003e\u003c/span\u003e and converges toward a universal minimum of approximately 2% of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\nu\\:}_{\\text{m}\\text{a}\\text{x}}\\)\u003c/span\u003e\u003c/span\u003e, below which distinct stable spectral configurations are not observed. This empirically derived organization reproduces and refines major chemical groupings, indicating that regularities commonly attributed to shell structure and orbital occupation can arise directly from invariant spectral organization.\u003c/p\u003e \u003cp\u003eAnalysis of hydrogen demonstrates that the same \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\nu\\:}_{\\text{m}\\text{a}\\text{x}}\\)\u003c/span\u003e\u003c/span\u003e-governed hierarchy observed across the periodic table is already present internally within its spectrum, indicating a nested and recursively organized oscillatory structure. These results show that atomic spectra constitute a low-dimensional, self-similar oscillatory hierarchy governed by an empirically accessible spectral invariant, providing a model-independent basis for spectral regularities and elemental classification.\u003c/p\u003e","manuscriptTitle":"Universal Spectral Scaling and Hierarchical Organization in Atomic Spectra","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-27 12:17:59","doi":"10.21203/rs.3.rs-8629139/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":"718f5b6b-a8ba-4bf8-844e-38821c09f8f9","owner":[],"postedDate":"January 27th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-02-12T14:42:40+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-27 12:17:59","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8629139","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8629139","identity":"rs-8629139","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}