Spectral Curvature Law for Periodic Anomalies

Spectral Curvature Law for Periodic Anomalies | 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PDF Research Article Spectral Curvature Law for Periodic Anomalies Cesar A. Cohen de Mello This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8177118/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract A long-standing puzzle of the periodic table is the family of ''irregular'' inner–transition and late transition elements (Ce, Eu, Gd, Yb, Th, U, Np, Cm), whose radii, redox windows, and magnetic moments break smooth quantum–number trends. From Madelung and Pauling to modern relativistic and strongly correlated treatments by Pyykkö, Grant, Allen & Martin, and Moore & van der Laan, these anomalies have been clearly identified but never reduced to a single, predictive scalar law valid across the table: existing descriptions are either local (case-by-case) or purely numerical, without an underlying curvature invariant. Here they arise naturally from \emph{curvature inversions} of the relativistic electronic Hessian a Kato–analytic operator including mass–velocity, Darwin, and spin–orbit terms up to $\mathcal{O}(c^{-2})$. Its minimal eigenvalue $\lambda_{\min}(Z)$ defines a universal curvature invariant: $\lambda_{\min}>0$ corresponds to localized ionic states, $\lambda_{\min}<0$ to delocalized $f$–$d$ hybridization, and $\lambda_{\min}=0$ marks a topological bifurcation $Z_c$ with quantized Berry change $\Delta C_{fd}=\pm1$. Critical scaling $\Phi_c!\sim!|Z-Z_c|^{-1/2}$ and spectral response $G(Z)!\sim!\lambda_{\min}^{-1}$ govern volume, magnetic, and cohesive observables as Elements at $\lambda_{\min}!\approx!0$ (Ce, Yb, U, Np) show mixed valence and high compressibility, whereas half–filled $f^7$ systems (Gd, Cm) remain rigid. Relativistic $5f$–$6d$ coupling in Th–U–Np smooths curvature gradients, producing continuous covalency. Thus, periodic anomalies emerge as geometric signatures of the Hessian spectrum, yielding a compact, predictive curvature law for chemical periodicity. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 26 Apr, 2026 Reviewers invited by journal 03 Dec, 2025 Editor assigned by journal 21 Nov, 2025 Submission checks completed at journal 21 Nov, 2025 First submitted to journal 21 Nov, 2025 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-8177118","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":555124130,"identity":"dc82182f-9e6a-4512-90c2-9b6bebb83333","order_by":0,"name":"Cesar A. 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Anomalies","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"journal-of-mathematical-chemistry","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jomc","sideBox":"Learn more about [Journal of Mathematical Chemistry](http://link.springer.com/journal/10910)","snPcode":"10910","submissionUrl":"https://submission.nature.com/new-submission/10910/3","title":"Journal of Mathematical Chemistry","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-8177118/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8177118/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"A long-standing puzzle of the periodic table is the family of ''irregular'' inner–transition and late transition elements (Ce, Eu, Gd, Yb, Th, U, Np, Cm), whose radii, redox windows, and magnetic moments break smooth quantum–number trends. From Madelung and Pauling to modern relativistic and strongly correlated treatments by Pyykkö, Grant, Allen \\\u0026 Martin, and Moore \\\u0026 van der Laan, these anomalies have been clearly identified but never reduced to a single, predictive scalar law valid across the table: existing descriptions are either local (case-by-case) or purely numerical, without an underlying curvature invariant.\nHere they arise naturally from \\emph{curvature inversions} of the relativistic electronic Hessian\na Kato–analytic operator including mass–velocity, Darwin, and spin–orbit terms up to $\\mathcal{O}(c^{-2})$. Its minimal eigenvalue $\\lambda_{\\min}(Z)$ defines a universal curvature invariant:\n$\\lambda_{\\min}\u003e0$ corresponds to localized ionic states,\n$\\lambda_{\\min}\u003c0$ to delocalized $f$–$d$ hybridization, and $\\lambda_{\\min}=0$ marks a topological bifurcation $Z_c$ with quantized Berry change\n$\\Delta C_{fd}=\\pm1$. Critical scaling $\\Phi_c\\!\\sim\\!|Z-Z_c|^{-1/2}$ and spectral response\n$G(Z)\\!\\sim\\!\\lambda_{\\min}^{-1}$ govern volume, magnetic, and cohesive observables as\nElements at $\\lambda_{\\min}\\!\\approx\\!0$ (Ce, Yb, U, Np) show mixed valence and high compressibility, whereas half–filled $f^7$ systems (Gd, Cm) remain rigid. Relativistic $5f$–$6d$ coupling in Th–U–Np smooths curvature gradients, producing continuous covalency. Thus, periodic anomalies emerge as geometric signatures of the Hessian spectrum, yielding a compact, predictive curvature law for chemical periodicity.","manuscriptTitle":"Spectral Curvature Law for Periodic Anomalies","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-05 05:50:18","doi":"10.21203/rs.3.rs-8177118/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-04-26T18:38:38+00:00","index":"","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-12-03T21:18:19+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-11-22T02:06:14+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-11-22T02:05:46+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Mathematical Chemistry","date":"2025-11-22T01:11:41+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-mathematical-chemistry","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jomc","sideBox":"Learn more about [Journal of Mathematical Chemistry](http://link.springer.com/journal/10910)","snPcode":"10910","submissionUrl":"https://submission.nature.com/new-submission/10910/3","title":"Journal of Mathematical Chemistry","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"a929e570-2592-440a-b81a-f7ba041fcd84","owner":[],"postedDate":"December 5th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-04-30T19:53:41+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-05 05:50:18","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8177118","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8177118","identity":"rs-8177118","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}