Nonlinear Boundary Degeneracy and the Limits of Scalar Diagnostics

Nonlinear Boundary Degeneracy and the Limits of Scalar Diagnostics | 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Article Nonlinear Boundary Degeneracy and the Limits of Scalar Diagnostics Takashi Matsuhisa This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9114370/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 Scalar order parameters are widely used to describe critical transitions in nonlinear systems, yet their structural validity is rarely stated explicitly. This paper identifies a sharp obstruction to scalar diagnostics that arises when regime boundaries are generated by interacting degeneracies. We show that if the reduced boundary ideal is non-principal, then the boundary-induced comparability structure necessarily contains incomparabilities, and no scalar index can be faithful (even up to monotone transformation) to the underlying boundary constraints. Our analysis yields a boundary-first classification. In the codimension-one principal regime, the boundary is a hypersurface with a canonical generator Δred (unique up to units and admissible reparametrizations), and scalar crisis diagnostics are structurally pinned down to a single channel. Beyond this regime, scalarization fails intrinsically: the failure is ideal-theoretic rather than a calibration artifact. We illustrate the mechanism by a minimal two-fold construction producing a non-principal boundary, and introduce the positive-degree Betti Profile Index BPI∗ as a minimal cohomological gatekeeper that becomes informative exactly when the scalar paradigm is no longer defensible. Applied Mathematics Operations Research Economic Theory boundary ideals and non-principal degeneracy discriminant channels crisis orders scalar diagnostics cohomological gatekeepers Betti Profile Index Full Text Additional Declarations The authors declare no competing interests. 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. 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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-9114370","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":605635977,"identity":"a2b49eb5-85b4-456b-b17a-1ea59bb825a5","order_by":0,"name":"Takashi Matsuhisa","email":"data:image/png;base64,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","orcid":"","institution":"IAMR KarRC RAS","correspondingAuthor":true,"prefix":"","firstName":"Takashi","middleName":"","lastName":"Matsuhisa","suffix":""}],"badges":[],"createdAt":"2026-03-13 11:46:02","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9114370/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9114370/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104682099,"identity":"7ca5de16-d120-4ac0-abdf-229a4b08bd00","added_by":"auto","created_at":"2026-03-16 02:55:50","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":347847,"visible":true,"origin":"","legend":"","description":"","filename":"NonlinearBoundaryDegeneracyandtheLimitsofScalarDiagnostics4.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9114370/v1_covered_2af318ec-ca56-4174-b8f5-9531cce545d2.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eNonlinear Boundary Degeneracy and the Limits of Scalar Diagnostics\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"MRI BSBH Foundation for Sci Res","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":"boundary ideals and non-principal degeneracy, discriminant channels, crisis orders, scalar diagnostics, cohomological gatekeepers, Betti Profile Index","lastPublishedDoi":"10.21203/rs.3.rs-9114370/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9114370/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eScalar order parameters are widely used to describe critical transitions in nonlinear systems, yet their structural validity is rarely stated explicitly. 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