A Symmetry-Induced Spectral Collapse in a Torus-Parameterized Non-Hermitian Matrix Family: With an Application to Kerr Ringdown Decay-Rate Scaling

A Symmetry-Induced Spectral Collapse in a Torus-Parameterized Non-Hermitian Matrix Family: With an Application to Kerr Ringdown Decay-Rate Scaling | 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 A Symmetry-Induced Spectral Collapse in a Torus-Parameterized Non-Hermitian Matrix Family: With an Application to Kerr Ringdown Decay-Rate Scaling Amin Al Yaquob This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8630045/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 give a complete spectral characterization of a 6×6 torus-parameterized non-Hermitian matrix family A(x,t₁,t₂) = −iI + B(x,t₁,t₂). Our main result is a symmetry-induced collapse: at the unique coupling x = 1, the shifted matrix B becomes real symmetric for every (t₁,t₂) ∈ T², forcing all eigenvalues of A to lie on the horizontal line Im(λ) = −1. We quantify the transition by the imaginary spread Δ(x), proving Δ(1) = 0 and Δ(x) > 0 for x ≠ 1, and we identify parameter locations attaining spectral-radius extrema on the torus. We further show the global spectral radius ρ(x) is non-monotone for small x, providing a concrete counterexample to monotonic expansion intuition in non-Hermitian coupling. We prove robustness: for x = 1 + ε with small |ε|, the imaginary spread satisfies Δ(1+ε) = O(|ε|). Beyond this specific family, we establish a mechanism theorem: any family A(x) = −iI + B(x) where B(x_c) is real symmetric at a unique x_c exhibits the same collapse phenomenon. We interpret the collapse as decay-rate synchronization in a damped-oscillator network and note a structural parallel with the suppression of Kerr quasinormal-mode damping as a/M → 1. Using 15 binary black hole mergers from GWTC-3 and GW250114, we find strong correlation (r = 0.905, ρ_s = 0.886) between remnant spin and ringdown lifetime—consistent with the spectral-flow interpretation of decay-rate compression near critical regimes. Theoretical Physics non-Hermitian matrix spectral collapse symmetry point eigenvalue transition quasinormal modes LIGO gravitational waves Full Text Additional Declarations The authors declare no competing interests. Supplementary Files spectralcollapsesupplementary.pdf validaterealdata.py 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. 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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-8630045","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":576240242,"identity":"10118bef-799e-49e1-a162-fc37c3990bc8","order_by":0,"name":"Amin Al Yaquob","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAv0lEQVRIiWNgGAWjYFACxgaGDwwWYCYz0VoYZzBIkKQFqJKHJC380w63PbapkUjsb88xfFzAYCen20BAi8TtxHbjnGMSiTPOvDE2nsGQbGx2gJA1txPbpHPYJBIbbuSYSfMwHEjcRkiLPEiLxT+JxPlEazEAaWFsk0jcQLQWQ6AWyd4+CeONZ54VG88wIMIvcrfTn0n8+GYjO+948sbHBRV2coS9DwWODQwJIHcSqRwE7BnAWkbBKBgFo2AUYAEAl41AoRX7xukAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0006-1590-3378","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Amin","middleName":"Al","lastName":"Yaquob","suffix":""}],"badges":[],"createdAt":"2026-01-18 08:26:18","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8630045/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8630045/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100646970,"identity":"40073bb4-5055-4558-bdf0-5630306075e6","added_by":"auto","created_at":"2026-01-20 05:02:19","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2596823,"visible":true,"origin":"","legend":"","description":"","filename":"spectralcollapsepaper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8630045/v1_covered_c526037f-8c64-4767-b846-5507847517fb.pdf"},{"id":100646938,"identity":"ed002796-1c79-4ab3-88d8-73fb2fb466dd","added_by":"auto","created_at":"2026-01-20 05:02:09","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":266131,"visible":true,"origin":"","legend":"","description":"","filename":"spectralcollapsesupplementary.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8630045/v1/f5b8992b73dca75dcb7ab95e.pdf"},{"id":100646834,"identity":"d3163ecb-a1c9-45fd-af8e-7f0c9e8998d1","added_by":"auto","created_at":"2026-01-20 05:01:33","extension":"py","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":16669,"visible":true,"origin":"","legend":"","description":"","filename":"validaterealdata.py","url":"https://assets-eu.researchsquare.com/files/rs-8630045/v1/ec02638ba4ff4c6dc18d0ea6.py"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA Symmetry-Induced Spectral Collapse in a Torus-Parameterized Non-Hermitian Matrix Family: With an Application to Kerr Ringdown Decay-Rate Scaling\u003c/p\u003e","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":"non-Hermitian matrix, spectral collapse, symmetry point, eigenvalue transition, quasinormal modes, LIGO, gravitational waves","lastPublishedDoi":"10.21203/rs.3.rs-8630045/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8630045/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe give a complete spectral characterization of a 6×6 torus-parameterized non-Hermitian matrix family A(x,t₁,t₂) = −iI + B(x,t₁,t₂). Our main result is a symmetry-induced collapse: at the unique coupling x = 1, the shifted matrix B becomes real symmetric for every (t₁,t₂) ∈ T², forcing all eigenvalues of A to lie on the horizontal line Im(λ) = −1. We quantify the transition by the imaginary spread Δ(x), proving Δ(1) = 0 and Δ(x) \u0026gt; 0 for x ≠ 1, and we identify parameter locations attaining spectral-radius extrema on the torus. We further show the global spectral radius ρ(x) is non-monotone for small x, providing a concrete counterexample to monotonic expansion intuition in non-Hermitian coupling. We prove robustness: for x = 1 + ε with small |ε|, the imaginary spread satisfies Δ(1+ε) = O(|ε|).\u003c/p\u003e\n\u003cp\u003eBeyond this specific family, we establish a mechanism theorem: any family A(x) = −iI + B(x) where B(x_c) is real symmetric at a unique x_c exhibits the same collapse phenomenon. We interpret the collapse as decay-rate synchronization in a damped-oscillator network and note a structural parallel with the suppression of Kerr quasinormal-mode damping as a/M → 1. 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