{"paper_id":"23327fb6-dc39-44ea-bc06-075e81b14278","body_text":"Φ-Optimal Hierarchical Brain Oscillations and β-Controlled Cognitive Dynamics: First-Principles Mathematical Foundations of the A7-HBM-ΩΦ Model | 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 Φ-Optimal Hierarchical Brain Oscillations and β-Controlled Cognitive Dynamics: First-Principles Mathematical Foundations of the A7-HBM-ΩΦ Model Yazeed Mohammed Al-Olofi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8408024/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 present the mathematical foundations of the A7-HBM-ΩΦ framework, unifying spectral hierarchy, neural geometry, and cognitive control. Through an optimality principle, we derive the golden ratio (Φ ≈ 1.618) as the optimal spectral spacing parameter that minimizes interference while maximizing communication efficiency. Biological constraints (0.5–200 Hz) naturally yield seven hierarchical frequency levels (δ, θ, α, σ, β, γ, Ω), each Φ-scaled from adjacent bands. We introduce a control parameter β ∈ [0,1] regulating information flow direction, with critical transitions at β = Φ⁻² ≈ 0.382 and β = Φ⁻¹ ≈ 0.618. The complete A7-HBM-ΩΦ-Bidi system comprises seven nonlinearly coupled Stuart-Landau oscillators with tri-diagonal β-dependent coupling. These mathematical foundations provide a rigorous basis for modeling hierarchical brain dynamics, enabling future investigations of microstate transitions, stability analysis, and clinical applications within a unified framework. Computational Neuroscience A7-HBM-ΩΦ golden ratio hierarchical oscillations control parameter β mathematical neuroscience brain dynamics coupled oscillators spectral scaling 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. 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-8408024\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":true,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":563179266,\"identity\":\"053283b6-2865-4e31-b3d7-863030b4fd40\",\"order_by\":0,\"name\":\"Yazeed Mohammed 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Through an optimality principle, we derive the golden ratio (Φ\\u0026thinsp;\\u0026asymp;\\u0026thinsp;1.618) as the optimal spectral spacing parameter that minimizes interference while maximizing communication efficiency. Biological constraints (0.5\\u0026ndash;200 Hz) naturally yield seven hierarchical frequency levels (δ, θ, α, σ, β, γ, Ω), each Φ-scaled from adjacent bands. We introduce a control parameter β \\u0026isin; [0,1] regulating information flow direction, with critical transitions at β\\u0026thinsp;=\\u0026thinsp;Φ⁻\\u0026sup2; \\u0026asymp; 0.382 and β\\u0026thinsp;=\\u0026thinsp;Φ⁻\\u0026sup1; \\u0026asymp; 0.618. The complete A7-HBM-ΩΦ-Bidi system comprises seven nonlinearly coupled Stuart-Landau oscillators with tri-diagonal β-dependent coupling. These mathematical foundations provide a rigorous basis for modeling hierarchical brain dynamics, enabling future investigations of microstate transitions, stability analysis, and clinical applications within a unified framework.\\u003c/p\\u003e \\u003cp\\u003e \\u003c/p\\u003e\",\"manuscriptTitle\":\"Φ-Optimal Hierarchical Brain Oscillations and β-Controlled Cognitive Dynamics: First-Principles Mathematical Foundations of the A7-HBM-ΩΦ Model\",\"msid\":\"\",\"msnumber\":\"\",\"nonDraftVersions\":[{\"code\":1,\"date\":\"2025-12-23 04:31:36\",\"doi\":\"10.21203/rs.3.rs-8408024/v1\",\"editorialEvents\":[{\"type\":\"communityComments\",\"content\":0}],\"status\":\"published\",\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"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\":\"d0fd7b2c-fe5f-4f47-9c8d-d10ad6bed5ba\",\"owner\":[],\"postedDate\":\"December 23rd, 2025\",\"published\":true,\"recentEditorialEvents\":[],\"rejectedJournal\":[],\"revision\":\"\",\"amendment\":\"\",\"status\":\"posted\",\"subjectAreas\":[{\"id\":59977274,\"name\":\"Computational Neuroscience\"}],\"tags\":[],\"updatedAt\":\"2025-12-23T04:31:36+00:00\",\"versionOfRecord\":[],\"versionCreatedAt\":\"2025-12-23 04:31:36\",\"video\":\"\",\"vorDoi\":\"\",\"vorDoiUrl\":\"\",\"workflowStages\":[]},\"version\":\"v1\",\"identity\":\"rs-8408024\",\"journalConfig\":\"researchsquare\"},\"__N_SSP\":true},\"page\":\"/article/[identity]/[[...version]]\",\"query\":{\"redirect\":\"/article/rs-8408024\",\"identity\":\"rs-8408024\",\"version\":[\"v1\"]},\"buildId\":\"8U1c8b4HqxoKbykW_rLl7\",\"isFallback\":false,\"isExperimentalCompile\":false,\"dynamicIds\":[84888],\"gssp\":true,\"scriptLoader\":[]}","source_license":"CC-BY-4.0","license_restricted":false}