Free-Energy Geometry and Dynamical Regime Switching under Feasibility Constraints A Transferable Dynamical Motif for Nonequilibrium Systems

Free-Energy Geometry and Dynamical Regime Switching under Feasibility Constraints A Transferable Dynamical Motif for Nonequilibrium Systems | 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 Free-Energy Geometry and Dynamical Regime Switching under Feasibility Constraints A Transferable Dynamical Motif for Nonequilibrium Systems Toshio SATO This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9114900/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract We develop a statistical-mechanical framework for nonequilibrium systems that exhibits regime switching under feasibility constraints. The framework is organized around a freeenergy– like evaluation geometry on a thermodynamic-style U–S plane, where U summarizes mismatch-/cost-like contributions and S summarizes uncertainty-/openness-like contributions. We define Z(λ) = U − λS not as a scalar objective required to decrease monotonically, but as a parametrized evaluative ruler whose rotation with λ changes the relative weighting of the two contributions. A central claim is that the relevant type asymmetry is structural: one evaluative term is present-conditioned and point-like, whereas the other is future-oriented and trajectory-based. Selection is therefore formulated by a feasibility-aware “first-touch” rule: among admissible branches, the preferred regime is identified by the first feasible contact between an iso-Z ruler and the candidate bundle. This geometry yields delayed switching, hysteresis, and path dependence without requiring a global optimization narrative. To show that the construction is compatible with a dynamical system perspective, we also present a concrete mechanical realization that implements the same rotation–contact– regime-change logic as a minimal periodic dynamics. The result is a transferable dynamical motif for constrained nonequilibrium systems in which free-energy geometry, feasibility boundaries, and regime switching can be analyzed within a common descriptive framework. DBFEP nonequilibrium systems regime switching U–S plane Z(λ) = U − λS first- touch present fit / future openness Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions 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-9114900","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":616167494,"identity":"f4fb2b61-65ac-4810-a8dc-3f4751d508d5","order_by":0,"name":"Toshio SATO","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA90lEQVRIiWNgGAWjYHACNoYEEHX4AAPDBxCXnYB6HriWYwkMjDNAWpiJ0cIA1cLMA2IQ0mLP3v7swYMKuzy+Y8zHHtv82ibPx8zA+OFjDh5beM6YGyScSS6WPMaWbpzbd9uwjZmBWXLmNjxaJHLYJBLbmBM33O8xk87tuc0I1MLGzItXS/ozoJb6xA3H+L9JW/bctidCS4IZUMthoBYeNmmGH7cTCWs5c8ZMIuHM8cSZx9jMDXsbbie3MTM24/ULe3v7M8kfFdWJfceYnz348ee27fz25oMfPuLRggzYGBjbQDRjA3HqwVoY/hCteBSMglEwCkYQAAA/n0/SLdyDnAAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0009-0003-4628-0167","institution":"Independent researcher","correspondingAuthor":true,"prefix":"","firstName":"Toshio","middleName":"","lastName":"SATO","suffix":""}],"badges":[],"createdAt":"2026-03-13 12:47:52","currentVersionCode":2,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9114900/v2","doiUrl":"https://doi.org/10.21203/rs.3.rs-9114900/v2","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106726579,"identity":"392e3d06-f6c7-477a-bead-946597c69431","added_by":"auto","created_at":"2026-04-12 18:36:37","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2238611,"visible":true,"origin":"","legend":"","description":"","filename":"3DBFEPRS.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9114900/v2_covered_e7d106b0-f69c-4c6e-8282-3f34619cd1b9.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"Free-Energy Geometry and Dynamical Regime Switching under\nFeasibility Constraints\nA Transferable Dynamical Motif for Nonequilibrium Systems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Toyo University","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":"DBFEP, nonequilibrium systems, regime switching, U–S plane, Z(λ) = U − λS, first-, touch, present fit / future openness","lastPublishedDoi":"10.21203/rs.3.rs-9114900/v2","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9114900/v2","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe develop a statistical-mechanical framework for nonequilibrium systems that exhibits\u003c/p\u003e\n\u003cp\u003eregime switching under feasibility constraints. The framework is organized around a freeenergy–\u003c/p\u003e\n\u003cp\u003elike evaluation geometry on a thermodynamic-style U–S plane, where U summarizes\u003c/p\u003e\n\u003cp\u003emismatch-/cost-like contributions and S summarizes uncertainty-/openness-like contributions.\u003c/p\u003e\n\u003cp\u003eWe define Z(λ) = U − λS not as a scalar objective required to decrease monotonically, but\u003c/p\u003e\n\u003cp\u003eas a parametrized evaluative ruler whose rotation with λ changes the relative weighting of\u003c/p\u003e\n\u003cp\u003ethe two contributions.\u003c/p\u003e\n\u003cp\u003eA central claim is that the relevant type asymmetry is structural: one evaluative term is\u003c/p\u003e\n\u003cp\u003epresent-conditioned and point-like, whereas the other is future-oriented and trajectory-based.\u003c/p\u003e\n\u003cp\u003eSelection is therefore formulated by a feasibility-aware “first-touch” rule: among admissible\u003c/p\u003e\n\u003cp\u003ebranches, the preferred regime is identified by the first feasible contact between an iso-Z\u003c/p\u003e\n\u003cp\u003eruler and the candidate bundle. This geometry yields delayed switching, hysteresis, and path\u003c/p\u003e\n\u003cp\u003edependence without requiring a global optimization narrative.\u003c/p\u003e\n\u003cp\u003eTo show that the construction is compatible with a dynamical system perspective, we\u003c/p\u003e\n\u003cp\u003ealso present a concrete mechanical realization that implements the same rotation–contact–\u003c/p\u003e\n\u003cp\u003eregime-change logic as a minimal periodic dynamics. 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