Hierarchical Geodesics in Quantum Gravity: A Thermodynamically Consistent UIRIM Framework-Revised

Hierarchical Geodesics in Quantum Gravity: A Thermodynamically Consistent UIRIM Framework-Revised | 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 Hierarchical Geodesics in Quantum Gravity: A Thermodynamically Consistent UIRIM Framework-Revised Venkatesan Narayanaswamy This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6621169/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 This monograph introduces the Universally Invariant Riemannian Idempotent Manifold (UIRIM) framework, incorporating hierarchical geodesics as a unified theoretical structure to systematically resolve fundamental open problems across physics and mathematics. Employing advanced variational optimization techniques, Koopman operator theory, Lie algebra stability criteria, and robust numerical validations, UIRIM addresses critical challenges in Quantum Gravity, establishing thermodynamic consistency at quantum-gravitational scales. The manuscript leverages this comprehensive formalism to offer novel quantum-gravity thermodynamic interpretations and proofs of significant mathematical conjectures, including the Riemann Hypothesis, Birch and Swinnerton–Dyer conjecture, Collatz conjecture, and ABC conjecture. Within this context, these conjectures acquire specific thermodynamic meanings: the Riemann Hypothesis defines critical entropy lines representing minimal entropy states; the Birch and Swinnerton–Dyer conjecture identifies algebraic conditions defining quantum-gravitational equilibrium states, which may include stable, unstable, or neutral states; the Collatz conjecture embodies iterative entropy convergence towards stable equilibrium; and the ABC conjecture sets analytic conditions necessary to ensure the stability and resilience of equilibrium states against perturbations. Thorough numerical simulations, statistical validations, and detailed empirical verifications collectively confirm the universality, robustness, and extensive interdisciplinary applicability of the UIRIM framework. Hierarchical geodesics naturally emerge as intrinsic thermodynamic trajectories optimizing Gibbs free energy and exergy, thus offering a fundamentally innovative mathematical approach capable of unifying quantum phenomena, gravitational physics, and classical and statistical thermodynamics. Mathematical Physics quantum gravity hierarchical geodesics thermodynamics Gibbs free energy Riemannian geometry geometric hierarchies Universally Invariant Riemannian Idempotent Manifold (UIRIM) Navier–Stokes equations Riemann hypothesis BSD conjecture Collatz conjecture ABC conjecture Koopman operator variational calculus mathematical physics Full Text Additional Declarations The authors declare no competing interests. Supplementary Files SupplementaryInformationAExamplesandExercises.docx Supplementary Information A – Numerical Worked Examples and Exercises with Solutions Illustrating UIRIM SupplementaryInformationBQGProof.docx Supplementary Information B - Quantum Gravity (Unification of Quantum Mechanics and General Relativity)—A Proof using the UIRIM Framework SupplementaryInformationC.docx Supplementary Information C: Detailed Analysis and Solutions of Schwarzschild, Kerr, Anti-de Sitter, and Petrov Type D Spacetimes via UIRIM SupplementaryInformationD.docx Supplementary Information D – Detailed proof of Riemann Hypothesis and Some Conjectures via UIRIM SupplementaryInformationE.docx Supplementary Information E - Proof of Existence and Regularity of Navier–Stokes Equations via UIRIM Figure1NavierStokesNumericalAnalysis.xlsx Figure3NSNumAnal.xlsx Figure3NSFullNumericalAnalysis.xlsx Figures4and5PetrosUIRIMNumericalSimulationData.xlsx Figure6RiemannZetaNumericalValidation.xlsx Figure8RiemannZeta3DNumericalValidation.xlsx Figure9andFigure10BSDNumericalValidation.xlsx NormalisedGPSAtomicClockData.xlsx UIRIMNumStatValidation.xlsx GPSAtomicClockRawData.xlsx QGUIRIMSensitivityAnalysis.xlsx QGUIRIMSimulationData.xlsx 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. 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