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

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.