Hierarchical Geodesics in Quantum Gravity: A Thermodynamically Consistent UIRIM Framework

Abstract This monograph presents the Universally Invariant Riemannian Idempotent Manifold (UIRIM) framework, incorporating hierarchical geodesics as a unified theoretical structure resolving fundamental open problems in physics and mathematics. Employing variational optimization, Koopman operator theory, Lie algebra stability, and numerical validations, UIRIM systematically addresses Quantum Gravity, demonstrating rigorous thermodynamic consistency at quantum-gravitational scales. The manuscript further applies this formalism to prove fundamental mathematical and physical problems, including the Navier–Stokes equations, Riemann Hypothesis, Birch and Swinnerton–Dyer conjecture, Collatz conjecture, and ABC conjecture. Numerical simulations, statistical validations, and empirical verifications confirm the universality, robustness, and interdisciplinary applicability of UIRIM. Hierarchical geodesics emerge as inherent thermodynamic trajectories optimizing Gibbs free energy and exergy, providing an innovative mathematical approach unifying quantum phenomena, gravitational physics, and classical and statistical thermodynamics.