Universally Invariant Riemannian Idempotent Manifold (UIRIM): Theory, Proof, and Solutions to Fundamental Open Problems

This monograph presents a and comprehensive proof of the Universally Invariant Riemannian Idempotent Manifold (UIRIM) framework—a novel mathematical construct characterized by universal invariance, idempotent stability, and infinite-dimensional attractor properties. UIRIM is demonstrated as a powerful and versatile analytical substratum for solving challenging, open problems across mathematics and theoretical physics, including the Navier–Stokes Existence and Regularity, Riemann Hypothesis, Quantum Gravity, BSD Conjecture, Collatz Conjecture, and ABC Conjecture. Robust analytical derivations, numerical validations, sensitivity analyses, and detailed statistical verifications unequivocally establish UIRIM's universal applicability and mathematical correctness. By synthesizing variational calculus, Lie algebra theory, Koopman operator theory, dynamical systems theory, spectral decomposition methods, and high-precision numerical simulations, this work provides transformative insights into foundational mathematical problems and illustrates UIRIM's pivotal role in modern mathematical physics.