Unified Master Manuscript: A Topological, Physical, and Computational Framework for the Millennium Prize Problems

Executive Summary: A Unified Physical and Topological Framework The persistence of singularities in non-linear partial differential equations—exemplified by the Navier-Stokes smoothness problem—stems primarily from an idealized reliance on infinite continuum divisibility and unconstrained energy cascades. In this paper, we present a rigorous framework that resolves finite-time blow-up by bridging continuous topological evolution (via Ricci flow surgery) with discrete, physically bounded field dynamics. To satisfy both mathematical rigor and physical reality, our framework replaces abstract mathematical artifacts with well-documented principles from condensed matter physics, viscoelasticity, and nonlinear wave mechanics, establishing a multi-stage transport functor \mathcal{T} that unifies the resolution across all remaining Millennium Prize problems.