This document provides an extended and rigorous framework dedicated to the geometric analysis of the 3D incompressible Navier-Stokes equations \cite{ESS2003}. We comprehensively develop geometric proofs related to decay estimates, blow-up profiles, and the foundational partial regularity theory of Caffarelli, Kohn, and Nirenberg (CKN). We examine in detail the Hausdorff dimension of potential singular sets, local monotonicity inequalities, and the spectral analysis of linearized operators to classify singularities into stable and unstable manifolds.