A Symbolic and Rigorous Mathematical Framework to Address the Navier–Stokes Existence and Smoothness Problem (v2.00)

This document presents version 2.00 of a symbolic and rigorous mathematical framework developed to address the Navier–Stokes Existence and Smoothness Millennium Problem. It introduces a novel dual-phase model of fluid motion using symbolic states—fu (uniform/stable motion), nfu (non-uniform/unstable motion), and S (singularity pathways)—arising from interactions between matter (M) and smoothness conditions (Sp, -Sp). The theory formalizes these transitions into three core equations and proposes the Lemma of Smoothness, outlining how smooth solutions can be restored under certain symbolic constraints. The mathematical formulation is supported with carefully constructed symbolic mappings, vector field interpretations, and ray diagrams, aiming to demonstrate bounded energy and smooth behavior. This version integrates all symbolic definitions with precise mathematical expressions, offering a potentially complete bridge toward resolving the existence and smoothness of solutions to the Navier–Stokes equations.