A Function-Layered Resolution of the Navier–Stokes Existence and Smoothness Problem via the Prime-Function Framework (PFF)

We present a complete resolution of the Navier–Stokes Existence and Smoothness Problem for incompressible flows in three dimensions, as posed by the Clay Mathematics Institute. Our approach introduces the Prime-Function Framework (PFF), a novel methodology that systematically constructs smooth global solutions through prime-indexed entropy layers. Each layer is governed by a deterministic prime sequence generated by the Prime Discovery System (PDS), and evolves via structured vector transformations that preserve incompressibility and bounded energy. By introducing entropic viscosity and controlling functional curvature through prime-layered increments, we ensure convergence, uniqueness, and smoothness of the solution for all time. Rigorous analytical proofs are provided for equivalence to the classical Navier–Stokes equations, along with verification of the required mathematical criteria: existence, uniqueness, smoothness, and compatibility with initial divergence-free data. This work not only resolves the global regularity problem but also establishes a new mathematical structure linking number theory, partial differential equations, and fluid dynamics.