Global Regularity of the Navier–Stokes Equations via the PBFD–Ω Framework (v4.1b)

Navier–Stokes Existence and Smoothness through the PBFD–Ω Framework (v4.1b)

A Structural–Analytic Proof and Open Verification Study

This upload presents the complete analytic framework and proof structure of the PBFD–Ω (Prime-Based Force Dispersion and Ω-Regularity) program, which addresses the existence and smoothness problem for the three-dimensional Navier–Stokes equations.  
It consolidates three key stages of the project as follows:

1. Navier_Stokes_Existence_and_Smoothness_Skeleton.pdf
   Provides the foundational structure outlining the Φ–Δ–Ω hierarchy and the initial formulation of the regularity hypothesis.

2. 3.2b_Navier_Stokes_Existence_and_Smoothness__PBFD_LES_and_Dimensional_Expansion_Framework.pdf
   Develops the dimensional-expansion and Large-Eddy-Simulation (LES) reduction framework that defines the PBFD dynamic operator basis and prime-indexed energy coupling.

3. PBFD_Omega_v4_1_FullIntegrated.pdf
   Contains the full analytic completion (v4.1b), the coercivity inequality (A2), residual-control lemmas, the time-averaged regularity theorem, and Appendices A–C describing the Open Verification Integration protocol.  
   This document formalizes a reproducible analytic pathway that allows independent verification of the results via numerical flux calibration and residual-energy evaluation.

Together, these works form a coherent and fully reproducible analytic program demonstrating that forced antisymmetric energy dispersion, inherent in the PBFD–Ω framework, ensures global regularity for all Leray–Hopf weak solutions of the Navier–Stokes equations.