Dynamic Coherence Closure (DCC) for 3D NSE: Part I Theoretical Foundation

The closure problem in high Reynolds number turbulence remains a central challenge in fluid mechanics, necessitating sub-grid scale (SGS) models to prevent numerical singularities. Traditional approaches, such as standard Large Eddy Simulation (LES) closures, often rely on a-posteriori stabilization via empirically-tuned viscosity, which fails to capture the intrinsic self-organization of turbulent energy ows. Drawing from the Dynamic Coherence Closure (DCC) framework, we propose an a-priori regularization mechanism where closure is dened not as a structural precondition, but as a continuous, bounded coherence state variable C(E) ∈ [0, 1] driven by the cumulative energy ow E. We embed the core governing law, the Dynamic Closure Equation (DCE), as an auxiliary dynamical constraint into the ltered NavierStokes equations (NSE). This sigmoidal law is rigorously derived from a variational principle (Appendix A) and is shown to be a mathematical necessity for smooth, self-limiting dissipative systems. A parameter-free statistical validation is performed against high-delity DNS datasets (Reλ ≈ 1000). We demonstrate that the theoretical inverse scaling C ∝ ε^−1 leads to a logarithmic standard deviation σln(C) ≈ 0.833, precisely matching published turbulence dissipation statistics from forced isotropic turbulence. This formulation yields a coherence-mediated viscosity, which provides dynamic regularization during turbulent transitions (C ≈ 0.5) while vanishing as the system approaches maximal coherence (C → 1). The resulting DCC framework provides a mathematically robust, physics-based alternative for closing the turbulent energy cascade and suggests a pathway toward ensuring the global regularity of NSE solutions. Aspects of this theoretical framework are subject to pending patent applications by Yunaverse, Inc.