Quadrupole-Depletion Closure: Scale-Invariant Gate and Rigidity for 3D Incompressible Navier--Stokes Global Regularity

This record deposits a closure manuscript for the 3D incompressible Navier--Stokes global regularity problem on $\mathbb{T}^3$ (and, with standard decay assumptions, on $\mathbb{R}^3$). The argument is built around the critical $L^3$ identity and a structural property of the pressure Hessian operator $\partial_j\partial_k\Delta^{-1}$. Navier--Stokes is\[\partial_t v + (v\cdot\nabla)v = -\nabla p + \nu \Delta v,\qquad \nabla\cdot v = 0,\]with pressure determined by\[-\Delta p = \partial_j\partial_k(v_j v_k),\qquad p = \mathcal{R}_j\mathcal{R}_k(v_j v_k),\]where $\mathcal{R}_j$ are the Riesz transforms. Multiplying the equation by $|v|v$ yields the critical identity\[\frac{d}{dt}\|v(t)\|_3^3 + 3\nu D_3(v(t)) = -3\int |v|\,v\cdot\nabla p\,dx,\]where\[D_3(v) = \int\left(|v|\,|\nabla v|^2 + \frac{(v\cdot\nabla v)^2}{|v|}\right)dx.\]Global regularity follows if the pressure work term is strictly dominated by dissipation, i.e.\[\left|\int |v|\,v\cdot\nabla p\,dx\right|\le c\,D_3(v)\quad\text{with}\quad c