Exact mean-covariance dynamics of the Weber field in the stochastic Lagrangian representation of the 3D Navier-Stokes equations

The Constantin-Iyer formula represents a smooth solution of the incompressible Navier-Stokes equations on ${\mathbb{T}}^3$ as $u=\mathbb{P}\mathbb{E}[(\mathrm{\nabla }{A}_t{)}^{\mathrm{\top }}(u_0\circ A_t)]$, the projected expectation of a stochastic Weber field. We separate this expectation into the product of the means and a centred covariance, and show that the covariance – together with the second moment of the inverse deformation gradient and the two-point covariance – satisfies a closed deterministic advection-diffusion equation: because every Lagrangian label is driven by the same Brownian path, the quadratic covariation collapses into a Laplacian and no moment hierarchy appears. From these equations we obtain an energy/production balance, pointwise and Holder covariance bounds, and an exact degeneracy of the two-point operator in the separation variable. A second group of results concerns the mean displacement, which solves a forced heat equation along the flow; its $L^2$ norm is controlled unconditionally by the initial energy, while an exact family of Navier-Stokes shear flows shows that no analogous Holder bound can hold uniformly down to $t=0$. We further isolate several obstructions, each backed by an explicit counterexample – the norm of an expected deformation gradient does not control the expected norm, even for genuine stochastic flows of smooth divergence-free drifts; a spatial Holder bound on the mean does not imply parabolic Campanato decay – and we prove a gauge-invariant quotient criterion equivalent to the Serrin norm of the velocity, together with a Liouville-type rigidity theorem for ancient solutions whose mean coordinate is affine of rank at least one. No claim is made regarding global regularity; the missing steps are stated explicitly as open problems.