Information Limits and Targeted Reconstruction of Pressure-Driven Rank Dynamics in Navier–Stokes Flow

Which spatial information determines pressure-driven changes of velocity-value dimension? We derive an exact chain from incompressible Navier–Stokes dynamics to a task-specific reconstruction criterion. Heat covariance identifies the exact affine velocity-value span. Pressure moves that span through the Grassmannian; its optimal linear-regression residual determines normal eigenvalue birth and the one-sided Bures–Wasserstein distance speed. A concave Ky–Fan envelope gives gap-free global tail dynamics. An explicit fixed-power family has identical complete second-order spatial correlations, zero complete third-order correlations, and identical first second-order time jets, but its birth coefficient is 7233/578 + 4 cos(Φ). The entire data fiber has a sharp uncertainty interval; one translation-invariant quartic scalar determines its position. A genuinely three-dimensional full-rank extension has second tail derivatives differing by exactly 16, ruling out the stated autonomous correlation state. Conversely, a collision-free class is identifiable from power data alone, and contracted third/fourth-order correlations are sufficient in general. The separation and targeted recovery are stable under explicitly bounded, rank-preserving perturbations with arbitrarily many additional modes. We prove observation-error and birth-error estimates, an adversarial estimation lower bound, and a certificate for the supplemented reconstruction. The full-rank separation persists qualitatively under small H⁴ perturbations. Exact Fourier algebra is checked by a separate grid-based pressure and initial-jet implementation; short-time integrations are rerun and documented separately. The completed task is instantaneous information assessment and robust targeted reconstruction, not a turbulence evolution closure or a global regularity criterion.