Heat-Covariance Rank Dynamics and Pressure-Driven Grassmann Motion in Incompressible Navier–Stokes Flow

We study the rank dynamics of a positive heat-semigroup covariance associated with a smooth incompressible Navier–Stokes velocity field. For every positive heat scale, the covariance rank equals the affine dimension of the velocity image and is independent of the observation point. This permits an intrinsic decomposition of pressure action into motion of the existing velocity-value subspace and genuine creation of new covariance directions. The first main result is a fixed-rank pressure–Grassmann identity: on every interval on which the heat-covariance rank is constant, the transverse pressure-gradient fluctuation is exactly the Grassmannian velocity of the existing velocity-value subspace. Consequently its optimal linear regression residual vanishes identically. The second main result treats a rank-touch time. The small covariance eigenvalues admit an h^2 expansion whose coefficients are the eigenvalues of a positive Schur-complement residual of the transverse pressure gradient after optimal linear prediction from the existing velocity fluctuation; directions with zero h^2 coefficient may open only at higher order. The trace of this residual is simultaneously the leading Bures–Wasserstein normal distance to the fixed-rank cone and the unavoidable first-order moving-subspace PCA reconstruction error after allowing the old subspace to rotate optimally. We also prove a rank-one rigidity statement on the periodic domain: if the velocity-value affine span has dimension at most one at an interior regular time, then the solution belongs on the entire connected regularity interval to a Galilean-translated parallel heat-flow branch. Thus a nontrivial isolated exact rank touch in three dimensions can only occur through rank two. The paper is structural: it does not claim a Navier–Stokes regularity theorem. Its contribution is an exact separation between pressure-driven support rotation and pressure-driven intrinsic dimension birth.