Projective Rank-one Closure for Terminal Navier–Stokes Saturation

This record contains a Coherence Geometry manuscript studying terminal rank-one saturation mechanisms in a dyadic analysis of the three-dimensional incompressible Navier-Stokes equations. It begins from a high-high OBCI closure module for comparable high-frequency interactions and analyzes the remaining determining-scale paraproduct strain branch using localized output Gram matrices. Positive spectral mass away from the top eigenvalue produces a coherence-rank defect and hence a square-function gap. The manuscript develops a terminal branch ledger that reduces possible nondepleted rank-one configurations through Beltrami depletion, finite-beat damping, orthogonal channel splitting, the middle-eigenvalue strain criterion, and velocity- and vorticity-direction criteria. The final moving-frame one-component branch has the form \[U=φ v, |v|=1.\] Rank-one output coherence turns this into a projective problem: the active output selects a projective direction \([v]\). If \([v]\) is flat, the branch is fixed-frame and trivial by incompressibility. If \([v]\) is nonflat and visible in the output space, it creates a secondary projective mode and hence a positive coherence-rank defect. The remaining axial or scalar-angle degeneracies are routed through the velocity-direction, vorticity-direction, planar/2D3C, or splitting alternatives. This manuscript is the second paper in a two-part Coherence Geometry sequence on Navier-Stokes terminal closure. It builds on CGI-RSR-000005, Rank-One Coherence Obstructions in High--High Navier-Stokes Interactions, available at https://doi.org/10.5281/zenodo.19970064. It is part of the Coherence Geometry Clay-problem research series and is categorized under the Navier-Stokes existence and smoothness problem. It is presented as a terminal-closure analysis of nondepleted rank-one saturation branches, not as an accepted resolution of the Navier-Stokes problem.