Rank-One Coherence Obstructions in High–High Navier–Stokes Interactions

Version 2.0 substantially revises the coherence analysis of comparable high–high interactions in the three-dimensional incompressible Navier–Stokes nonlinearity. The nonlinear output is decomposed using genuine output projectors, including separated output frequency or angular packets and output helicity or polarization, while input angular, helical, and radial labels are retained only as bookkeeping within each actual output branch. A Gram matrix of the resulting nonlinear output contributions defines an extended output-coherence matrix whose spectrum measures rank-one coherent summation and genuinely independent output directions.The revised analysis preserves the exact Gram-matrix inequality, Beltrami depletion identities, and finite-pattern heat-damping mechanism, while correcting the earlier assumption that orthogonality of input helicity or radial labels implies orthogonality of nonlinear output vectors. Exact helical triads show that distinct input channels may collapse onto the same output frequency and polarization.The resulting high–high alternative separates genuine output-rank defect, Beltrami depletion, and unrenewed finite-pattern damping from an unresolved nondepleted near-rank-one dominant-output sector. Accordingly, Version 2.0 should be read as a high–high reduction theorem rather than a complete high–high closure theorem.CGI Internal ID: CGI-RSR-000005