A Conditional Structural Theorem for Self-Generated Tangent Saturation in the Three-Dimensional Fourier–Leray Navier–Stokes Nonlinearity

We prove a conditional structural theorem for normalized Fourier snapshots of the three-dimensional Navier–Stokes equations on the torus, using the true Fourier–Leray quadratic nonlinearity. The hypotheses, denoted CRIT and SAT, are explicit respectively as bounds on the normalized spectral geometry and as a lower bound for a projected self-generated tangent response; neither hypothesis is asserted to be dynamically selected. The tangent is decomposed mode by mode into a diagonal multiplier component and an orthogonal defect. A corresponding covariance defect measures the failure of that multiplier to commute with the quadratic nonlinearity. For all sufficiently large frequency scales n, CRIT and SAT imply a four-way alternative: a polarization defect, a mixed modal defect, a cutoff covariance contribution, or a volumetric ultra-thin branch. In the last branch there are Ω(n³) output modes with occupation at most C n⁻⁵, multiplier size at least c√n, and normalized projected linearized response at least cn. A weighted Tonelli–Cauchy–Schwarz argument then yields Ω(n⁶) ordered nonzero mixed interactions (k,p). This is not a count of geometrically distinct triads, and no bound |p|,|k−p| = O(n) is obtained. We also record exact frozen-projector dynamical identities, without deriving a further dynamic dichotomy. The theorem neither proves selection of CRIT and SAT nor establishes global regularity for three-dimensional Navier–Stokes.