Collision-Robust Soft Projectors for Navier–Stokes Strain Geometry: Gap-Free Variation Bounds, Orlicz Tail Leakage, and a Coherence Certificate at the Enemy-Mode Threshold

This record provides a self-contained LaTeX source and compiled PDF for a “projector package” that isolates a recurrent bottleneck in geometric Navier–Stokes regularity routes: instability at strain eigenvalue collisions and direction-blind enemy-mode conditions.

We introduce a gap-free soft spectral projector P_\tau(S)=\exp(\tau S)/\mathrm{tr}(\exp(\tau S)) for the strain tensor S=\tfrac12(\nabla u+\nabla u^\top), and prove a collision-robust variation estimate

\|\nabla P_\tau(S)\|_{L^2}\ \le\ 2\tau\,\|\nabla S\|_{L^2},

which avoids any division by the spectral gap. This replaces the classical Davis–Kahan/Kato gap denominator that typically causes circularity near collisions.

We then provide an Orlicz-controlled high-frequency leakage rate for the strain tail under the Orlicz function \Phi(s)=s^2\log(1+s^2), yielding an explicit decay envelope of order 1/\log(1+N^2) for frequency truncations. Using these ingredients, we certify a Fork Certificate D2 (gap-free, with an explicit bound) for a projector-gradient dispersion functional integrated against a dissipation clock.

A second contribution is a sharp negative theorem: an eigenvalue-only “enemy-mode” condition does not enforce positive vortex-stretching coercivity (Claim X). We give a kinematic counterexample at the sharp threshold \delta=1/\sqrt{6}, and state the minimal repair required (an explicit alignment hypothesis).

Finally, we define a soft collision metric \Gamma_\tau derived from \mathrm{tr}(P_\tau(S)^2) and show an algebraic coherence certificate: at the barrier value \delta=1/\sqrt{6}, the enemy-mode condition forces \lambda_1=\lambda_2 on the high-vorticity set, which yields an explicit positive lower bound \Gamma_\tau\ge \Gamma_{\min}(\tau)>0 and discharges the rewritten Fork Certificate D3 without any spectral-gap denominators.

This record is intended as a machine-auditable, finite endpoint package: it closes the collision-robust projector variation bottleneck, exposes the exact remaining open frontier (deriving alignment from dynamics), and provides full artifact manifests for reproducibility.