M22c The Kernel Bound, Euler Flows, and Kolmogorov Scaling in Law Space - Closing the 2D Gap, Isolating the 3D Obstruction, and the Inviscid Law-Morphing Programme

This monograph continues the law‑morphing programme for fluid dynamics developed in M22a–b, with three main advances in the analysis of Navier–Stokes and Euler flows in law space. (I) The 2D kernel bound is proved.In two dimensions, the vortex‑stretching term (w . V)w vanishes identically. The pressure source reduces to a form controlled by the incompressibility isomorphism, and the morphed pressure kernel is shown to localize. As a result, the kernel bound |w . Vp| <= K Fo Kw is proved in 2D, yielding global regularity in the law‑morphing framework as a corollary. (II) The 3D obstruction is isolated.In three dimensions, the algebraic trace of vortex stretching introduces a genuinely new pressure source. This contribution is identified as the unique obstruction to closing the kernel bound globally. A symmetric stretching tensor Sw is introduced, and the 3D problem is reduced to a single analytic inequality controlling the pressure generated by ||Sw||^2 along the law trajectory. The kernel bound is proved in the turbulent sector theta > pi/2, while the laminar sector remains the sole open regime. (III) Euler flows and scaling laws in law space.The inviscid case v = 0 is analysed. Without viscous damping, the law‑morphing self‑regulation mechanism fails: curvature growth is not offset by any dissipative term, providing a precise law‑space explanation for possible Euler blow‑up. Onsager’s conjecture is reformulated geometrically: the critical Hölder exponent alpha = 1/3 corresponds to the Manifold sector theta_c = 2 arccos(1/3), the tetrahedral angle. Kolmogorov’s 4/5 law emerges as a native rank‑3 identity, with the 1/3 exponent arising naturally in log‑amplitude coordinates. Taken together, these results close the 2D case, identify the exact 3D obstruction, and unify Navier–Stokes regularity, Euler blow‑up, Onsager’s threshold, and Kolmogorov scaling within a single geometric framework. The laminar‑sector kernel bound is isolated as the remaining analytic summit.