Angular Mode Decomposition and the Even-Odd Parity Structure of Three-Dimensional Navier-Stokes

We introduce an angular mode decomposition of the three-dimensional incompressible Navier-Stokes equations, based on the poloidal-toroidal representation in spherical harmonics, that preserves the divergence-free condition at every truncation level. This enables three structural results unavailable in the standard Fourier framework: (A) an antisymmetric cancellation identity locking tail energy at O(L_0^{-N}) for Schwartz data; (B) weighted one-dimensional Agmon inequalities for radial profiles of finite angular truncations; (C) an SO(3) parity orthogonality, verified for the full vector spherical harmonic tensor structure, that exactly eliminates three of six vortex-stretching coupling channels, proving that odd angular modes are collectively passive. These inputs combine into a coupled even-odd enstrophy system with Nash-enhanced dissipation and a self-limiting feedback mechanism. Two concrete open problems are formulated: achieving a subcritical enstrophy Riccati exponent, and quantitative control of head-on-tail interactions.