A First-Principles Anatomy of the Three-Dimensional Navier–Stokes Regularity Problem: Scaling Rigidity, the Two-Scale Obstruction, and the Double-Edged Role of Angular Momentum Conservation

Abstract We trace, from first principles, the structure of the obstruction to deciding global regularityversus finite-time singularity for the three-dimensional incompressible Navier–Stokes (NS)equations. The account is a synthesis and re-derivation of a chain of largely known results,organized so that each link exposes exactly where the next obstacle lies. We establish,in order: (i) viscosity enforces a parabolic scaling rigidity, admitting only the β = 12 exactself-similar mode, which is trivial at finite energy (Neˇcas–R˚uˇziˇcka–ˇSver´ak); (ii) in theaxisymmetric no-swirl setting, material conservation of ωθ/r is incompatible with smoothexact self-similarity, forcing H¨older C1,α profiles, as realized in Elgindi’s Euler singularity;(iii) in one-dimensional models the viscosity-versus-blowup competition collapses to a singlescaling exponent μ, with μ < 12 permitting viscosity-surviving self-similar blowup—realizedfor the dissipative generalized Constantin–Lax–Majda equation and reproduced here by adynamic-rescaling computation; (iv) in three dimensions with swirl, the intrinsic anisotropyof collapse conflicts with viscous parabolic pinning, generating a two-scale structure—thescaling instability—measured by an effective-dimension gap that vanishes with viscosity(Hou); (v) at exact dimension three the residual aspect-ratio instability is purely viscous inorigin; and (vi) the decisive structure is the double-edged conservation of angular momentumΓ = r uθ: its geometric 1/r2 amplification drives a closed, self-reinforcing blowup loop,while its maximum-principle boundedness furnishes the strongest known regularity handle.We conclude that pure NS sits on a genuine knife-edge that no simple internal mechanismdecides, and we argue that a definite external vorticity regulator—the role played by electromagneticself-regulation in the space-fluid program—is the natural resolvent in the physicalsetting. We do not claim a proof of regularity or of blowup; the contribution is a coherentfirst-principles map of the crux and its physical interpretation.