Regulatory Geometry and RG-Class Systems: Scale-Local Evolution, Localised Singularities, and Finite-Band Criteria / Global Regularity of Three-Dimensional Navier–Stokes Equations via Enstrophy Budget Exhaustion

Some things are right, some are wrong, but science loses things by only publishing the right - knowing why you're wrong is as helpful as knowing why you're right. This is a two-paper structure: A (rejected) paper on the concept of 'regulatory geometry' and 'RG-class' systems; a unified framework for nonlinear dissipative systems where the interplay between dissipation, nonlinear transfer, and forcing admits a robust geometric description in frequency space. This structure implies, using Navier-Stokes as an example, that any finite-time blow-up must be preceded by energy concentration in a finite band of low-to-intermediate shells, ruling out pure ultraviolet blowup. A second (unsubmitted) paper building on the system as applied specifically to Navier-Stokes; decomposing the Leray-Hopf solutions onto dyadic shells reveals a natural diffusive timescale and that sustained energy over such requires persistent nonlinear transfer from neighbouring scales. This requires paying an enstrophy budget that the finite dissipation budget implicit in the Leray energy inequality means cannot be paid ad infinitum.