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 something by only pretending everything submitted was right to begin with. Knowing why you're wrong can be as important as knowing why you're right. This is a two-paper structure: A rejected paper on the 'RG-class' system; a unified framework for dissipative nonlinear equations in which the interplay between dissipation, nonlinear transfer, and forcing admits a robust geometric description in frequency space. This structure implies 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 blow-up scenarios. A second (unsubmitted) paper then built on the rejected RG-class paper with a conditional result of regularity for the Navier-Stokes equations in the Leray-Hopf framework; decomposing the velocity field onto dyadic shells a natural diffusive time scale emerges that persists to sustain nonlinear transfer across shells. This incurs an enstrophy cost which cannot be paid by the finite budget implied by the Leray energy inequality.