Part I of the "Boundary Rigidity in Fluid Dynamics" – Global Regularity for Band-Limited Navier–Stokes Equations

Part I of the “Boundary Rigidity in Fluid Dynamics” seriesGlobal Regularity for Band-Limited Navier–Stokes Equations This short technical note establishes global-in-time regularity for a spectrally truncated (band-limited) version of the three-dimensional incompressible Navier–Stokes equations. When the velocity field is restricted to a fixed Fourier band, the system reduces to a finite-dimensional Galerkin model. Standard energy estimates together with the Bernstein inequality yield uniform enstrophy bounds, excluding finite-time blow-up. The result is elementary and does not address the full Navier–Stokes regularity problem. Its purpose is structural and pedagogical: to make explicit that any potential singularity mechanism in the unrestricted equations must involve access to arbitrarily high frequencies. This paper provides the foundational example for the broader “Boundary Rigidity” framework developed in subsequent parts of the series. Series links: Part III of the "Boundary Rigidity in Fluid Dynamics" Series - Boundary-induced regularity note: 10.5281/zenodo.18842688 Part III of the "Boundary Rigidity in Fluid Dynamics" Series - Constraint-Dependent Phase Randomization and Windowing Sensitivity in Spectral Flux Estimates: 10.5281/zenodo/18842852