Density of Forces Producing Navier-Stokes Blowup

Abstract

Starting from the compact, smoothly forced Navier--Stokes blowup solution constructed by OpenAI, we study the distribution of the singular data it generates. For fixed viscosity and time horizon on the three-dimensional torus, smooth forces producing classical breakdown from rest by that time are dense in the inherited time-integrated spatial $H^s$ topology if and only if $s<1/2$ (the norms are defined in Section~\ref{sec:intro}). The positive result follows from an exact insertion into every regular reference trajectory. A localized vector potential removes the background around a concentrated singular packet, so all nonlinear cross terms vanish and the modified force remains smooth through the singular time. We give the full support construction, derivative estimates, fractional Sobolev scaling and classical-lifespan argument. The inserted trajectories converge strongly in the energy and dissipation norm. A separate critical-force bootstrap, followed by an $H^1$ estimate and high-Sobolev continuation, supplies the regular open set needed for non-density at and above $s=1/2$. Further results describe extended-data projections, every fixed smooth initial-velocity slice, mixed force norms, infinite-dimensional variations and interior no-slip realizations. The construction varies the force; it does not classify singular initial velocities for a single prescribed force.