Singular Forces on the Whole Space: Sobolev Density Thresholds and Energy Approximation

We derive whole-space distribution results from the established compact, smoothly forced Navier–Stokes blowup construction of OpenAI. For every fixed positive viscosity and deadline, smooth forces causing classical breakdown from rest on ${\mathbb{R}}^3$ are dense in the relative $L_t^1{H}_x^s$ topology exactly when $s<1\mathrm{/}2$, and in the relative $L_t^2{H}_x^s$ topology exactly when $s<-1\mathrm{/}2$. The positive results preserve every fixed initial velocity in $H_{\sigma }^{\mathrm{\infty }}({\mathbb{R}}^3)$. They follow from a compact divergence-free removal of a regular background, followed by exact insertion of a single whole-space singular packet. Separate critical estimates prove non-density; low spatial frequencies are controlled explicitly, without a periodic mean or a Poincaré inequality. The conclusions hold for spacetime-compact forces and for the rapidly decaying data class in the Clay formulation. We also obtain density in the usual completed energy-force spaces, strong energy-and-dissipation approximation of regular trajectories, and exact equality of cell observations on a prescribed finite family of whole-space grids. Initial-state spaces, forcing spaces and numerical error regularity are distinguished throughout. No claim of fixed-force instability, loss of weak existence or new formal verification is made.