Every actual motivic stage on the empty scheme is zero

A sheaf of modules on the empty scheme has only its forced zero value on the empty open set. Thus all complexes and their derived images are equivalent to zero, and the full perfect category is equivalent to the zero stable category. It is idempotent complete. Its cofiber Grothendieck group has only the generator $[0]$, and the cofiber sequence $0\to 0\to 0$ imposes $[0]=[0]+[0]$, so this group is zero. Applying [F1] under Choice gives ${\pi }_{0}K(\mathrm{\varnothing })=0$.

By [F2], ${\pi }_0{F}^0=0$, so the unit $\eta :\mathbb{S}\to F^0$ is nullhomotopic. For any integer $r$, the unit law identifies the composite $F^r\simeq \mathbb{S}\wedge F^r\stackrel{\eta \wedge 1}{\to }{F}^0\wedge F^r\to F^r$ with $1_{F^r}$. The composite is nullhomotopic because smash preserves zero maps. Hence $1_{F^r}=0$ in the homotopy category, so the zero maps between $F^r$ and the zero spectrum are inverse there, and $F^r\simeq 0$. This holds for all integers $r$.