An explicit missed solid-ball region

Let $f$ be a nonzero smooth bump, positive at $(1,0)$, with

Consider the relative open collar

For every $(t,\mathbf{x})\in F$ and $(v,\mathbf{z})\in U$,

No backward null relation is possible. Therefore $b=0$ throughout $U$ for every source $-f\mathrm{d}\varphi$, and the bulk and surface derivative terms both vanish there. The scalar derivative is supported in $F$, disjoint from $U$. Consequently

The time-zero ball associated with $(1,0)$ contains $B(0,1/4)$. Thus the proposed universal solid-ball lower-support assertion is false for the flat realization.

A standard wave support theorem does not repair this assertion: the sphere-versus-ball distinction is exactly the relevant three-dimensional propagation distinction Ivrii.