Proposition: Solid-ball lower support is false

There are nonzero $f$ for which a relatively open initial collar inside a backward causal ball is disjoint from ${\mathrm{supp}}_{\mathrm{fun}}{B_f}^{\mathrm{fl}}$.

Proof. Choose a nonzero bump positive at $(1,0)$, with

Use the relative collar

For $(t,\mathbf{x})\in F$, $(v,\mathbf{z})\in U_0$,

There is no backward null incidence, so $U_0\cap N_F$ is empty. All first derivative kernels vanish there, including the surface contribution. Yet the time-zero ball from $(1,0)$ contains $B(0,1/4)$. A translation and positive rescaling places the same diagnostic in a sufficiently large retained chart.