Exact initial-surface support

For this realization, the surface-ball alternative is decided exactly by (eq:exact-surface). A ball belongs to functional support precisely when it is contained in ${\mathcal{S}}_F$.

If a point ${\mathbf{z}}_0$ of a proposed ball is outside ${\mathcal{S}}_F$, the continuous function

satisfies $d_F({\mathbf{z}}_0)>0$. Choose $0<\epsilon <d_F({\mathbf{z}}_0)/4$. For $|\mathbf{z}-{\mathbf{z}}_0|<\epsilon$ and $0\le v<\epsilon$, no backward null cone from $F$ meets $(v,\mathbf{z})$. All advanced sources, their local bulk derivatives, and their surface traces vanish there, for every smooth scalar configuration.

This is an explicit missed open region. Its cause is absence of propagation, not cancellation of a nonzero delta by the bulk. The alternative's reference to a “cancelling configuration class” must therefore allow this zero-propagation case.