Theorem: Full exclusion from sphere and observation support

Suppose

and every initial null sphere from each positive-time point of $F$ lies in ${\mathrm{supp}}_{\mathrm{fun}}{B}_f$. If both actual orderings hold, then

Proof. Suppose $R\le t+s$.

If $t=0$, then $R\le s$ makes $x\in J^{-}(y)$. Since $x\in {\mathrm{supp}}_{\mathrm{fun}}{B}_f$ and $y\in {\mathrm{supp}}_{\mathrm{fun}}{W}_g$, $B_f\gtrsim W_g$ fails.

Let $t>0$. If $R<t-s$, then $y\in J^{-}(x)$, and $W_g\gtrsim B_f$ fails. Otherwise $t-s\le R\le t+s$. Lemma (eq:sphere-ball) supplies a point $z=(0,\mathbf{z})$ on the initial null sphere of $x$ with $|\mathbf{z}-\mathbf{y}|\le s$. Thus

and $B_f\gtrsim W_g$ fails.

The argument includes tangencies, $R=0$, $s=0$, and coincidence. Every case contradicts at least one ordering.