Corollary: Bridge for the flat realization

For the explicitly specified flat realization, (SB) holds for all retained first-stage pairs.

Proof. The preceding support results supply all hypotheses of Theorem thm:geometric. If $F,G$ are nonempty, compactness and (eq:full-exclusion) give a positive uniform margin

Thus $R$ is bounded away from zero and both direct-cone deltas are disjoint from the observation support.

IX-E-2's Fourier-defined kernel vanishes in the open region $R>t+s$; its common-regulator proof also supplies smooth time dependence with values in spatial distributions. The margin gives the same zero restriction at any initial-time boundary met by these smooth tests. There is no additional apex or boundary contact to assign. Hence $fg\mathrm{Im}K=0$. If either test is zero, the conclusion is immediate.