= Finite-jump boundary crossing identity
{title2=$\mathbb P(1-S+X_t>t\text{ for all }t<1)=1-S$}
For the <uniform-arrival residual mass martingale> with total weight $0<S<1$, the first crossing of level one happens continuously: its only jumps are downward, and an arrival almost surely misses the finitely many deterministic possible crossing times. Stop at that crossing or at $r<1$. The stopped values lie between zero and one, and their limit as $r\uparrow1$ is the crossing indicator. <Optional stopping> and <dominated convergence> identify its <probability> as $S$. At $S=1$ the strict boundary inequality already fails at time zero.
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