Finite-jump boundary crossing identity ($\mathbb{P}(1-S+X_t>t\text{ for all }t<1)=1-S$)

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 $r\uparrow 1$ 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.