Survival factorizations and the price of random time

Anytime-valid inference relies on Ville's inequality and on optional stopping, both of which are stated for stopping times. In practice the time at which a result is reported is often chosen after seeing the data and is not a stopping time. We show that every random time $\tau$ on a discrete-time filtered probability space admits a canonical survival factorization $S_t=(1-F_t)M_t$ of its conditional survival probability into a hazard clock $F$ and a nonnegative martingale factor $M$, and that the dichotomy $M_t\equiv 1$ versus $M_t\neq 1$ is well posed for this canonical pair. In terms of the factorization, the e-process guarantee $\mathbb{E}[E_\tau]\le 1$ and optional stopping hold with an explicit correction at every random time; the correction is a functional of $M$, and it vanishes if and only if $M\equiv 1$. Ville's threshold guarantee holds at every random time, and only the expectation guarantee is lost. The expected anticipation debt $\mathbb{E}[\log M_\tau]$ of a reported time is a relative entropy of the joint law of the path and the reported time against the measure induced by its clock, is at most one nat, and is zero precisely on the pseudo-stopping class. Choosing the reported time from a class of $k$ after seeing the data has an exact value, at most $\log k$ discounted by the overlap of the class, positive even for a class of stopping times at each of which the e-process guarantee holds. On an exhaustively enumerated model the regime boundary is sharp, and Monte Carlo separates the stopping class by the conditional survival distribution.