Corollary: History-conditional battery certificate

Before assessment block $k$, condition on the complete history, including current checkpoints and chosen probe laws. Suppose the next example blocks are conditionally fresh and independent and the checkpoints are fixed during the block. Allocate predictable error budgets ${\delta }_{k,j}$ with

almost surely. The corresponding Hoeffding radii hold simultaneously across the trajectory with probability at least $1-\delta$.

Proof. Apply the conditional-block argument of Psy03-B thm:conditional, using the conditional bounded-variable discipline in Psy02-B thm:sampling CheungPsy03B,CheungPsy02B. Average each conditional failure probability and sum them.