The forty-shot limitation, with its exact scope

An exact binomial procedure can be sharper without changing this small-error limitation for its specified generic conversion. For binary oriented scores, let failure probabilities $p_j$ be defined so that

For one-sided confidence allowances ${\alpha }_j>0$ summing to ${\alpha }_{\mathrm{s}}$, a zero-failure upper bound is

For other failure counts use the usual upper binomial inversion. The resulting certified infidelity is at most $\frac{1}{2}{\sum }_j{u}_j$, with those inversions used for the observed counts.