Proposition: Simultaneous binomial certificate

Choose positive tail allowances $a_{Aj}^{+},a_{Aj}^{-}$ whose sum over all tasks, settings, and tails is at most ${\alpha }_{\mathrm{s}}$. Write $u_{Aj},{\ell }_{Aj}$ for the corresponding limits. Then

can replace the fidelity endpoints in Theorem thm:main, with the same simultaneous guarantee.

For an acceptance-only procedure, upper limits alone suffice, with ${\sum }_{A,j}{a}_{Aj}^{+}\le {\alpha }_{\mathrm{s}}$.

Proof. Within a setting, the failure count is binomial under the conditional contract. At a fixed true $p$, the event that its upper limit is below $p$ is contained in a lower-tail event of probability at most $a^{+}$. The lower-limit failure is similarly an upper-tail event of probability at most $a^{-}$. A union bound gives all probability intervals simultaneously. Equation (eq:failureidentity) and the calibration allowance give the fidelity intervals. The remaining channel conversion and transfer steps are those already proved.