Corollary: Zero-failure feasibility

For the acceptance-only binomial procedure with equal counts $n$, equal allowances $a={\alpha }_{\mathrm{s}}/(3K)$, zero calibration and transfer, and no observed failures, generic acceptance at $0<\epsilon <1$ is possible exactly when

This scales as $6{\epsilon }^{-2}\log (3K/{\alpha }_{\mathrm{s}})$ for small $\epsilon$.

Proof. At zero failures, $u=1-a^{1/n}$. The inferred infidelity is $3u/2$. The generic channel upper bound is at most $\epsilon$ exactly when $u\le {\epsilon }^2/6$, which rearranges to (eq:binomsamples). The asymptotic follows from $-\log (1-z)\sim z$.