Proposition: Forty-shot limitation for the stated binomial conversion

With $40$ trials per setting and any positive one-sided allowances satisfying ${\sum }_{A,j}{a}_{Aj}^{+}\le {\alpha }_{\mathrm{s}}<1$, the generic binomial conversion cannot accept $\epsilon =0.1$, even at zero failures and zero allowances of the calibration and transfer types.

Proof. Fix one task and write its three statistical tail allowances as $a_j$. Their sum is at most ${\alpha }_{\mathrm{s}}$. The convexity of $a\mapsto 1-a^{1/40}$ gives

Bernoulli's inequality gives $0.99^{40}>1-40(0.01)=0.6>1/3$, so $3^{-1/40}<0.99$. Therefore $\frac{1}{2}{\sum }_j{u}_j>0.015>0.0025$, whereas generic acceptance at $0.1$ requires inferred infidelity at most $0.0025$. Nonzero failure counts only increase the upper binomial limits.