Proposition: Forty shots cannot pass this Hoeffding certificate

For one label, $n_x=n_y=n_z=40$, and $0<{\alpha }_{\mathrm{s}}<1$, the certificate of Theorem thm:main cannot accept the threshold $\epsilon =0.1$, even with zero calibration and transfer allowances and the best possible empirical score.

Proof. Because scores lie in $[-1,1]$, $\hat{F}\le 1$. Here

Thus the best possible lower fidelity is $F_L=\max \{0,1-R\}$, and the smallest possible upper error is

Since $\log (6/{\alpha }_{\mathrm{s}})>\log 6>1$, this number is strictly greater than $0.1$. Nonnegative allowances only weaken acceptance.