Power under a declared noise model

For three independent binomial settings with true error probabilities $q_i$, fixed $n_i$, and a predeclared acceptance rule, its acceptance probability is exactly

This expression is a reproducible prospective power calculation. A go/no-go design may require this probability to exceed a stated value, for example $0.9$, under a stated design channel. It does not prove that the apparatus has that channel.

Under an independently supported Pauli-channel hypothesis, Proposition prop:pauli gives the stronger conversion $d_{\diamond }=1-f$. That may greatly improve shot requirements. The hypothesis cannot be manufactured from a favorable identity overlap alone.