Proposition: Setting-dependent drift defeats naive pooling

If the $x,y,z$ blocks test ${\mathrm{Ad}}_X,{\mathrm{Ad}}_Y,{\mathrm{Ad}}_Z$, respectively, all oriented parities pass with certainty, so naive pooling gives $\hat{F}=1$. Each channel nevertheless has error one.

Proof. Each Pauli unitary preserves the correlation in its own axis. The observed signs are $1,-1,1$. Every nonidentity Pauli unitary sends a Bell state to an orthogonal Bell state. There is no common tested channel across the blocks.