Proposition: An unknown logical Pauli is a separate obstruction

Suppose the observer is given radiation from one of the four channels

but no information identifying $P$. For a single decoder required to work uniformly,

This remains true with unlimited laboratory memory.

Proof. Averaging the four input Paulis gives $\mathcal{T}$. The averaged decoded channel is constant, so its error is at least $3/4$ by Lemma lem:constant. Convexity of the diamond norm bounds that averaged error above by the maximum of the four individual errors. Ignoring radiation and preparing $I_2/2$ attains $3/4$.