Proposition: Assembly for fixed classical sector weights

Suppose a prepared interface is

with input-independent probabilities $p_{\alpha }$, and orthogonal sector supports in both accessible and residual outputs. If each sector satisfies

and has a specified decoder with error $e_{\alpha }$, then

where $\mathcal{D}$ is the sector-controlled decoder.

Proof. Orthogonality in the traced factor removes cross-sector terms. The channels are orthogonal direct sums with weights $p_{\alpha }$. Use $\tau ={⨁}_{\alpha }{p}_{\alpha }{\tau }_{\alpha }$, the triangle inequality on referenced outputs, and block-controlled decoding.