The surviving source algebra

Let $\Gamma$ be a finite named matter interface and $\varrho \in \mathcal{D}({\mathcal{H}}_{\Gamma })$. Registers declared classical are jointly block diagonal in their distinguished bases, including their correlations with other factors.

For destructive basis readout of a factor $S$, with complement $B$, the output channel is

The name $S$ is absent from the output interface. Forgetting $R$ gives ${\mathrm{Tr}}_S\varrho$, by summing the diagonal blocks.

Choose a unital $\ast$-subalgebra

on declared surviving names, with a fixed embedding into the complete current interface. Fix

The embedding is suppressed in formulas. Equivalently, the source input is the restricted state functional

The notation $j[f]$ permits a declared real linear combination of smearings, but only the resulting fixed finite matrix is used. Calling it a current is a declared interpretation, not a hypothesis needed to evaluate the trace.

A consumed factor, an inaccessible coherent carrier, spectator cardinality, and implementation history are not source variables. A produced record is eligible as a source only when its surviving register is explicitly named.