Lemma: Deterministic-table reduction

Every finite-alphabet law of the form (eq:classicalfactor) is a mixture of deterministic encoding and decoding tables with mixture randomness independent of $X,J$.

Proof. Condition on $\lambda$. Before the fresh input is used, sample an encoding-table entry for every possible $x$ according to $a_{\lambda }(\cdot \mid x)$, and a decoding-table entry for every possible $(m,j)$ according to $d_{\lambda }(\cdot \mid m,j)$. Use independent table samples conditional on $\lambda$.

Each table realization is deterministic. Averaging its entries reproduces the two kernels and their composition. Average also over $\lambda$.