Lossless separation for finite IID sources and finite DMCs

If $H(P_S)=0$, each nonzero term $-P_S(s)\ln P_S(s)$ is nonnegative, so exactly one source letter has probability one. Choose that letter $s_0$ and any channel input letter $x_0$, encode every source block as $x_0^{n_k}$, and decode every output to $s_0^k$. This is a deterministic joint code with zero error.

Now assume $0<H(P_S)<\rho C(W)$. Choose $\eta >0$ with $H(P_S)+2\eta <\rho C(W)$. By [F4], choose source codes with message counts $M_k$, source block error tending to zero, and ${\mathrm{lim\hspace{0.17em}sup}}_k{k}^{-1}\ln M_k\le H(P_S)+\eta$. Then ${\mathrm{lim\hspace{0.17em}sup}}_k{n}_k^{-1}\ln M_k\le (H(P_S)+\eta )\mathrm{/}\rho <C(W)$.

For any joint code, ${\widehat{S}}^k$ is a function of $Y^{n_k}$, so [F7] gives $I(S^k;{\widehat{S}}^k)\le I(S^k;Y^{n_k})$. The encoder is deterministic and the DMC output law given $S^k=s^k$ is $W^{n_k}(\cdot \mid f_k(s^k))$; averaging its finite conditional entropies by the value of $X^{n_k}=f_k(S^k)$ gives $H(Y^{n_k}\mid S^k)=H(Y^{n_k}\mid X^{n_k})$. By [F8], $I(S^k;Y^{n_k})=I(X^{n_k};Y^{n_k})$, which is at most $n_{k}C(W)$ by [F2] and [F5]. If $p_k=\mathrm{Pr}[{\widehat{S}}^k\mathrm{\ne }{S}^k]$, then [F6] and [F8] give $kH(P_S)\le n_{k}C(W)+\ln 2+p_{k}k\ln \mathrm{\mid }\mathcal{S}\mathrm{\mid }$. Dividing by $k$ and taking $k\to \mathrm{\infty }$ along a reliable code sequence proves $H(P_S)\le \rho C(W)$.

Choose a rate $R_c$ strictly between $(H(P_S)+\eta )\mathrm{/}\rho$ and $C(W)$. By [F3] and the supremum property, some maximal-error achievable rate $R^{\mathrm{\prime }}>R_c$ exists, so its channel-code sequence has ${\mathrm{lim\hspace{0.17em}inf}}_n{n}^{-1}\ln L_n\ge R^{\mathrm{\prime }}$ and maximal errors tending to zero. Since $n_k\to \mathrm{\infty }$, eventually $\ln L_{n_k}\mathrm{/}{n}_k>R_c>\ln M_k\mathrm{/}{n}_k$. Restrict each such code to $M_k$ messages; mapping any discarded decoder output to a retained message preserves every retained message's former correct-decoding event, so maximal error does not increase.

Compose the source encoder and decoder from step 1.2 with the restricted channel code from step 2.1. The source-code index can have any distribution, but its channel decoding error is at most the maximal error. The joint block-error event is contained in the union of the source-code error event and the channel message-error event, so its probability is at most the sum of those two probabilities and tends to zero.

Step 1.1 gives zero-error achievability when the source entropy is zero; steps 1.2, 2.1, and 3.1 give reliable achievability under the strict positive-entropy threshold; and step 1.3 gives the necessary condition. Thus the stated strict-threshold conclusions hold, with no claim at the positive-entropy equality boundary.