Lossless Wyner-Ziv endpoint is the Slepian-Wolf rate

Any feasible test channel has nonnegative Hamming distortion with expectation zero, so $X=\widehat{X}=g(Y,U)$ almost surely. Every positive-mass conditional law of $X$ given $(U,Y)$ is therefore a point mass; [F4] and [F7] give $H(X\mathrm{\mid }U,Y)=0$, and [F2] gives $I(X;U\mathrm{\mid }Y)=H(X\mathrm{\mid }Y)$. The admissible choice $U=X$, $g(y,u)=u$ has zero distortion and the same objective, so the Wyner-Ziv minimum is exactly $H(X\mathrm{\mid }Y)$.

In the two-source Slepian-Wolf region of [F3], the subset $\{X\}$ requires $R_X\ge H(X\mathrm{\mid }Y)$. The rate pair $(R_X,R_Y)=(H(X\mathrm{\mid }Y),H(Y))$ satisfies the other two inequalities because $H(Y)\ge H(Y\mathrm{\mid }X)$ follows from [F5] and the first identity in [F6], while $H(X\mathrm{\mid }Y)+H(Y)=H(X,Y)$ is the second identity in [F6]. These are the operational rates of the fixed-length code convention in [F8], so the minimum X-encoder rate with the Y-encoder sending its lossless sequence is $H(X\mathrm{\mid }Y)$, matching step 1.1.