= Three-input two-point additive-noise channel
Let the input <alphabet> be $\{0,1,2\}$ and let the output be $Y=X+Z$, where <independent> noise is uniform on two distinct integers. Translating the output <alphabet> reduces the noise to $\{0,d\}$, $d\geq1$. The reachable output <alphabet> has size $3+\min(d,3)$: it is the union of two three-element intervals shifted by $d$.
For $d=1$, every row of the <transition matrix> has conditional <information entropy> one <bit>, so $I(X;Y)\leq\log_2 4-1=1$. Input <probabilities> $(1/2,0,1/2)$ make the output uniform and attain one <bit> per use. For $d\geq3$, the two translated input <alphabets> are disjoint and every output identifies its input. A uniform input then attains $\log_2 3$ <bits> per use.
For $d=2$, <reflection> <symmetry> and <concavity of information entropy> permit restricting to input <probabilities> $(s/2,1-s,s/2)$. Conditional on output <parity>, outputs whose labels are <even integers> have <probabilities> $(1/4,1/2,1/4)$ and outputs whose labels are <odd integers> have <probabilities> $(1/2,1/2)$. Consequently $I(X;Y)=h_2(s)+s/2$. Its strictly negative <second derivative> and stationary equation $\log_2((1-s)/s)+1/2=0$ give $s=\sqrt2/(1+\sqrt2)$ and <channel capacity> $\log_2(1+\sqrt2)$. All three values follow from overlap of translated input <alphabets>, rather than from output <alphabet> size alone.
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