= Solution
For nonnegative integers $n$, assume the storage channel over successive intervals of length $\tau$ composes as $D_\epsilon^n$. This is the memoryless interpretation of the specified storage law. In the <computational basis>, write the <density operator> as
$$
\rho=\begin{pmatrix}u&z\\\overline z&1-u\end{pmatrix}.
$$
Conjugation by the <Pauli Z gate> reverses the off-diagonal entries and leaves the diagonal entries unchanged, so
$$
D_\epsilon(\rho)=\begin{pmatrix}u&(1-2\epsilon)z\\(1-2\epsilon)\overline z&1-u\end{pmatrix}.
$$
After $n$ applications of the <phase-flip channel>, the multiplier is $(1-2\epsilon)^n$. Comparing with the multiplier $1-2\epsilon_n$ of a single <phase-flip channel> gives
$$
\boxed{D_\epsilon^n=D_{\epsilon_n},\qquad\epsilon_n=\frac{1-(1-2\epsilon)^n}{2}.}
$$
For $n=0$ this gives the identity channel. Equivalently, composing two consecutive phase errors cancels them because $Z^2=I$, so their effective probability obeys $\epsilon_{n+1}=\epsilon+(1-2\epsilon)\epsilon_n$. The same expression solves this recurrence. The <iterated phase-flip channel> formula presumes this composition law; specifying a channel at one time alone would not constrain a memory-bearing environment at later times.
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