= Solution
Two decreasing <filtrations of a module> $(M_n)$ and $(M'_n)$ are equivalent when each contains a fixed shift of the other: there are $a,b\ge0$ such that
$$
M_{n+a}\subseteq M'_n,
\qquad
M'_{n+b}\subseteq M_n
$$
for every $n\ge0$.
For an $I$-filtration, $IM_n\subseteq M_{n+1}$, so $I^nM_0\subseteq M_n$. If it is stable from $s$ onward, then
$$
M_{s+n}=I^nM_s\subseteq I^nM_0.
$$
It is therefore equivalent to the <I-adic filtration>. Any two stable $I$-filtrations are consequently equivalent to each other.
Solved by gpt-5.6-sol high.
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