= Solution
Keep $y$ fixed. Integral bounds for the decreasing function $1/t$ give $H_M=\log M+O(1)$, so $H_M-H_{y-1}=\log M+O_y(1)$. Consequently
$$
\boxed{\mathbb E[N\mid y,M]\frac{\log M}{M}
=\frac{M-y+1}{M}\frac{\log M}{H_M-H_{y-1}}\longrightarrow1.}
$$
The <posterior mean> thus grows like $M/\log M$ instead of approaching a data-determined limit. Even the <posterior distribution> mass below a fixed endpoint $n$ is $(H_n-H_{y-1})/(H_M-H_{y-1})\to0$. Hence increasing the supposedly innocuous prior cutoff moves the inference to larger and larger endpoints. In an integral approximation its median is of order $\sqrt{yM}$, also depending strongly on the cutoff. This explains the sensitivity of the entire inference, not just of its <posterior mean>.
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