Single-observation location minimax bound (${\inf }_T{\sup }_a\mathbb{E}(T(a+ϵ)-a{)}^2=\mathrm{Var}(ϵ)$)

For a known mean-zero noise law with finite variance ${\sigma }^2$, observing only $Y=a+ϵ$ with unrestricted $a\in \mathbb{R}$ has minimax risk ${\sigma }^2$ under squared-error loss. The observation itself attains this risk. For the lower bound, use a flat prior distribution on $[-A,A]$ and condition an easier experiment on $|ϵ|\le M$. Away from the prior endpoints, the Bayesian posterior noise has its original truncated law. Let $A\to \infty$ and then $M\to \infty$ in the resulting Bayes risk lower bound. No normal distribution or density for the noise is required.