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Note: See the edit at the bottom for an argument showing that there is an unbiased algorithm which has variance strictly lower than $1/12$ for all $x \in [0,1]$.
We can at least prove that if $x$ is chosen uniformly from $[0,1]$, then the average variance must be at least $\pi^2/64 - 1/12$. There is a dithering algorithm that achieves this average-case ...
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