Exact Minimax One-Bit Unbiased Compression: Heavy-Tail Necessity and Finite-Randomness Approximation

A pointwise-unbiased one-bit compressor reconstructs every real input in expectation while transmitting one bit. For a scalar source $P$ with CDF $F$, mean $m$, and $\mathcal J(P)=\int_{\mathbb R}\sqrt{F(r)(1-F(r))}\,dr$, we prove that the infimum of the source-averaged reconstruction second moment