Unbiased rational Bernoulli sampling by bit rejection

Input integers $n\ge 1$ and $0\le m\le n$, a stream of independent fair bits as in Disjoint streams of independent fair bits produce independent uniform variables, and an optional positive integer attempt cap $M_{\max }$ (or $M_{\max }=\mathrm{\infty }$). If $m=0$, return $0$ without reading bits; if $m=n$, return $1$ without reading bits. Otherwise let $k$ be the least nonnegative integer with $N=2^k\ge n$. For each attempt, read a fresh non-overlapping block of $k$ bits and interpret it as a uniform integer $R\in \{0,\dots ,N-1\}$. If $R\ge n$, reject the block and start another attempt. If $R<n$, return $1$ when $R<m$ and $0$ otherwise. If a finite cap is reached before an accepted block, return status inconclusive and no Bernoulli value.

The finite-cap branch is not a sample with a substituted default value. Its success law and failure probability are proved separately. The algorithm assumes independent fair-bit blocks; an LCG or keyed deterministic stream does not satisfy that premise by definition.