Lei proposed an algorithm Algorithm ${\mathcal{A}}_3$ in 2023 to generate an exact discrete uniform distribution from an unknown biased Bernoulli source. The present paper does not claim a new extraction algorithm. Its contributions are analytical: first, we provide a Fourier-analytic proof of the uniformity mechanism based on roots of unity and coefficient extraction; second, we derive explicit acceptance-probability and expected-runtime bounds, with a rigorous treatment of composite moduli; third, we show how independent accepted ${\mathcal{A}}_3$ digits yield a continuous $\mathrm{Uniform}(0,1)$ limit through a base-n expansion and quantify the cost of finite-digit simulation.