Proposition: A finite stochastic implementation

The joint law at the frozen instance can be generated using six independent fair bits and deterministic classical operations. Two bits generate $X$ uniformly. Two further independent pairs generate the two noise signs: flip coordinate $i$ exactly when both bits of its pair are one.

Proof. Each flip has probability $1/4$, and the flip events are independent of one another and of $X$. Thus the induced transition probabilities are $9/16$, $3/16$, or $1/16$, according to whether zero, one, or two coordinates differ. This is exactly (eq:fourmatrix). Each of the $64$ seeds is equiprobable, so the joint table has denominator $64$.