Lemma: Uniform rule marginals

Conditional on any admitted switch schedule,

for every $t,r$. The complete rule process is independent of the point sequence.

Proof. At a switch the transition matrix has zero diagonal and off-diagonal entries $1/3$. It is doubly stochastic and preserves the uniform distribution. At other steps the transition is the identity. Induct from the uniform initial rule. All variables constructing the rule process are independent of the point-generating variables.