Proposition: Faithful scalar coding preserves the descent question

Let $q:Z\to Q$ be a quotient map and $F:Z\to \mathbb{R}$. Put

The associated binary law factors through $q$ if and only if $F$ does.

The same statement holds for a finite family selected by a retained clock code, when equality is required on all clock basis inputs.

Proof. If $F$ is constant on quotient fibers, so are the probabilities. Conversely equality of the two probabilities on a fiber implies equality of $b(F)$. The real function $b$ is injective, so $F$ is equal on that fiber.

For a retained clock code, prepare each clock basis state in turn. This isolates the corresponding binary law.