Lemma: Finite futures of the frozen component order

For every datum of ${\mathcal{P}}_{3q}$, the component order is a partial order. Each experiment contributes at most three components, and

Proof. Mutual reachability is an equivalence relation. Reachability between its equivalence classes is well defined: changing either representative adds paths within a component. It is reflexive and transitive, and mutual comparison of two classes means that they are the same class.

Every path remains within its experiment identifier. The corresponding directed graph has three vertices, so its component quotient has at most three elements. The future of a component is contained in that quotient.