Proposition: Order and finite futures

The relation ${⪯}_D$ is a well-defined partial order. Its carrier is countably infinite. Each experiment contributes between one and three components, and

Every chain has at most three distinct components.

Proof. Reflexive reachability is transitive, and mutual reachability is an equivalence relation. Changing representatives only adds paths within the relevant components, so the quotient order is well defined. Reflexivity and transitivity descend. Comparison in both directions puts two representatives in the same component, giving antisymmetry.

A path stays within one identifier. Each three-vertex graph has between one and three components, so there are countably many nonempty finite blocks. Every future and chain lies in one block.