Lemma 6.2

Lemma 6.2 (One-state extension dichotomy). Suppose two such fresh roots have the same behavior.

1. If the behavior is not already a state of $Q$, their missing and $\mathsf{SELF}$ positions agree, and their old target agrees at every other port.

2. If the behavior is an old state $q\in Q$, each row is a guarded unroll of $q$: it copies every missing and strict-successor port, while a self-loop of $q$ may target either $q$ or the fresh variable $X$.