Theorem: Shared-generator identification obstruction

Suppose two mechanism families have, after every allowed history and setting,

where $b_0,b_1\in [b_{-},b_{+}]$, $c\ge 0$, and apparatus and initial accessible states are common. If $c\le b_{+}-b_{-}$, the families contain a pair with identical complete records.

Proof. Choose $b_1=b_{-}$ and $b_0=b_{-}+c$. The conditional generators agree everywhere in the experiment. Theorem thm:process applies.