Theorem: Shared-amplitude nuisance overlap

Let $b_0,b_1\in [b_{-},b_{+}]$, $a\ge 0$, and suppose at every stage, setting, and history

with common apparatus. If $a\le b_{+}-b_{-}$, a null–alternative pair has identical complete records. No sample size distinguishes that pair.

Proof. Choose $b_1=b_{-}$, $b_0=b_{-}+a$. All conditional generators and instruments agree. Apply Theorem thm:process.