Lemma: Isolated residual

If $\lambda$ is simple, every other eigenvalue is at least $\Delta >0$ from $\mu$, and normalized $v$ has residual $r=\Vert (H-\mu I)v\Vert <\Delta$, then

after optimal phase alignment.

Proof. Expand in the eigenbasis. Off-target squared norm is at most $r^2/{\Delta }^2$. For off-target norm $s$, aligned distance squared is $2(1-\sqrt{1-s^2})\le 2s^2$.