Canonical nuisance-valuation rate on every fixed effective ball

Write $L_n=\log n$, $y_n=n^{2/9}$, $Y_n=\lfloor y_n\rfloor$, and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le Y_n\}$. Fix a finite nonempty head family $P_h$, physical intervals satisfying $1\le I_\sigma^{\rm lower}<I_\sigma^{\rm upper}\le U$ with $U\ge1$, guard ledgers, and a cutoff $W>1$ containing all head primes. For any radius $a\ge0$ there are a nonnegative rate $\epsilon_n$ and a threshold, with $\epsilon_n\log L_n\to0$, such that the following holds uniformly for every finite band set and canonical guard-deleted bridge with separated intervals and nonempty remaining cells. At any effective parameter $z$ with $\|z\|\le a$, write $\mu_z$ for the tilted law and $N_c$ for a nuisance statistic. Then eventually

The radius is chosen after the cutoff and is unrestricted; this rate is the marked nuisance input for Schur-complement estimates.