Cutoff-before-mesh reference sharp inverse

Write $L_n=\log n$, $y_n=n^{2/9}$, $Y_n=\lfloor y_n\rfloor$, and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le Y_n\}$. There exist a mesh tolerance $\tau>0$, an inverse constant $C_{\rm ref}>0$, and a cutoff $W_0$. For every $W\ge W_0$, every finite head set, and every regular relative mesh $M$ with $\delta>0$ and $\delta+M.\mathrm{ratio}\le\tau$, choose a nonempty set of positive cells with lower endpoints greater than $1/8$, upper endpoints at most $7/8$, and half their total width at least $1/8$. For all sufficiently large $n$, every bridge with sample size $n$, cutoff $W$, and the canonical partition of this mesh has an interval certificate and an invertible projected arithmetic reference sharp operator $T$ on its sharp gauge space satisfying

The cutoff and inverse constant precede the mesh quantifiers, so the bound is independent of the number of cells and the moving low-band center.