Balanced raw correction densities have the uniform squared-logarithmic rate

Write $L_n=\log n$, $s_n=n/\log n$, $y_n=\lfloor n^{2/9}\rfloor$, and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le y_n\}$. Fix natural numbers $W,K_0$, real $\beta$, and $c,\delta_*>0$. Let $h_n$ be the upper-tail length associated to $c$, and choose the canonical balanced coefficient $\alpha_n$ for width $W$, multiplicity $K_0+1$, coefficient $\beta$, and logarithmic scale $L_n$. Then, for all sufficiently large $n$ and every attained active nonexceptional rough label $\ell$,

Here $C_{\rm raw}$ is the source's explicit uniform raw-row correction density constant, independent of $n$ and $\ell$. The conclusion is a simultaneous bound over all active correction rows.