Explicit raw valuation-prefix covariance bound

Write $L_n=\log n$, $y_n=n^{2/9}$, $Y_n=\lfloor y_n\rfloor$, and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le Y_n\}$. For a head pattern $P$, let $S_P(A,C;n)$ be the positive $Y_n$-smooth integers in $(\lfloor An\rfloor,\lfloor Cn\rfloor]$ having exactly the valuations prescribed by $P$. Fix $H$, $0<A<C$ with $C>0$, and $W>1$ containing every head prime. There exist $K,G>0$ and $N_0$ such that for $n\ge N_0$, $\lfloor An\rfloor<k\le\lfloor Cn\rfloor$, $p\in\mathcal P_{n,W}$, and any nonempty $S=S_H(A,C;n)$ with uniform law $\mu$,

Here $s_W$ is the source prime-power cutoff scale (cutoffScale).

The estimate exposes the beyond-four-prime-power tail separately from the principal reciprocal row bound.