Centered divisibility-prefix bound on a raw structured cell

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 a pattern $P$ and $0<A<C$. Uniformly for integers $k,d$ with $\lfloor An\rfloor<k\le\lfloor Cn\rfloor$, $0<d\le Y_n^4$, all prime factors of $d$ at most $Y_n$, and $\gcd(d,\operatorname{modulus}(P))=1$, there exist $K>0,N_0$ such that for $n\ge N_0$, writing $\mathbb E_S$ for the source uniform average on $S=S_P(A,C;n)$,

The centered estimate preserves the marked modulus factor needed when summing prime-power contributions.