A fixed raw structured cell has the sharp full-valuation profile

Write $L_n=\log n$, $s_n=n/\log n$, $y_n=\lfloor n^{2/9}\rfloor$, and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le y_n\}$. For a prime $p\in\mathcal P_{n,W}$ put $k_p=\lfloor\log_p(y_n^4)\rfloor$ and $V_n(p)=\sum_{j=1}^{k_p}M_n(p^j)$, where $M_n(D)$ is the paper Dickman divisibility main term. Fix a head pattern $P$, real $0<A<C$, and $W>1$ containing all head primes. There exist $C_v>0,N_0$ such that for every $n\ge N_0$ and $p\in\mathcal P_{n,W}$, the structured cell $S=\operatorname{structuredCell}(P,\lfloor An\rfloor,\lfloor Cn\rfloor,y_n)$ is nonempty and

Nonemptiness is part of the conclusion, and the estimate concerns the genuine full valuation.