Moving prefixes have uniform sharp full-valuation profiles

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 pattern $P$, $0<A<C$, $r_{\min}>0$, and $W>1$ containing all its head primes. There exist $C_v>0,N_0$ such that for $n\ge N_0$, every endpoint $\lfloor An\rfloor<k\le\lfloor Cn\rfloor$ with physical prefix fraction at least $r_{\min}$, and every $p\in\mathcal P_{n,W}$, the prefix structured cell $S$ is nonempty and

This uses the same truncated Dickman main term as a fixed cell, uniformly over the moving upper endpoint. The physical prefix fraction means exactly $(k/n-A)/(C-A)$.