Moving prefixes have sharp normalized Dickman divisibility profiles

Write $L_n=\log n$, $s_n=n/\log n$, $y_n=\lfloor n^{2/9}\rfloor$, and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le y_n\}$. Fix a head pattern $P$, $0<A<C$, and $r_{\min}>0$. There exist $K>0,N_0$ such that whenever $n\ge N_0$, $\lfloor An\rfloor<k\le\lfloor Cn\rfloor$, and the physical prefix fraction is at least $r_{\min}$, the prefix cell $S=\operatorname{structuredCell}(P,\lfloor An\rfloor,k,y_n)$ is nonempty and

This holds simultaneously for every integer $d>0$ with $d\le y_n^4$, all prime factors at most $y_n$, and $\gcd(d,P.\mathrm{modulus})=1$. The constant and threshold do not depend on the moving prefix or divisor. The physical prefix fraction means exactly $(k/n-A)/(C-A)$.