Uniform signed marked profiles for fixed finite tilted cells

For a divisor $D$, let $M_n(D)$ denote the common paper divisibility main term; write $w_{p,r}=(r+1)/p^r$ and $w_{p,q,r,s}=w_{p,r}w_{q,s}$. Fix a finite family of head patterns $H_c$ and positive endpoints $0<A_c<C_c\le C_{\max}$. For every cutoff $W>1$ containing all head primes and every box size $B\ge0$, there exist a nonnegative sequence $E_n$ and threshold $N_0$ with $E_n\to0$ and $E_n\log\log n\to0$. For every $n\ge N_0$ and every valuation tilt with $|\eta_p|\le B$ on the moving prime band, the structured cells with these endpoints and smoothness cutoff $y_{\mathbb N}(n)$ are nonempty. Each of their tilted laws $\mu_c$, interpreted under the common upper value bound, satisfies

for distinct moving-band primes $p,q$ and all natural exponents $r,s$ with $p^rq^s\le y_{\mathbb N}(n)^4$. The error is chosen before the tilt and is common to the entire finite family. The moving prime band is $\{p\text{ prime}:W<p\le\lfloor n^{2/9}\rfloor\}$.