Uniform agreement of tilted valuation means across finitely many raw cells

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 $P$. For coefficients $\eta$, the tilted cell law assigns mass proportional to $\exp(L_n^{-1}\sum_{q\in\mathcal P_{n,W}}\eta_q v_q(m))$. Fix a finite cell index set, patterns $H_c$, and real endpoints $0<A_c<C_c\le C_{\max}$ with $C_c>0$. Fix $W>1$ containing every head prime, and any coefficient bound $B\ge0$. There are a nonnegative sequence $\epsilon_n$ and $N_0$ such that $\epsilon_n\log L_n\to0$ and, for $n\ge N_0$, all $p\in\mathcal P_{n,W}$, all $|\eta_q|\le B$ on this band, and all nonempty cells, their tilted laws $\mu_c^\eta$ satisfy

The rate is uniform over the finite family and the full coefficient box, with no relation imposed between the box size and the cutoff.