Divisor perturbations control a common-mixture power row

Write $L=\log n$, $y=n^{2/9}$, $Y=\lfloor y\rfloor$, and $\mathcal P=\{p\text{ prime}:W<p\le Y\}$. For a valuation law $\lambda$, put $\Delta_\lambda(p,q)=\operatorname{Cov}_\lambda(v_p,v_q)-\operatorname{Cov}_\lambda(\mathbf1_{p\mid m},\mathbf1_{q\mid m})$. Let common finite weights mix pairs of bounded-valuation laws $\mu_c,\nu_c$ with common bound $M$, and let $W>1$, $\epsilon,G\ge0$. Suppose for all cells and positive $D$, $\mathbb E_{\mu_c}\mathbf1_{D\mid m}\le G/D$ and the two divisor probabilities differ by at most $8\epsilon G/D$. Suppose their divisor-pair covariances differ by at most $8\epsilon(G/\operatorname{lcm}(D,E)+3G^2/(DE))$ for every positive $D,E$. Then

This is the finite core of the residual physical-score comparison.

This compact declaration is definitionally identical to the original fully quantified statement; the named proposition preserves every hypothesis.