Uniform-order Proposition 8.7 with varying active mass

Fix positive $c_{\rm mesh}$, physical intervals with endpoints in $[1,U]$ where $U\ge1$, natural guard budgets and a ledger family. There exist $\tau>0$ and $W_0$ such that, for every $W\ge W_0$ and every real mass sequence $q_n$ eventually at least $1$, the following holds. Choose a finite nonempty head family supported exactly on primes at most $W$, five nonnegative target, initial-error, mass, frozen-weight and active-weight constants, and a positive cell-margin floor. Then a radius $a>0$ and $C_{\rm post}\ge0$ can be chosen before any permitted mesh of positive $\delta$ and width $\delta+\eta\le\tau$.

For each such mesh, eventually every canonical bridge with separated nonempty surviving physical cells, canonical scale-separated partition, width $\delta+\eta$, and baseline $q_nT$ for a barycentric target $T$ above the margin floor has a Proposition 8.7 path certificate. The hypotheses are target envelopes for $\Delta$, $N\ge0$, active mass at most $C_{\rm mass}N$, initial marked errors at most $C_{\rm initial}N/(pL)$ throughout the prime band, literal band balance $\Delta_j$ equal to the initial marked-band residual, separated head patterns, a feasible frozen layer, frozen and active sample bounds $C_{\rm fixed}/L,C_{\rm active}/L$, and an integer quota equal to frozen plus active mass. The choice order is

This provides literal marked-band balance for active masses varying with $n$, with the initial mesh tolerance and cutoff independent of the mass family.