Canonical valuation mean bound and pairwise agreement rate

Write $L_n=\log n$, $y_n=n^{2/9}$, $Y_n=\lfloor y_n\rfloor$, and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le Y_n\}$. Fix a finite nonempty head index set, patterns $P_h$, physical intervals $I$, guard constants $C_{\rm prom},C_{\rm bank}$ and a ledger $G_n$ at each $n$. Fix $W>1$ with all head primes at most $W$, and $A_{\rm coef}\ge0$. Consider any finite band set and bridge $B$ with sample size $n$, cutoff $W$, separated physical intervals, and nonempty cells after deleting the ledger guards; require its sample data to equal the canonical sample data for these choices. Let $\xi$ be any parameter whose effective prime coefficients have absolute value at most $A_{\rm coef}$, and let $\mu_c^\xi$ be its medium law on cell $c$. There exist $A_{\rm val}\ge0$, a nonnegative sequence $\epsilon_n$, and a common threshold with $\epsilon_n\log L_n\to0$, such that eventually, for every band prime $p$ and cells $c,c\prime$,

A single reciprocal first-moment constant and a vanishing mean-agreement rate work for every later bridge and admissible parameter.