Uniform signed profiles for the actual guarded medium-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 nonempty finite head set with patterns $H_h$, two fixed physical intervals $I$, a common upper bound $C_{\max}$, and guard ledgers with fixed promotion and bank parameters. Fix $W>1$ containing all head primes and a nonnegative effective prime-coefficient bound $A$. Assume each upper interval endpoint is at least one and at most $C_{\max}$. There exist a nonnegative sequence $E_n$ and $N_0$, independent of the later choice of band index set and bridge data, such that $E_n\to0$ and $E_n\log\log n\to0$. For any such bridge at $n\ge N_0$ whose head patterns, physical endpoints, guard set, and cutoff are exactly the prescribed ones, and whose effective prime coefficients are bounded by $A$, all component laws $\mu_c$ satisfy

for distinct moving-band primes $p,q$. Here $\mu_c$ is the actual guarded medium-tilted cell law. This attaches the abstract marked profiles to the concrete bridge using only structural identifications. The moving prime band is $\{p\text{ prime}:W<p\le\lfloor n^{2/9}\rfloor\}$.