Occupation of every canonical merged ratio cell

Let $M$ be a regular relative mesh with $\delta>0$, $n>1$, nonzero cutoff $W$, and scale separation, so its canonical prime partition is defined. Fix a ratio $\rho>1$. Assume the fixed-ratio prime-interval occupation property at cutoff $W$ and ratio $\rho$: for every natural $A\ge W$, the interval $(A,\lfloor\rho A\rfloor]$ has a prime. Then for every exponent band and every cell index through its last merged ratio cell,

The possibly short final fragment need not contain a prime separately because it is merged with the preceding full cell.

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