Prime-counting lower bound for a canonical 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 $\rho>1$. For an exponent band and raw cell index $j$ strictly below its last raw ratio cell, let $A$ be the multiplicative ratio cutoff at the beginning of that cell. If $N_{\rm cell}$ is the cardinality of the corresponding merged ratio cell, then

A full raw prime interval injects into the merged cell, which may additionally contain the final short fragment.

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