Logarithm-weighted port numerator bound

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 further $W\ge2$ and that $W$ exceeds the uniform reciprocal-prime-sum cutoff. Let $s\ge0$ and let $p$ be a canonical tangent prime. Write $A$ for the current ratio-cell lower endpoint, $Y$ for its exponent-band upper endpoint, $N_p(s)$ for the source port numerator envelope, and $C_M$ for the uniform reciprocal-sum constant. Then

The two incident harmonic tails cost at most twice the ordinary logarithmic gap plus the explicit reciprocal-sum error.

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