Within-cell displacement of the fully-real fractional correction

Let $X>0$, $2\le a\le b\le X$, $a\le m<b$ be natural numbers with $\log X/\log a\le5$, and let $s\in[m,m+1]$ be real. Write $I_{\mathbb R}(q,z)$ for the fully-real base-free fractional integral, and $\mathcal O_{X,a}(m)$ for the integrated cell oscillation ledger. Then

The bound transfers pointwise kernel oscillation to the continuous correction integral.