Summed real-hyperbola oscillation ledgers

Let $X>0$ and $2\le a\le b\le X$ be natural numbers with $\log X/\log a\le5$. Denote by $\mathcal O_{X,a}(m)$ the canonical integrated fully-real hyperbola cell fractional-oscillation ledger, using common cap $a^5$. Then

This supplies a summable deterministic bound for within-cell fractional corrections.