Arbitrary-interval theta-weight variation at inverse-log-square scale

Let C≥0 and let X,a,b be natural numbers with 3≤a≤b≤X and log X/log a≤5. Define the natural-quotient theta weight Q_X(m)=roughSaiasNaturalMain(⌊X/m⌋,m)/log m. Then its discrete variation on the half-open integer interval [a,b), weighted by Cm/(log m)⁴, satisfies the explicit bound below. Unlike the upper-selector estimate, this covers arbitrary faces of the hyperbola decomposition without assuming X≤a².