Explicit inverse-log-square majorant for the interval shift budget

For natural A,B,y,d with y≥2, d>0 and A≤B, let C⋆ denote roughSaiasSharpDefectConstant and use the sharp endpoint rate η(y)=10C⋆/(log y)². Denote by Bη(A,B;y,d) the source interval fixed-divisor budget: 6+(B−A)(log d+2)/(d log y)+Pη(⌊A/d⌋,⌊B/d⌋;y)+Pη(A,B;y)/d, where Pη(a,b;y)=η(y)(a+b)+5(b−a)/log y. Then this abstract budget is bounded by the explicit sharp budget below. No condition d≤A or upper restriction on log B is required for this deterministic comparison.