Fixed-divisor interval shift from a Saias endpoint approximation

Assume the compact bounded-variation translation principle: for every function f of variation at most 2 on [−5,5] and a,b∈[0,5], the integral over v∈[0,5] of |f(a−v)−f(b−v)| is at most 2|a−b|. Let η be an endpoint error rate and Y₀ a threshold such that the source Saias endpoint error at (X,y) has magnitude at most η(y)X whenever y≥max(Y₀,2), X>0 and log X≤5 log y. Fix natural numbers y≥max(Y₀,2), 0<d≤A≤B, with log B≤5 log y. Write Ψ(X,y) for the count of y-friable positive integers at most X, A_d=⌊A/d⌋ and B_d=⌊B/d⌋. Define Pη(a,b;y)=η(y)(a+b)+5(b−a)/log y for ordered natural endpoints. Then the genuine friable interval shift satisfies the following cancellation-preserving bound; its right side is roughSaiasIntervalFixedDivisorShiftBudget.