Head-free smooth interval lower bound from divisor shifts

Fix $W$, $K>0$, and a natural threshold $Y_0$. Let $P_W$ and $d_W$ denote the rough-head modulus and density. Assume that for every $z\ge Y_0$, positive divisor $d\mid P_W$, $d\le X$ and $\log X\le5\log z$,

For natural endpoints with $Y_0\le y$, $W\le y$, $y\ge2$, $P_W\le A\le B$ and $\log B\le5\log y$, let $S(A,B;y)$ be the smooth interval and $S_W^*(A,B;y)$ its head-free subset. Then

This makes the loss in the finite Möbius removal of head primes explicit.