Short-interval prime upper bound from a smooth divisor weight

For $Y\ge X\ge 2$, a smooth Selberg divisor weight of level $D=X^{1/10}$ equals one on every prime in $(Y,Y+X]$. Its second moment can be expressed through an Euler product $H$ whose zeta-factor bound contributes $O(1/\log D)$ after integration against rapidly decreasing Fourier transforms. Consequently