  We introduce a modification of the linear sieve whose weights satisfy strong
factorization properties, and consequently equidistribute primes up to size $x$
in arithmetic progressions to moduli up to $x^{10/17}$. This surpasses the
level of distribution $x^{4/7}$ with the linear sieve weights from well-known
work of Bombieri, Friedlander, and Iwaniec, and which was recently extended to
$x^{7/12}$ by Maynard. As an application, we obtain a new upper bound on the
count of twin primes. Our method simplifies the 2004 argument of Wu, and gives
the largest percentage improvement since the 1986 bound of Bombieri,
Friedlander, and Iwaniec.
