  We show the primes have level of distribution $66/107\approx 0.617$ using
triply well-factorable weights. This gives the highest level of distribution
for primes in any setting, improving on the prior record level $3/5=0.60$ of
Maynard. We also extend this level to $5/8=0.625$, assuming Selberg's
eigenvalue conjecture. As applications of the method, we obtain new upper
bounds for twin primes and for Goldbach representations of even numbers $a$.
For the Goldbach problem, this is the first use of a level of distribution
beyond the square-root barrier, and leads to the greatest improvement on the
problem since Bombieri-Davenport from 1966. Our proof optimizes the
Deshouillers-Iwaniec spectral large sieve estimates, both in the exceptional
spectrum and uniformity in the residue $a$, refining Drappeau-Pratt-Radziwill
and Assing-Blomer-Li.
