Primes in arithmetic progressions to large moduli, and Goldbach beyond the square-root barrier

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.