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

We show the primes have level of distribution $66\mathrm{/}107\approx 0.617$ using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level $3\mathrm{/}5=0.60$ of Maynard. We also extend this level to $5\mathrm{/}8=0.625$, 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.