We show the primes have level of distribution 66/107 using triply well-factorable weights, and extend this level to 5/8 assuming Sel- berg’s eigenvalue conjecture. This improves on the prior world record level of 3/5 by Maynard. As a result, we obtain new upper bounds for Goldbach representations of even numbers. This is the first use of a level of distribu- tion beyond the ’square-root barrier’ for the Goldbach problem, and leads to the greatest improvement on the problem since Bombieri-Davenport from 1966.