Proof of the weak Goldbach Conjecture

I think this blog post of Terry Tao, as well as the comments following it (including some from Helfgott) answer this question as completely as one could reasonably hope.

https://terrytao.wordpress.com/2012/05/20/heuristic-limitations-of-the-circle-method/

https://plus.google.com/114134834346472219368/posts/8qpSYNZFbzC