Local Maxima of Number of Goldbach Decompositions

Theorem
Let $\mathbb E$ be the set of even positive integers.

Let $G: \mathbb E \to \N$ be the mapping defined as:
 * $\forall n \in \mathbb E: \map G n =$ the number of Goldbach decompositions of $n$.

Then $G$ has local maxima when $n$ is a multiple of $6$.