Define $G_n=K_{\pi(2n)-1}\circ C_1 $ to be the complete graph on $\pi(2n)-1$ vertices composed with the self-loop to yield a complete graph with self loop edges and where $\pi(k)$ is the prime counting function returning the number of prime numbers up to $k$. Let each $v_i\in V(G_n)$ be labeled with $p_{i+1}$, the $i^{th}$ odd prime. Let each $v_iv_j\in E(G_n)$ be labeled by $\frac{p_{i+1}+p_{j+1}}{2}$, the average of the prime vertex labels.
Goldbach's Conjecture states every even number grater than 2 can be expressed as the sum of two primes in at least 1 way. This is equivalent to saying that every integer greater than 1 has at least one pair of (possibly equal) primes that are equidistant from it. This is then equivalent to saying that every integer greater than 1 is the average of at least one pair of primes.
This then directly implies that all natural numbers $2<x\leq n$ should appear as the label of at least one edge in $G_n$. However, doing out up to $G_7$, there appears to be an even stronger fact; if we express the implication of Goldbach as $$GC\implies\{3,4,\dots n\}\subseteq L[E(G_n)]$$ Where $L(v_iv_j)$ is the label of the edge, then we can express this new observation as $$\forall n\geq3,\phantom{22} L[E(G_n)]=\{3,4,\dots, p_{\pi(2n)}\}\text{ or }\{3,4,\dots, p_{\pi(2n)-2},p_{\pi(2n)}\}$$ Meaning that when constructing these $G_n$, the actual list of edge labels extends well past $n$ and is always either everything up to the last prime before $2n$, or is missing the integer 1 less than that final prime. Is this a general trend? Is this in fact stronger than the Goldbach Conjecture or am I missing how GC implies this much as well? I admit to not being a good enough programmer to check this out further, but if this holds, could it worked backwards through the same equivalencies to be phrased like a stronger GC?