# Are those limits defined in terms of a positive integer actually independent thereon?

Under Goldbach's conjecture, denote by $r_{0}(n)=\inf\{r\gt 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and by $k_{0}(n)=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$ for $n$ large enough. Say $n$ is $k$ -central if and only if $k_{0}(n)=k$.

Let's define $\delta_{k}^{+} : =\lim\sup\dfrac{\log r_{0}(n)}{\log\log n}$ when $n$ runs through $k$ -central numbers and tends to $\infty$, $\delta_{k}^{-}$ similarly with this time a $\lim\inf$, and $\delta_{k}$ as the limit as $n$ tends to $\infty$ of $\dfrac{1}{\pi_{k,C}(n)} \sum_{C<m\leq n,k_{0}(m)=k}\dfrac{\log r_{0}(m)}{\log\log m}$ for $C$ an absolute positive constant, where $\pi_{k,C}(x)$ is the number of $k$ -central numbers greater than $C$ not exceeding $x$. Does one have $\delta_{k}=\frac{\delta_{k}^{+}+\delta_{k}^{-}}{2}$ for all $k>0$? Are those quantities actually independent on $k$? Is there a good reason to believe that for all $k>0$ one has $\delta_{k}^{+}=2$?