Assume Goldbach's conjecture throughout this question. Then every integer $$n>1$$ admits a non negative integer $$r$$ such that both $$n-r$$ and $$n+r$$ are prime. Let $$r_{0}(n)$$ the infimum of those integers $$r$$ and define the fundamental primoradial ratio $$R_{n}$$ as the ratio $$\frac{n+r_{0}(n)}{n-r_{0}(n)}$$. I proved in a comment to a former question of mine that $$r_{0}(n), so that one has $$R_{n}<3$$.

Is $$R_{n}$$ provably maximal for $$n=22$$? Is it true that $$R_{n}<2$$ as soon as $$n$$ is greater than $$22$$? This would entail $$r_{0}(n).

Finally is it true that $$\forall\varepsilon>0,\exists m, n>m\Longrightarrow R_{n}<1+\varepsilon$$? This would entail $$r_{0}(n)=o(n)$$.

Edit August 5th 2022: could we prove that $$r_{0}(n)\lesssim x_{n}$$ where $$x_{n}$$ is the positive solution of the following equation in $$x$$:

$$x=\frac{n+x}{\log (n+x)}$$

which seems to hold numerically, we would get $$r_{0}(n)=o(n)$$ and $$\lim_{n\to\infty}R_{n}=1$$. This is inpired by a theorem of Selberg on RH stated in https://www.sciencedirect.com/science/article/pii/0022314X80900311

Moreover, setting $$\alpha_{0}:=\inf\{\alpha\mid r_{0}(n)=O(n^{\alpha})\}$$, can we get a non trivial upper bound for $$\alpha_{0}$$, for example assuming GRH?