A proof of the inequality $latex \varepsilon_{m}\le 2\varepsilon_{moy}$

Given $n\ge 14$ a positive integer, let’s call the difference between two consecutive potential typical primality radii of $n$ a « Goldbach gap ». There are exactly $N_{1}(n)$ such Goldbach gaps. Let’s denote by $\eta(n)$ the number of distinct Goldbach gaps, and let’s write $\varepsilon_{i}$ , $1\le i\le \eta(n)$ the $i$ -th such Goldbach gap in the increasing order. Thus $\varepsilon_{1}=\varepsilon_{min}$ and $\varepsilon_{\eta(n)}=\varepsilon_{max}$ . Moreover, let’s denote by $w_{i}$ the multiplicity of $\varepsilon_{i}$ . One has $\displaystyle{\sum_{i}^{\eta(n)}w_{i}\varepsilon_{i}=P_{ord_{C}(n)}}$ and $\displaystyle{\sum_{i}^{\eta(n)}w_{i}=N_{1}(n)}$ . Writing $p_{i}:=\dfrac{w_{i}}{N_{1}(n)}$ , one gets $\varepsilon_{moy}=\displaystyle{\sum_{i=1}^{\eta(n)}p_{i}\varepsilon_{i}}$ . Let’s define $s_{i}$ for $i$ ranging from $1$ to $\eta(n)$ so that $s_{1}=\dfrac{1}{2}$ , $s_{\eta(n)}=\dfrac{1}{2}$ and $s_{i}=0$ if $i\not\in\{1,\eta(n)\}$ . One has $\vert\varepsilon_{moy}-\varepsilon_{m}\vert=\vert\displaystyle{\sum_{i}^{\eta(n)}\varepsilon_{i}(p_{i}-s_{i})\vert}=\vert(p_{1}-\dfrac{1}{2})\varepsilon_{1}+(p_{\eta(n)}-\dfrac{1}{2})\varepsilon_{\eta(n)}+\displaystyle{\sum_{i=2}^{\eta(n)-1}p_{i}\varepsilon_{i}}\vert$ . Hence $\vert\varepsilon_{moy}-\varepsilon_{m}\vert\le\displaystyle{\sum_{i=2}^{\eta(n)-1}p_{i}\varepsilon_{i}}$ , and $\vert\varepsilon_{moy}-\varepsilon_{m}\vert\le \varepsilon_{moy}$ . Therefore $\varepsilon_{m}\le 2\varepsilon_{moy}$ .