As usual, under Goldbach's conjecture, let's define for a large enough composite integer $n$ the quantities $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$.
The sequence of prime gaps in the interval $[n-r_{0}(n),n+r_{0}(n)]$ sorted in decreasing order can be listed as $g_{1}(n),\cdots,g_{k_{0}(n)}(n)$ and forms a partition of $2r_{0}(n)$. As such, the sequence $(h_{i}(n))_{1\leq i\leq k_{0}(n)}$ defined by $h_{i}(n)=\frac{g_{i}(n)}{2}$ is a partition of $r_{0}(n)$ in $k_{0}(n)$ parts. Let's call this partition the fundamental partition associated to $n$.
Among all partitions of $r_{0}(n)$ in $k_{0}(n)$ parts, is the fundamental partition associated to $n$ the one that maximizes entropy in Shannon's sense?
