Define the fundamental primality radius of an integer $n>1$ as $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$.
Can we intepret the average prime gap around $n$ as an absolute temperature $T$ and use the value $\frac{2\pi^{5}}{15}$ of Stefan constant $\sigma$ in Planck units to get the following upper bound for $r_{0}(n)$:
$$r_{0}(n)<\sigma\log^{4}n\text{ ?}$$
Edit: this would be consistent with swapping upper and lower indices and reversing the inequality in https://arxiv.org/abs/1412.5029 if we set $\log_{a}^{b} n:=(\log^{\circ a} n)^{b}$.
Could such a process be proven legit, we might proceed recursively to end up with:
$$\max_{p_{n+1}\leq X} g_{n}\asymp\log_{a}^{b}X\log_{b}^{a}X$$
and maybe $$\max_{n+r_{0}(n)\leq X} r_{0}(n)\asymp\log_{a}^{c}X\log_{c}^{a}X$$
With $(a,b,c)=(1,1,2)$ (those indices appearing in both upper and lower position they have to be integers, so that the "compositional power"actually makes sense).