Let $n$ be a large enough composite integer, and consider an arithmetic function $f$ that maps $n$ to the sum of prime gaps making a closed interval $J_{f}(n)$ containing $n$ whose extremities are prime. In what follows I'll assume the truth of Goldbach's conjecture and consider the map $f:n\mapsto 2r_{0}(n)$ where $r_{0}(n)=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ but we can set a general theoretical framework.
Let $G_{f}(n)$ be the multiset of prime gaps $g_{i}(n)$ the sum of which is $f(n)$, $w_{g_{i}}$ their respective "weight", that is the number of times $g_{i}$ appears in $G_{f}(n)$ where we omit $n$ to avoid too heavy notations.
Now consider the "prime gap transform of $f$ evaluated at $n$" defined by $\displaystyle{T_{f}(n):=\prod_{g\in G_{f}(n)}p_{g/2}^{w_{g}}}$ that associates to $n$ and $f$ an integer characterizing the "type" of sequence of gaps making $J_{f}(n)$ sorted in increasing order.
Taking $f:n\mapsto 2r_{0}(n)$, can we take advantage of some fixed point theorem to show that the equality$T_{f}(n)=f(n)$ occurs infinitely often? If yes, could we deduce from that the truth of the twin prime conjecture?
Edit: as $f(n)$ is a sum of prime gaps, it is necessarily even. Then it would suffice to show that $T_{f}(n)$ is itself even infinitely often, that is, for infinitely many $n$, to entail the truth of the twin prime conjecture.
