The question Does this infinite primes snake-product converge? made me come up with the following idea : given a strictly increasing sequence of positive integers $u_n$, define the $k$-th order gap transform of $(u_n)$ as the sequence $(g^{(k)}(u_n))_n$ where $g^{(k+1)}(u_n)=p_{g^{(k)}(u_n)+1}-p_{g^{(k)}(u_n)}$ and the $0$-th order gap transform being the identity map.

Is there a canonical inner product $\langle,\rangle$ such that for all $(u_n)$ one has $\langle g^{i}(u_n),g^{j}(u_n)\rangle=\delta_{ij}$?

This could help tackle the considered question by studying the gap transforms of the sequences of general term $4n$, $4n+1$ and $n^2$ and shed some light on Proth-Gilbreath conjecture as well.