# Gap transform of an integer sequence

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.