An interesting pattern in the differences between prime numbers.

You have re-discovered Gilbreath's conjecture, namely that creating a sequence in which each term $s_n$ is the difference between the consecutive primes $p_{n+1}-p_n$, and then repeating this process for the newly created sequence, and so forth, will always yield sequences that begin with $s^k_0=1$, for the $k^{th}$ sequence.

Andrew Odlyzko verified the conjecture for $k=3.4 \times10^{11}$ iterations, however, there are no proofs of this conjecture as of today.

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Your calculations regarding the sequence ${2^n}$ have, unfortunately, nothing to do with Gilbreath's conjecture. It is easily shown that the difference between any consecutive terms $s_n$ and $s_{n+1}$ is:

$$ s_{n+1}-s_n=2^{n+1}-2^n=2^n, $$

yielding the element in the next sequence $q_n=2^n=s_n$.