  A well-known conjecture of Gilbreath, and independently Proth from the 1800s,
states that if $a_{0,n} = p_n$ denotes the $n^{\text{th}}$ prime number and
$a_{i,n} = |a_{i-1,n}-a_{i-1,n+1}|$ for $i, n \ge 1$, then $a_{i,1} = 1$ for
all $i \ge 1$. It has been postulated repeatedly that the property of having
$a_{i,1} = 1$ for $i$ large enough should hold for any choice of initial
$(a_{0,n})_{n \ge 1}$ provided that the gaps $a_{0,n+1}-a_{0,n}$ are not too
large and are sufficiently random. We prove (a precise form of) this postulate.
