A Zeta Function of Differences: Sequences Without Rules and the Surprise of Prime Gaps

After publishing a reformulation of Goldbach's conjectures through differences of primes, a question remained: what exactly does it mean that the next prime cannot be predicted? These notes follow that question in the order in which its answers arose. We prove that every even number has infinitely many representations as a difference of four distinct primes, completing the three-prime theorem for odd numbers. We prove that no rule computing the next prime gap from a fixed number of preceding gaps, whatever its form, can generate the sequence of gaps, and we separate the two senses in which a polynomial could govern the gaps: as a machine that produces them and as an equation satisfied by the constant $\rho$ that concatenates them. Since the known tools of transcendence detect an excess of order that $\rho$ lacks, we introduce a zeta function of differences for any increasing sequence, $D_A(s)=\sum_n(a_{n+1}-a_n)a_n^{-s}$, whose simplest case is the Riemann zeta function; we prove a general residue theorem, its continuation to the whole plane for the Fibonacci numbers, and its continuation to $\operatorname{Re}s>0$ for irregular sequences with gaps of $1$ or $2$, and we pose the location of its wall for the primes as the central open question. A natural boundary for the generating function of the gaps, reached independently, turns out to follow from an earlier result, and our route yields a general criterion. Finally, we define surprise as what remains to be known when searching is not allowed, and measure it along five axes: the order of the prime gaps carries a weak, local memory that fades slowly with height.