A fractal algorithm encodes prime number patterning.

This article focuses on number theory by modeling the deterministic, symbolic, and fractal patterns encoding prime numbers and their gaps, revealing structured prime generation beyond traditional statistical distribution theories. We present a novel mathematical machine with a novel fractal algorithm for encoding the pattern of prime numbers and gaps. This framework leverages three fractal processes “move”, “copy”, “change” with deterministic rules to construct six walksets. Walksets An and APn act as collectors, Bn and BPn determine prime or composite, Cn starts with all natural numbers and CPn contains the fractal process with always periodic patterns. Primes and gaps arise as emergent phenomena within this rule-based system – proving that this Ansatz holds for all prime numbers and gaps. We call this type of mathematical machine “Synchronous Factory Automaton” SFA. It was implemented in Java.