A fractal algorithm shows prime number patterning.

This article focuses on modeling the deterministic, symbolic, and fractal patterns underlying 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 modeling the pattern of prime numbers and gaps. This framework leverages three fractal processes with deterministic rules to construct six walksets. Walksets A_n and AP_n act as collectors, B_n and BP_n determine prime or composite, C_n starts with all natural numbers and CP_n 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 the machine “Synchronous Factory Automaton” SFA. It was implemented in Java.