A fractal algorithm shows prime number patterning

We present a novel mathematical machine with a novel fractal algorithm for modeling the distribution 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.