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. This deterministic framework leverages symbolic substitution and recursive refinement to construct walksets whose periodic symbolic structure encodes all viable prime candidates as labeled walksets. Apparent irregularities in prime gaps emerge not from randomness but from layered symbolic complexity and low correlation. Through fully constructive propagation, primes arise as emergent phenomena within a rule-based system—demonstrating that this Ansatz holds universally for the entire set of prime numbers. The algorithm's recursive symbolic refinement yields exhaustive coverage and structure-preserving propagation, ensuring that no prime escapes the generative envelope of the machine. We call the machine “Synchronous Factory Automaton” SFA.