Vanishing Fractality: An endogenous, deterministic, non-stationary S-adic model for the sieve of Eratosthenes

This paper presents a theoretical and computational analysis of the Sieve of Eratosthenes modeled as a deterministic, non-stationary S-adic automaton. We demontrate that the sequence of prime numbers emerges as the limit set of an endogenous process where morphisms are controlled by the automaton’s internal state. By identifying a 6–symbol substructure and analyzing its incidence matrix, we derive a recursive growth factor of p – 1. We prove the existence of a Stability Zone [n + 1, 2n - 1] in which symbolic information is “unmutable”. Using a “Frozen Window” technique allows for reliable experimental verfication up to n = 250,000 within the “Stability Zone”. A central result is the evidence of vanishing fractality: while the system begins as a Cantor-type set with high fractal order (approximaltely 0.92), the fractal dimension D=ln(p–1)/ln(p) converges to 1 as n → ∞. This transition provides a deterministic explanation