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

Vanishing Fractality: A Deterministic, Endogenous, Non-Stationary S-adic Automaton for the Sieve of Eratosthenes This research presents a novel reformulation of the Sieve of Eratosthenes, shifting its definition from a procedural algorithm to a deterministic, non-stationary S-adic automaton. Unlike traditional approaches that treat prime distribution as a stochastic phenomenon, this work demonstrates that the prime-composite classification emerges endogenously from the internal symbolic dynamics of a growing periodic tape. Key Contributions of Version 37: Discovery of Hierarchical Grammar: This version highlights a breakthrough in understanding the symbolic structure of the number line. We show that the Sieve is a sequence of alphabet renormalizations. Beyond the initial four-letter alphabet a, b, c, d, the system recursively constructs higher-order meta-letters. A central finding is the emergence of the third-order meta-letter e=aabac at prime p=5, which serves as the "base pattern" for subsequent variations. The Phenomenon of Vanishing Fractality: We introduce and define "vanishing fractality"—a dynamic process where the prime candidate set behaves as a fractal dust at small scales but converges to a dimension of D=1 as n→∞. This provides a constructive, geometric account of the transition from the highly structured distribution of small primes to the apparent randomness observed at large scales. The Stability Zone & Hydra Effect: We provide a formal proof for the "Stability Zone" n+1, 2n−1, a zone of immutability that advances through the number line. Within this framework, the Hardy–Littlewood combinatorial factors are derived deterministically from the spectral properties of a prime-dependent transition matrix (Mp). Full Reproducibility: Theoretical claims are verified through a "Frozen Window" experiment up to n=250, 000. To support the principles of Open Science, the complete Java source code and resulting experimental datasets (. csv) are available on GitHub to ensure full peer-review transparency and reproducibility. Links: Source Code & Data: https: //github. com/cerebrummi/stabilityzone ORCID: https: //orcid. org/0000-0001-6713-583X Subject Area: Number Theory, Symbolic Dynamics, S-adic Systems, Fractal Geometry, Automata Theory.