On the Asymptotic Saturation and Local Linearity of Trinary-Encoded Prime Gap Series

This paper presents a novel computational framework for analyzing deterministic structures within long-range prime gaps. By transforming sequential prime gap intervals into a discrete trinary data stream (the "Primarcode" framework), we isolate a highly ordered, macroscopic phase behavior that remains hidden under standard statistical treatments. Through systematic empirical validation across six orders of magnitude (from 10^4 to 10^9), the resulting symmetry centers are aggregated using localized density clustering. Linear regression of the phase drift against a standardized 210-raster reveals an exceptional and increasing local linearity, with goodness-of-fit metrics systematically exceeding R² > 0.9999 in higher ranges. Furthermore, we evaluate the evolution of the drift slope m_reg across scaling thresholds using multiple non-linear regression models. The statistical validation (via RMSE and R²) confirms that the slope evolution is optimally described by an asymptotic saturation model (m(x) = a + b / x^α) rather than unbounded logarithmic decay models. This indicates that the macroscopic phase ultimately stabilizes into a fixed, universal linear drift. The accompanying repository contains the full, production-ready Python source code using NumPy and Scikit-Learn to ensure complete computational reproducibility of the tabulated results, datasets, and regressions.