Asymptotic Modeling of the Recursive Prime Permutation Mapping for Consecutive Prime Numbers

Abstract

The spatial distribution and frequency trajectories of prime numbers within structured permutation bases pose fundamental challenges in analytic number theory and symbolic dynamics. In this study, the deterministic and asymptotic properties of prime frequencies across multi-layer permutation mapping structures are investigated under the framework of Recursive Prime Permutation Mapping (RPPM). By categorizing prime permutation trajectories across k-digit sequence spaces (k ∈ {8, 9, 10, 11, 12}), a closed-form asymptotic model governed by an exponential growth component and a logarithmic damping term is constructed. Using Direct Nonlinear Least Squares (NLLS) optimization, the predictive capacity of the model is evaluated for k = 12 across 64 individual permutation dynamics categorized into four functional layers (P1–P4). The model achieves an overall Mean Absolute Percentage Error (MAPE) of 0.2266% in the 5–11 calibration and 12-step out-of-sample validation experiment. The empirical results indicate that individual prime permutation dynamics exhibit stable asymptotic growth trajectories and that the proposed formulation provides an analytical framework for high-dimensional numerical sequence forecasting.