Spectral Dynamics, Variational Optimization, and Critical Line Stability in Prime Gap Prediction.

Abstract

This paper investigates the analytical structure of a recursive transition operator for the deterministic prediction of gaps between consecutive prime numbers ($p_i - p_{i-1}$), focusing on the variational method, self-consistent field Lagrangians, and functional spectral analysis. Starting from a balanced reformulation of Chebyshev's explicit formula—incorporating a geometric phase symmetry factor of $1/2$—we demonstrate how an action functional with second-order corrections in $1/\ln(p)$ establishes a rigorous dynamic confidence band. Furthermore, we introduce a functional framework where the Riemann Hypothesis (RH) emerges not as an a priori assumption, but as a necessary dynamic consequence of self-adjoint operator stability. Empirical validations confirm absolute scale invariance and a $100\%$ success rate across magnitudes up to $10^{15}$.