Spectral Orbits, Zero Mirrors, and the Imaginary Complement: A Zero Pairs Interaction Functional Framework for Prime Distribution

Source abstract

This work introduces a unified mathematical framework within the Zero Pairs Interaction Functional (ZPIF) framework for understanding the distribution of prime numbers through three novel concepts: spectral orbits, zero mirrors, and the imaginary complement. We demonstrate that the non-trivial zeros of the Riemann zeta function follow closed spectral orbits with defined phase angles, revealing an organized structure behind prime distribution. A remarkable symmetry is uncovered: for every zero γn, there exists a mirror zero γN+1-n such that the sum of their fractional parts equals approximately 1. This symmetry is further confirmed through logarithmic and inverse transformations. The framework provides a natural explanation for the stability of prime gaps and the coherence of zero interactions without requiring external regulation. Numerical simulations using the first 10,000 non-trivial zeros confirm the convergence and stability of the proposed model with high precision.