Analysis of Zero Pairs Interaction Functional Using Zigzag Zero–Fractional Zero, Zero Behavior Motions, and Prime Distribution within Spectral Alignment and Coherence

This work presents a comprehensive analysis of the Zero Pairs Interaction Functional (ZPIF) within the Unified Spectral Alignment and Coherence (USAC) framework, integrated with the Zigzag Zero–Fractional Zero (ZZFZ) formulation, to investigate the full spectrum of Zero Behavior Motions (ZBM) and their direct connection to Prime Distribution (PD).Unlike classical approaches that treat zeta zeros as static and independent, this framework redefines zeros as active spectral entities that interact with themselves before interacting with others. These self-interactions include quadratic, higher-order polynomial, and fractional interactions (division, root extraction, and logarithmic mappings), followed by cross-mode interactions with other zeros.The theory introduces a unified spectrum of interconnected phenomena: zigzag motion, aberration, energy stability, cavitation, spectral reflection, spectral constriction, unstable spectral oscillation, spectral fissure, temporal succession, spectral pulsing, spectral bursting, axial orientation, spectral cleavage, boundary reflection, slicing and reformation, layered spectral stratification, and spectral unfolding.The entire system operates under a closed, energy-conserving regime that guarantees spectral stability and coherence across all scales. The framework offers a natural, mathematically rigorous explanation for the distribution of prime gaps, the nature of dark energy, the emergence of consciousness, and the geometric structure of spacetime. Numerical simulations using the first 100 non-trivial zeta zeros confirm convergence and stability.This work establishes ZPIF-USAC-ZZFZ as a new foundational paradigm that unifies spectral theory, number theory, and mathematical physics, opening new directions in quantum gravity, information theory, and consciousness studies.