ZPIF-USAC: A Unified Mathematical Framework for Fractional Self-Fragmentation of Zeros with Applications to Prime Gaps, Dark Energy, Neural Dynamics, and Spacetime Geometry

Source abstract

This work presents the ZPIF-USAC (Zero Pairs Interaction Functional – Unified Spectral Alignment and Coherence) framework, a comprehensive mathematical theory that introduces fractional self-fragmentation of zeros as a fundamental property of the Riemann zeta function. Unlike classical approaches that treat zeros as passive, independent entities, ZPIF-USAC reveals that each zero is an active spectral entity capable of splitting into multiple fractional components and later reuniting through natural alignment dynamics. The framework introduces three revolutionary concepts: (1) the Fractional Alignment Parameter $\mathcal{A}_{nm}^{(\alpha)}$, which generalizes spectral alignment to arbitrary fractional orders $\alpha$ and reveals that optimal coherence occurs at $\alpha = 1/2$; (2) the Self-Fragmentation Coefficient $F_n$, which describes the intrinsic ability of each zero to fragment into multiple components; and (3) the Reunification Coefficient $R_{nm}$, which governs the tendency of fragments to reunite into coherent structures. These concepts are unified through the Fractional ZPIF-USAC Functional: $\operatorname{ZPIF}_{SF}^{(\alpha)}(x) = \sum\limits_n \gamma_n^{2\alpha} |c_n|^2 + \sum\limits_n F_n \sum\limits_{k=1}^{M_n} \gamma_{n,k}^{2\alpha} |c_{n,k}|^2 + \sum\limits_{n \neq m} R_{nm} \, c_n c_m$ This functional represents the first unified framework that combines fractional interactions, self-fragmentation, reunification dynamics, and active spectral properties into a single mathematical structure. The framework is rigorously applied to four fundamental domains. For prime gaps, ZPIF-USAC predicts the statistical distribution through fractional alignment dynamics. For dark energy, the fractional coherence term $\mathcal{C}^{(\alpha)}$ provides a natural model for cosmic acceleration. For neural dynamics, it describes synchronization patterns in the brain through alignment dynamics. For spacetime geometry, it induces a fractional spectral metric that bridges quantum mechanics and general relativity. Numerical simulations using the first 100 non-trivial zeta zeros confirm convergence to a unique stable state. The results demonstrate that fractional interactions of order $\alpha = 1/2$ are the most efficient in generating spectral coherence, and that self-fragmentation provides a natural mechanism for the emergence of complex structures in the universe. This work establishes ZPIF-USAC as a new paradigm in spectral theory, number theory, and mathematical physics, revealing that fractional self-fragmentation and natural alignment are fundamental principles governing the behavior of spectral systems and their applications to prime gaps, dark energy, neural dynamics, and spacetime geometry.