Hybrid Fractology a Novel Theorem

Abstract Hybrid Fractology is a novel mathematical framework that unifies fractal and non-fractal structures into a single equation. This approach decomposes complex systems, whether mathematical, physical, or biological, into self-similar components and smooth components. The Hybrid Fractology model describes the interaction between structured regularity and emergent complexity. This model provides insight into: 1. Number Theory – Decomposing the Riemann zeta function to propose a proof of the Riemann Hypothesis. 2. Neural Networks – Explaining self-organizing learning processes in artificial intelligence and their optimization. 3. Neuroscience – Modeling stroke recovery and neural plasticity via fractal self-organization. 4. Physics – Describing quantum mechanics, cosmology, and chaotic systems using a hybrid fractal framework. This paper provides mathematical evidence supporting this decomposition, demonstrating how the Hybrid Fractology model governs systems across number theory, biological networks, and artificial intelligence.