Self-Replicating Intelligent Systems (Mathematical Foundations through Infinite Series with Multiple Ratios)

This book was written to establish a rigorous mathematical foundation for Self-Replicating Artificial Intelligence Machine Systems (SRAIMs) through the framework of Infinite Series with Multiple Ratios (SRMs). It develops the conceptual and formal basis of self-replicating intelligent systems, connecting deep mathematics to generative intelligence, autonomous architectures, advanced data structures, computational modeling, and long-range scientific applications. Rather than offering a purely speculative vision, the book proposes an original theoretical structure capable of supporting future research at the intersection of mathematics, artificial intelligence, and complex systems. By combining formal development with technological perspective, this volume is intended for researchers, advanced students, and professionals seeking new mathematical approaches to intelligent systems and emerging computational paradigms.