Algebraic Resonance

Introduction

DEO AI Volume 1

introduces the core concepts of Dynamic Equilibrium Operation (DEO) and Algebraic Resonance. It proposes that mathematical operations can connect to structural dynamics if they are based on a relationship between a zero-point and unity. By treating zero and unity as identity elements, this framework interprets complex rotation as a dynamic structure involving variable augmentation and cancellation. This theory combines geometry, complex functions, calculus, and physics concepts, like wave equations, to model algebraic resonance.Here, zero-centered operations maintain symmetry and dynamic equilibrium, forming the basis for complex rotation in wave functions. Applied to number systems, this approach generates Base Number Waves (BNW), leading to the Spiral Ascension Dimension Model (SADM). Algebraic Resonance uses this model to reveal "path" and "memory" in operations, offering new perspectives on nonlinear outputs in neural networks.

Key Content of Volume 2

Application in Artificial Neural Networks

Volume 2 applies DEO theory to neural networks by interpreting complex rotation as balanced augmentation and cancellation of real and imaginary parts. This Base Number Wave (BNW) system provides insights into multidimensional interactions within neural networks, addressing the limitations of traditional methods in handling nonlinear interactions.DEO mathematics provides a framework that integrates complexity theory, helping neural networks to process nonlinear data interactions, dynamic equilibrium, and fractal structures for optimized weight calculations.

Dynamic Equilibrium Operation Fractal Space (DEOFS)

The DEO Fractal Space (DEOFS) offers AI a flexible structure where each computation connects within a fractal pattern, maintaining system-wide balance. This fractal approach allows the system to adapt to new environments while providing stable outputs, handling higher complexity through repetitive patterns.

Dynamic Equilibrium Operation Memory

Dynamic Equilibrium Operation Memory enables AI to retain previous pathways and outcomes, supporting consistency and stability even in unfamiliar situations. This memory balances past learning with new inputs, enabling AI to make stable, goal-driven decisions.

Operational Meta-Stability

Operational Meta-Stability supports a flexible equilibrium, allowing the system to adjust autonomously to shifts in goals or environments while sustaining stability. This flexibility enables the system to adapt efficiently to unexpected patterns or changes.

Adaptive Resonant Realignment

Adaptive Resonant Realignment allows the system to align its resonance state with new goals or environments, relying on internal resonance instead of external feedback. This method supports balance while enhancing emergent properties.

Emergent Properties and Spontaneous Pattern Management

With DEO and Algebraic Resonance, AI can generate spontaneous patterns that align with new data or goals, treating them as emergent solutions instead of errors. This approach manages these patterns autonomously, enhancing the system's adaptability.