## Introduction: To consolidate all the concepts developed in our work, we present a unified axiomatic equation that integrates: - **The Dual-Non-Dual (D-ND) model** - **Unified Information Theory** - **The principles of emergent gravity and polarization dynamics** - **The key components of the quantum operating system, including non-relational potential, possibilistic density, quantum fluctuations, NT (Null-All) states, and non-local transitions** ## Unified Axiomatic Equation

## Abstract: We present a novel approach to improving the Barnes-Hut algorithm for N-body simulations by integrating it with a Dual-Non-Dual (D-ND) quantum framework within a Quantum Operating System (QOS). This integration incorporates concepts from Unified Information Theory, particularly the emergent gravity paradigm and the dynamics of polarization. By introducing quantum fluctuations, possibility densities, and non-relational potentials, we enhance both the performance and accuracy of the algorithm. The framework utilizes a proto-axiomatic state to guide spatial decomposition and force calculations, potentially improving computational efficiency without compromising physical precision.

## 1. Introduction The **Quantum Emergence Model** aims to unify concepts from quantum mechanics, information theory, and cosmology through the introduction of an **emergence operator** \( E \) and an **initial null-all state** \( |NT\rangle \). This approach makes it possible to describe the transition from an undifferentiated, non-dual state to emergent, differentiated states, providing a theoretical basis for understanding the origin of complexity, the arrow of time, and the structure of the universe.

**Enunciated:** In the **Quantum Emergence Model**, evolution from an undifferentiated (non-dual) state to differentiated (dual) states is governed by the following fundamental axiom: 1. Given an undifferentiated initial state \( |NT\rangle \) in a Hilbert space \( \mathcal{H} \), and an emergence operator \( E \) acting on \( \mathcal{H} \), the system evolves in time through a unitary operation \( U(t) \). This process leads to a monotonic increase in the complexity measure \( M(t) \), reflecting the inevitable emergence and differentiation of states.

## 1. Introduction. The **Quantum Emergence Model** aims to unify concepts from quantum mechanics, information theory and cosmology through the introduction of an **emergence operator** \( E \) and an **initial null-all state** \( |NT\rangle \). This approach makes it possible to describe the transition from an undifferentiated, non-dual state to emergent, differentiated states, providing a theoretical basis for understanding the origin of complexity, the arrow of time, and the structure of the universe.

## 1. Introduction The **quantum emergence model** aims to unify concepts from quantum mechanics, information theory, and cosmology through the introduction of an **emergence operator** $$E$$ and an **initial null-everything state** $$|NT \rangle$$. This approach enables the description of the transition from an undifferentiated, non-dual state to emerging, differentiated states, providing a theoretical basis for understanding the origin of complexity, the arrow of time, and the structure of the universe.

The Logical Pipeline of Relations between Entities aims to formalize Axioms Autologically through a combinatorial movement across three levels. The layers combine to provide a nonlinear movement consistent with the exponentiality of revealed possibilities. Plausible effects are considered, and the result is followed until we understand how to do everything necessary for a new version: