The LOEANE (Linearity of Existence and Non-Existence) theorem offers a novel mathematical framework that redefines the binary notions of existence and non-existence by modeling them as points on a continuous spectrum. This article presents the key equations central to the LOEANE theorem, which articulate the dynamic interplay between positive energy, negative energy, and the processes of matter creation and annihilation. Through the Reality Equation and supporting inequalities, the theorem reveals an inherent asymmetry that favors existence, resonating with observable cosmological and quantum properties. Furthermore, operator-based quantum mechanical formulations are introduced to describe the underlying particle dynamics that support the theorem's core principles. The LOEANE theorem provides a conceptual and mathematical structure that deepens our understanding of physical reality, energy balance, and the non-binary nature of ontological states in the universe.