The UMA Monad Theory: A Number Philosophy Toward a Unified Theory of 0 and ∞

This thesis begins with a fundamental questioning of the definitions of "0" and "∞" in modern mathematics, which deals with addition and multiplication. By extending the conventional number line into a three-dimensional spherical model, we perform an informational expansion of "existence," such as "R = 0, i = ∞." While existing frameworks like the Riemann sphere and Wheel algebra attempt to address division by zero based on the premise of the existence of numbers, this study takes a more foundational position: numbers are materialized only after designing the geometric symmetry of existence itself. Rather than a rigorous mathematical proof, this paper serves as a sketch of spatial design philosophy, anticipating a new geometric perspective toward the unification of general relativity and quantum theory. Part 1: Spatial Models and the Redefinition of the Zero Point Based on Leibniz's philosophy, the "Monad" is established as the minimum unit of existence. By introducing "Five Universe Models" that monitor geometric symmetry, "0" is reinterpreted not as a mere absence but as a dimensional "portal" (exit). This section argues for the necessity of the "Universal Mapping Absolute (UMA) Monad," an absolute zero point that integrates addition and multiplication without contradiction. Part 2: The Spherical Model and the Existence Axis The conventional number line is extended into a three-dimensional spherical model by adding an "Existence axis" to the real number axes (additive and multiplicative circles), reconceptualizing numbers as three-dimensional coordinates. It is conjectured that the standard number line is a "mapping" extracted from a specific region within this vast information space, and the implications of the lost information are discussed. Part 3: Procedures of Reflux Calculus As a specific operational procedure, this section explains the process of converting numerical values into binary and mapping them onto the existence axis. Through this, a conceptual framework for arithmetic operations including division by zero and infinity---which remain undefined in conventional mathematics---is presented along with a simplified computational program. Finally, it is predicted that quantum theory is a discipline dealing with the existence axis, demonstrating the potential for a new geometric perspective toward unification with the theory of relativity.