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.