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

This thesis begins with a fundamental doubt regarding the definitions of "0" and "∞" in modern mathematics dealing with addition and multiplication. It extends the standard number line into a three-dimensional spherical model, reconceptualizing numbers as three-dimensional coordinates. By performing an information expansion of "existence," such as "R = 0, i = ∞" and "R = ∞, i = 0," it explores the possibility of defining singularity operations, including division by zero. This work serves as a philosophical sketch of number theory, representing a preliminary stage toward rigorous mathematical proof. Part 1: Spatial Models and the Redefinition of the Zero Point: Based on Leibniz's thought, the "Monad" is established as the minimum unit of existence. By introducing "Five Universe Models" that monitor geometric symmetry, this section reinterprets "0" not as a mere absence but as a dimensional "exit (portal)" and discusses the necessity of an absolute zero point, the "Universal Mapping Absolute (UMA) Monad," which integrates addition and multiplication without contradiction. Part 2: The Spherical Model and the Existence Axis: The space of numbers is modeled as a solid sphere, adding an "existence axis" to the real number axes (the additive and multiplicative circles). It proposes that the standard number line is a "mapping" that extracts only specific regions from this vast information space and discusses the resulting lost information. Part 3: Procedures of Reflux Calculus: As a specific calculation procedure, this section explains the process of converting numbers into binary and mapping them onto the existence axis. Through this, arithmetic operations including division by zero and infinity---typically undefined in standard mathematics---are defined as "reflux calculus" based on consistent rules. These behaviors are demonstrated alongside a simple calculation program. Conclusion and Outlook: It is predicted that if the model in this thesis is axiomatized in the future, it will provide a new geometric perspective for the unification of relativity and quantum theory and grant fundamental meaning to the origin of non-commutative geometry.