An Asymmetric Invertibility Law Constructed Independently of the Classical Group Axioms

This paper investigates an asymmetric invertibility law constructed without assuming the classical axioms of group theory. The central motivation is to examine whether a transformation analogous to the familiar transposition of terms in an equation can be formulated independently of conventional algebraic structures. Rather than presupposing commutativity, associativity, identity elements, or classical inverse elements, this work focuses on the directional behavior of reconstruction: recovering an arrival point from a given starting point, and recovering a starting point from a given arrival point. A mathematical model is developed using a positive real domain, a fixed positive constant, and an auxiliary record that tracks the number of zero-addition events. The model introduces a forward operation governed by a baseline function and an exponential factor. Within this framework, the paper establishes several results: An explicit reconstruction formula for the arrival point. A characterization of the range of the forward operation. Conditions under which arrival-point reconstruction is uniquely solvable. A demonstration that reverse reconstruction is not globally unique. The existence of a positive maximum for the reverse reconstruction function. An analysis of how the fixed constant affects the forward operation. A reciprocal symmetry property of the baseline function. An additive rule governing the auxiliary event record. These results establish a mathematical distinction between forward reconstruction and reverse reconstruction within the proposed model. The paper does not claim to have established a complete algebraic system or a formal logical independence theorem. Instead, it provides a concrete mathematical framework in which asymmetric reconstruction can be studied without invoking the classical group axioms. The broader objective is to develop a foundation for understanding asymmetric invertibility as an independent mathematical structure, with further work required to formalize the underlying transposition law and extend the model beyond the positive real domain. This work also introduces Inversion Theory as the name of the proposed mathematical framework. It represents an independent system of mathematical operations characterized by asymmetric invertibility and directional reconstruction. It is not defined as a variant of group theory, but as a distinct algebraic structure whose fundamental laws and properties are intended to be developed independently.