Asymmetric Singular Coordinate System (ASCS): A Structural Reformulation of the Number Line via Boundary Singularity and Non-Linear Asymptotics

The traditional Cartesian coordinate system exhibits limitations when modeling systems with phase discontinuities or boundary shocks. This paper proposes the Asymmetric Singular Coordinate System (ASCS) featuring three core characteristics: the origin acts as a boundary singularity severing symmetry, the positive domain expands linearly toward infinity, and the negative domain initiates from negative infinity adjacent to the boundary before increasing non-linearly to asymptotically approach a saturation bound at extreme distances. The study establishes custom algebraic operations, analyzes differential behaviors, and demonstrates its application as a threshold-sensitive activation function for artificial neural networks.