Mathematical Foundations of Non-Identity Calculus: Rough Operator Algebra (ROA) and Seonggil Torsion (STCT) in Process-Relational Geometry

To overcome the axiomatic limitations of smooth manifolds and absolute identity that form the foundation of modern mathematics, this paper proposes the mathematical foundations of Process-Relational Geometry, which posits events and transformations, rather than points, as the fundamental entities of reality. To rigorously formulate the rate of change indynamical, quantum, and complex systems where the strict identity of objects is not conserved, we establish the fundamental operational rules of Non-Identity Calculus based on Rough Operator Algebra (ROA) and the Seonggil Torsion-Curvature Tensor (STCT). The classical infinitesimal limit is replaced by the informational stability of the rewriting process, and through torsion-corrected difference operations, we derive the violation of linearity, the Non-Identity Leibniz rule, and the path-dependent chain rule. This framework provides novel analytical tools for Seonggil Matrix Theory (SMT) and Seonggil Field Equations (SFE) toward macro-micro integration.