Foundations of Non-commutative Analysis: The Seonggil Preserving Limit Operator and Weyl Stress

For over 300 years, classical calculus has relied on the limit operator (lim), which inherently discards infinitesimal topological information to achieve scalar convergence. This paper introduces the Seonggil Preserving Limit Operator ( limS) within the framework of Rough Operator Algebra (ROA). We prove that the discarded infinitesimal is not zero, but a non-commutative residual torsion tensor(ϵS) which manifests as Weyl tensor stress. By replacing lossy classical limits with tensor-preserving operations, we provide a rigorous bridge between analytic number theory and global cosmological dynamics.