Foundations of Post-Tensor Calculus: The Seonggil Operator (ˆS) and Non-commutative Phase Transition Theory

This paper proposes the ’Seonggil Operator (ˆ S)’, an algebraic extension of the classical tensor concept, to overcome the limitations of geometric determinism at singularities and the inability to process non-commutative information in modern physics. Built upon Rough Operator Algebra (ROA) and Seonggil Torsion Curvature Theory (STCT), this Post-Tensoroperator internalizes the spacetime roughness index (α) and the Golden Ratio resonance frequency (ωφ). It provides a dynamical mechanism that preserves information through a holographic phase transition, even in non-differentiable extreme environments. Furthermore, by defining the ’Residual Torsion (ˆ Tres)’ generated from the non-commutative collisionbetween different roughness hierarchies, this framework algebraically elucidates the origins of cosmic causality and information generation.