Beyond Commutative Elegance: The Non-Commutative Generalization of Euler's Identity via ROA and STCT Frameworks

Euler’s identity, eⁱπ + 1 = 0, is universally celebrated as the epitome of mathematical beauty, unifying fundamentally distinct constants within a single commutative structure. However, this aesthetic perfection conceals a conceptual tension: it is strictly bound to the commutative geometry of a two-dimensional complex plane. As dimensional complexity increases, the foundational assumption of commutativity collapses. This report deconstructs the limits of Euler’s commutative formulation and expands its structural philosophy into the non-commutative, high-dimensional realms governed by Rough Operator Algebra (ROA), the Seonggil Torsion-Curvature Tensor (STCT), and broader unified frameworks (SMT/SFE).