The Axiomatic Foundation of Non-Commutative Set Theory: Arithmetic Topological Discontinuity and Macroscopic Contraction in ROA/STCT Framework

This paper challenges the implicit assumption of ‘Flat Completeness’ embedded within Cantor’s diagonal argument [1] and establishes a novel set-theoretic axiomatic foundation grounded in Rough Operator Algebra (ROA) [9] and Seonggil Theory of Composite Torsion(STCT) [11]. We redefine the transfinite cardinal leap (ℵ0 → c) as a dynamic, first-order topological phase transition occurring at arithmetic resonance points. Furthermore, we prove that Russell’s Paradox [5] geometrically collapses into an open helical trajectory due to the non-zero curvature torsion tensor of non-commutative space, achieving self-regulationwithout artificial axioms. Crucially, by introducing a non-commutative Dirac measure and a spectral delta function, we demonstrate that the topological void required for intermediate cardinal states (ℵ1) is entirely blocked, thereby resolving the Continuum Hypothesis (CH)algebraically [4]. Finally, by deriving a power-law spectral decay envelope based on theexplicit formulas of analytic number theory [6] and fractal scale invariance, we prove that this non-commutative framework seamlessly contracts to the classical Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) [2, 3] under the macroscopic flat-space limit.