Advanced Rough Calculus (ARC): Breaking the Continuum Hypothesis via Rough Operator Algebra and Topological Torsion

Classical calculus, founded on the Cauchy-Weierstrass ϵ−δ limit and the hypothesis of an infinitely divisible continuous manifold (α = 0), encounters severe pathological breakdowns when confronting fractal geometries, quantum discretization, and cosmological singularities. In this paper, we establish a novel mathematical framework termed Advanced Rough Calculus (ARC), built upon the foundational principles of Rough Operator Algebra (ROA) [6]. By replacing infinitesimal limits with algebraic phase-transition resolutions and introducing topological torsion tensors, we resolve historical paradoxes such as the non-differentiability of the Weierstrass function [1] and reformulate the foundations of differential and integral calculus. Furthermore, we demonstrate that ARC circumvents the Continuum Hypothesis by transforming the continuous manifold into a countable space of coarse-grained resolvable states. Finally, we extend ARC to the Seonggil Field Equations (SFE) framework [7], demonstrating that cosmological dark matter and dark energy emerge naturally as manifestations of internal torsion tension and boundary topological residuals, respectively.