Advanced Rough Calculus (ARC) and the Dynamic Rough-Speed-of-Light (DRSL) Framework: Emergent Lorentz Invariance and Spacetime 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. Extending ARC to the Seonggil Field Equations (SFE) framework [8], we show thatcosmological dark matter and dark energy emerge naturally as manifestations of internal torsion tension and boundary topological residuals. Finally, we resolve the foundational tensions of Einstein’s constancy of the speed of light by introducing the Dynamic Rough Speed-of-Light (DRSL) framework, proving that c is not an absolute fundamental constant, but an emergent low-energy symmetry (α → 0) governed by spacetime roughness and topological torsion tensors.