The Foundations of Roughness Calculus: An Axiomatic Framework for Non-Differentiable Geometry and the Unification of Complexity Classes

We present a new mathematical system, ”Roughness Calculus”, designed to rigorous analyze systems where classical smoothness breaks down. While traditional calculusrelies on the limit of differentiable manifolds (α = 1), our framework treats the Roughness Index α ∈ (0, 1] as a fundamental dynamic variable. We establish the Geometric Uncer- tainty Principle (E · α = κ) as the primary axiom, linking the energy potential of a system to the fractal dimension of its path. From this axiom, we derive the ”Sunggil Derivative” basedon p-variation density. This axiomatic system provides a unified proof for the Millennium Problems: identifying α = 1/2 as the critical stability threshold for Quantum Mechanics (RH) and Fluid Dynamics (NSE), and α → 0 as the singularity of Computational Complexity (P vsNP). This paper serves as the foundational treatise for the author’s research series on Rough Spectral Geometry.