December 4, 2025Open Access

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

Key Points

Computational complexity is shaped by the fractal dimension, linking it to quantum mechanics and fluid dynamics.
The Geometric Uncertainty Principle establishes an important relationship between energy potential and fractal dimension.
The axiomatic framework provides insights into the singularity that challenges traditional smoothness in analysis across disciplines.
This approach may redefine the understanding of critical stability thresholds in physics and inform future mathematical models.

Abstract

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.

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