Rough Operator Algebra: A Unified Framework for Non-Commutative Geometry and the Resolution of Singularities via α-Extended Operators

Traditional linear algebra and differential geometry rely heavily on the assumption of “smooth- ness” (α = 1). However, this Tyranny of Smoothness encounters fundamental limitations when addressing the inherent roughness of nature, such as quantum uncertainty, turbulence, and grav- itational singularities. In this paper, we propose a new mathematical system, Rough Operator Algebra (ROA), which incorporates the roughness index α ∈ (0, 1] as a fundamental variable of algebraic operations. By adopting the Geometric Uncertainty Principle (E · α = κ) as a primary axiom, we define the α-Extended Operator (⊛ ) that enables operations between matrices of different dimensions and controls non-commutativity via geometric area terms. Furthermore, we derive the Sunggil Field Equation and demonstrate that singularities are not endpoints of information but topological phase transitions into roughness noise, thereby resolving the black hole information paradox.