The UF Operator – Definition, Construction, and Boundaries

Abstract

This paper defines the UF operator (Unrestricted Function) as a new algebraic framework for implicit domain restriction. The operator allows functions to be written without explicit conditional statements by embedding domain restrictions into the algebraic structure via a formal undefined value ⊥ and a set of propagation axioms. We present a complete axiomatic system, including a forbidden-simplification rule that preserves the filtering mechanism. A comprehensive catalog of algebraic filters is provided for continuous intervals, discrete sets, modular arithmetic conditions, and non-elementary sets such as the integers (constructed via the Gamma function) and lattice sets (constructed via Theta functions and Weierstrass ℘). The framework is extended to integration, summation, and contour integration; a detailed analysis of the branch-cut failure in the complex plane is included. Finally, we formulate the UF Universality Conjecture as an open problem. This paper is self-contained and includes full appendices.