00: The Recursive Zero and the Layered Abstraction of Number

Title:00: The Recursive Zero and the Layered Abstraction of Number Description:This paper introduces a structurally distinct mathematical entity: recursive zero, denoted as 00. Unlike classical zero (0), which marks termination and nullity, 00 functions as a recursive neutral — a boundary that enables logical recombination and transformation rather than collapse. We demonstrate the operational distinction between 0 and 00 through two formal models: a recursive logic engine and an automata-based system. The paper also explores the implications of this distinction across computation, formal logic, automata theory, algebra, quantum systems, and programming language design. The extended number line is revealed to possess a structural asymmetry across the 00|0 boundary, where the domain left of 00 remains abstract and open, and the domain right of 0 is operationally closed. This asymmetry offers a new lens through which to understand definability, closure, and recursion in foundational mathematics. This is a foundational contribution intended for researchers in mathematics, theoretical computer science, formal logic, and symbolic systems. It may also be of interest to philosophers of mathematics and AI theorists exploring machine-structured reasoning.