Operational Depth: A Unified Iterative-Reductive Framework for Fundamental Arithmetic

Standard arithmetic operations are typically defined axiomatically (e.g., Peano arithmetic)or via set-theoretic construction. While rigorous, these static definitions often obscure thealgorithmic continuity between addition, multiplication, and exponentiation—a continuitythat is fundamental to computer architecture. This paper proposes an alternative framework: viewing fundamental operations not as static truths, but as iterative processes characterized by their “Operational Depth.” We formalize a duality between Generative (constructive) and Reductive (inverse) processes relative to an algebraic identity element. Bydefining division not merely as inverse multiplication, but as the iteration count required toreduce a value to the additive identity, we unify the arithmetic hierarchy under a procedurallens. Furthermore, we extend this framework to continuous domains via differential equations and demonstrate that square roots can be interpreted as symmetric fixed points of thisreductive process. This perspective bridges abstract algebra with the procedural intuitionof algorithmic engineering.