On the Separation of Number and Operation: A Minimal Arithmetic Machine

This paper proposes a minimal foundation for arithmetic by separating numerical generation from independently specified arithmetic operations. Four primitive generative acts—progression, reflection, refinement, and turning—are proposed as the foundations of the natural, integer, real, and complex number systems, with breadth, opposition, depth, and phase as their corresponding structural characteristics. The paper distinguishes number generation from numeral representation, numerical identity from operational expressions, and primitive operations from relations among operations. Addition, subtraction, multiplication, and division constitute an independently defined operational layer. Together, the numerical and operational layers form the minimal arithmetic machine. Across twenty sections, equality, order, operational laws, divisibility, parity, primality, rationality, algebraicity, equations, polynomials, and higher algebraic structures are discussed as subsequent relational or organizational layers. The guiding principle is to define the minimum at the foundation and derive further structure through the interaction of numerical identities and operations.