The Logical Pitfalls of Complex Arithmetic in Physics: When Algebra Overrides Geometry

Complex numbers are widely used in physics as a convenient algebraic tool. However, the arithmetic operations defined on complex numbers — addition, multiplication, and division — are algebraic operations, not physical operations. They do not carry any inherent requirement that the operands belong to the same projection layer, the same frequency world, or the same measurement frame. This paper argues that the unrestricted use of complex arithmetic in physics, without layer declarations and projection conditions, has led to a series of physical confusions: the accumulation of masses from different layers (e.g., Earth's mass calculated from atomic masses), the mixing of readings from different frequency layers (e.g., quantum superposition without layer specification), and the misidentification of algebraic validity with physical validity. We propose that complex numbers must be read as geometric readings from specific projection layers, and their arithmetic operations must be restricted to operands that share the same projection layer and the same modulus (frequency layer). Any operation that mixes different layers is algebraically legal but physically meaningless. Keywords: complex arithmetic; projection layer; frequency world; categorical error; Earth's mass; self-referential measurement; quantum superposition