Exponential Division and Structural Irreversibility: The Inverse Operation a ⊘ b = log₂(b/a) and What Its Domain Restrictions Encode

This paper analyzes the inverse of exponential multiplication, defining exponential division as a ⊘ b = log₂(b/a) for a, b ∈ ℝ₊. Unlike its forward counterpart a ⊗ b = a · 2^b, which is defined across all real numbers, the inverse operation is restricted to strictly positive reals — a domain asymmetry that is not a technical inconvenience but a structural property with measurable consequences. It is proved that exponential division is a right inverse but admits no left inverse, establishing a fundamental directionality within the operation pair. The inverse is interpreted as a measurement of informational distance in a doubling metric, connecting it to Shannon's binary entropy framework. The zero-absorption…