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 property of the forward operation is analyzed as a formal model of path information loss: once a system reaches the absorbing state, the inverse cannot reconstruct the trajectory that led there. The paper concludes with applications to coherence decay modeling and discusses the implications of structural irreversibility for systems undergoing state collapse.