Exponential Multiplication: A Binary Operation for Harmonic Scaling Systems

I introduce exponential multiplication, a binary operation defined as a ⊗ b = a · 2^b, and analyze its algebraic properties. This operation provides a formal framework for modeling systems that exhibit harmonic doubling patterns, including musical octaves, human vocal ranges, the visible light spectrum, binary computing architectures, and biological cell division. We prove fundamental properties including non-commutativity, non-associativity, identity elements, absorption, additive-associativity, and right distributivity. The operation yields the notable result 1 ⊗ 1 = 2. Five core properties are formally verified in Lean 4 with Mathlib (zero sorry, confirmed live.lean-lang.org). A cross-domain analysis reveals that sound and light obey the same scaling operation at frequency scales separated by a factor of ~10^12.