The Bridge Equation: A Structural Identity Connecting the Born Rule, RMS Power, and the Gaussian Integral

Abstract A single algebraic operation--the multiplication of a complex quantity by its conjugate, z · z* = |z|²--appears across quantum mechanics, electrical engineering, signal processing, and pure mathematics, in each case serving the same structural function: converting a phase-bearing amplitude into a real-valued observable. This paper identifies the operation explicitly, traces it through five domains where it is taught under different names, and demonstrates that the n-dimensional Gaussian integral ∫e⁻|x|² dⁿx = πⁿ² provides a unifying capstone: the exponential base e acts as the operator applied to the bridge equation |x|², while the circle constant π emerges as its dimensional output. Euler's identity eiπ = −1 is reinterpreted as this operation in its most compressed form. No new mathematics is introduced. The contribution is pedagogical: recognizing a single structural identity that connects results typically taught in isolation.