Z-U Space Duality: A Mathematical Correspondence with General Relativity and Quantum Mechanics

This paper examines the mathematical correspondence between the Z-U dual iteration system derived from the balance equation ln f + (π/2)f = 0 and several core concepts in general relativity and quantum mechanics. The equation defines a unique positive real constant f = (2/π)W₀(π/2) ≈ 0.47454, leading to a pair of dual iterative systems: the Z-space iteration z_{n+1} = (z_n + e^{-π z_n/2})/2 and the U-space iteration u_{n+1} = √(u_n) e^{-π u_n/4}. These two spaces are connected by the transformation u = e^{-π z/2} and share the same fixed point f. We demonstrate that several mathematical elements of this structure—such as the geodesic mapping, nonlinear terms, domain exclusions, path superposition, projection transformation, shared fixed points, and multivalued inverses—correspond precisely to concepts in general relativity (geodesics, spacetime curvature, singularities, horizons) and quantum mechanics (wave functions, measurement collapse, quantum entanglement, uncertainty principle). This correspondence offers a unified mathematical perspective on the foundations of these physical theories.