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

This paper examines mathematical analogies between the Z-U dual iteration system derived from the balance equation ln f + (π/2)f = 0 and certain structures 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^{-πz_n/2})/2 and the U-space iteration u_{n+1} = √u_n e^{-πu_n/4} [1, 2]. These two spaces are connected by the transformation u = e^{-πz/2} and share the same fixed point f. We observe that several mathematical elements of this structure—such as the exponential mapping, nonlinear terms, domain exclusions, and multivalued inverses—exhibit formal similarities to concepts in general relativity (geodesics, curvature, singularities, horizons) and quantum mechanics (wave functions, measurement projection, uncertainty). These observations are purely mathematical and do not constitute a physical theory.