Structural Incompleteness of the Banach–Tarski Paradox: A Proof via Gaussian Complex Manifolds and Magnitude Conservation

This paper presents a formal disproof of the physical realizability of the Banach–Tarski paradox by introducing a complex-valued metric to 3D manifolds. We propose that points within a manifold are defined by a Gaussian structural identity z = a + bi. We prove that while the Axiom of Choice (AC) allows for the reassembly of a sphere into two identical volumes, it fails to conserve the imaginary magnitude of the system. This leads to a ”Structural Residue” of approximately 5.15%, which represents the fundamental limit of geometric duplication.