Fixed Points in Quantum Metric Spaces: A Structural Advantage over Fuzzy Frameworks

Abstract

We prove an existence and uniqueness theorem for fixed points of contraction maps in the framework of quantum metric spaces, where distinguishability is defined by the $L^2$ norm: $d_Q(\psi_1,\psi_2) = \|\psi_1 - \psi_2\|$. The result applies to normalized real-valued Gaussian wavefunctions under continuous contractive evolution preserving the functional form. In contrast, while fuzzy metric spaces admit analogous fixed point theorems, they lack interference, phase sensitivity, and topological protection. This comparison reveals a deeper structural coherence in the quantum framework -- not merely technical superiority, but compatibility with the geometric richness of Hilbert space. Our work extends the critique of fuzzy logic into dynamical reasoning under intrinsic uncertainty.