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

TL;DR

This paper establishes a fixed point theorem in quantum metric spaces using the $L^2$ norm, highlighting structural advantages over fuzzy frameworks, especially in interference and phase sensitivity.

Contribution

It introduces a fixed point theorem for contraction maps in quantum metric spaces, demonstrating their structural coherence and advantages over fuzzy metric spaces.

Findings

Fixed points exist and are unique for contraction maps in quantum metric spaces.

Quantum metric spaces preserve interference and phase sensitivity.

Fuzzy metric spaces lack interference and topological protection.

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.