Fixed Points in Quantum Metric Spaces: A Structural Advantage over Fuzzy Frameworks
Nicola Fabiano
TL;DR
This work develops a fixed-point theory for quantum metric spaces using the $L^2$-based distance $d_Q$ between normalized Gaussian states. It proves existence and uniqueness of fixed points for contractions that preserve Gaussian structure, with global convergence of iterates. The paper argues that quantum metric spaces intrinsically capture interference, phase, and topological features absent in fuzzy metric frameworks, offering a deeper, physically grounded foundation for reasoning under uncertainty. It also positions these results within analysis, operator theory, and AI foundations, and contrasts them with fuzzy logic critiques. The findings suggest that quantum geometric reasoning provides a more coherent framework for modeling intrinsic uncertainty in cognitive and physical systems.
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: $d_Q(ψ_1,ψ_2) = \|ψ_1 - ψ_2\|$. 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.