Enhanced Uncertainty Modeling through Neutrosophic MR-Metrics: A Unified Framework with Fuzzy Embedding and Contraction Principles

Abstract

This paper explores the fundamental connections between Neutrosophic MR-Metric Spaces (NMR-MS) and classical Fuzzy Metric Spaces (FMS). We present three key theoretical contributions: (1) an embedding theorem showing how any FMS can be systematically incorporated into an NMR-MS framework, (2) a fixed point theorem for contraction mappings in complete NMR-MS that generalizes the fuzzy Banach contraction principle, and (3) a characterization of sequence convergence in NMR-MS that reveals its stricter requirements compared to FMS. Through concrete examples and applications in machine learning classification, robotic path planning, and medical image reconstruction, we demonstrate how the additional structure of NMR-MS - particularly its explicit handling of truth ($\mathcal{T}$), falsity ($\mathcal{F}$), and indeterminacy ($\mathcal{I}$) components offers enhanced modeling capabilities     for uncertain systems. The compatibility conditions between the MR-metric ($M$) and neutrosophic components are shown to be crucial for maintaining theoretical consistency while enabling practical applications.