NEUTROSOPHIC MR-METRIC SPACES: THEORY, COMPACTNESS, AND APPLICATIONS IN MACHINE LEARNING

Abstract

This paper introduces a comprehensive framework for Neutrosophic MR-Metric Spaces (NMR-MS), combining the ternary structure of MR-metrics with the uncertaintymodeling capabilities of neutrosophic logic. We establish the fundamental theoretical foun-dations, including a first-order axiomatization of NMR-MS and a detailed analysis of com-pactness properties. Our results demonstrate that while the full theory T_{NMR} is not compact, the restricted theory T_{NMR}^{K} for bounded metrics is compact and admits ultraproduct constructions satisfying Loś's Theorem. Building on this theoretical framework, we developa novel Neutrosophic Metric Learning paradigm that generalizes standard metric learningby incorporating truth, indeterminacy, and falsity memberships.