The Mathematics of Anomalous Stability: Fractional Landau Inequalities and Their Role in Deep Learning

This study advances the mathematical understanding of \textbf{fractional Landau inequalities} by connecting fractional calculus with the stability of deep neural operators. We address key challenges in optimizing constants, understanding function space geometry, and applying these ideas to neural networks. Our work refines existing fractional Taylor estimates to produce sharper gradient bounds for functions in high-dimensional spaces, extending classical inequalities to fractional Sobolev spaces. For fractional orders between 2 and 4, we introduce novel geometric measures \textbf{fractional curvature} and \textbf{fractional torsion} to capture non-local behavior, leading to tighter and more dimensionally aware bounds. These results are further generalized to deep neural networks, where we prove stability under input perturbations using fractional smoothness. Applications span fractional partial differential equations, operator learning, and anomaly detection in complex systems. By unifying classical gradient analysis with fractional dynamics, this framework provides new tools for studying systems with anomalous diffusion or irregular geometries.