Operationalizing Generalized Symmetries: A Falsifiable Dictionary for Anyon Halos and Stretched Exponential Splitting in Moiré Superlattices

The experimental realization of Fractional Chern Insulators (FCIs) in twisted moiré superlattices has opened a new frontier for topological physics, yet the classification of their underlying generalized symmetries remains theoretically abstract. This study operationalizes the framework of non-invertible symmetry defects into concrete, falsifiable experimental protocols. We propose a structural isomorphism between the “Anyon Density-Wave Halo” in condensed matter and “Wet Hair” in high-energy holography, positing that both phenomena represent the non-local encoding of symmetry charges. Using computational simulations parameterized for twisted MoTe$_2$, we demonstrate that the spatial radius of the anyon halo serves as a robust diagnostic for the quantum dimension of the defect. Sensitivity analysis confirms that this distinguishability persists across linear, logarithmic, and power-law scaling models ($p \ll 10^{-100}$) and remains robust against correlated disorder mimicking twist-angle inhomogeneity. Furthermore, we validate the “Stretched Exponential” ground state splitting predicted for systems with long-range interactions, confirming that the scaling exponent $\gamma=1.5$ is recoverable even in limited-range datasets ($L=20-50$) accessible to current experiments. These findings provide a rigorous “Halo-Hair Dictionary” for identifying non-Abelian topological orders using standard Scanning Tunneling Microscopy (STM) and transport techniques.