Revealing the Topology invariance of vectorial vortex beam in complex media
Orbital angular momentum (OAM), a topological degree of freedom of light, is theoretically invariant under continuous deformations; yet, its physical observability degrades precipitously in complex media, creating a fundamental "topology-observability gap." Here, we propose a novel paradigm for topological measurement based on the non-separable coupling between polarization and topological features in vectorial vortex beam. By constructing a topological non-separability measure derived from global Stokes fields, and integrating it with a physics-guided machine learning calibration framework that combines Bayesian Gaussian process regression with XGBoost-driven adaptive model selection, we achieve high-fidelity identification of topological features up to 200. Crucially, this robustness persists even when beam intensity and phase structures are completely distorted by extreme complex media, including strong atmospheric turbulence, oceanic turbulence, and high-temperature jet exhausts. This approach overcomes the dual bottlenecks of limited accessible OAM modes and susceptibility to perturbations that constrain conventional methods. Our work not only bridges the fundamental divide between topological theory and physical observability, but also establishes a robust framework for the reliable deployment of high-dimensional OAM in real-world complex environments, promising exciting advancements for wireless optical communications, remote topological sensing, and classical analogs of quantum information protocols.