Action Manifold Smoothing: A Lipschitz Pathway Perspective on High-Dimensional Reinforcement Learning

High-dimensional continuous control remains challenging in deep reinforcement learning, where algorithms like TD3 and SAC often collapse. We propose a unifying Lipschitz Pathway framework that decomposes instability into four amplification stages, namely action parameterization ($L_1$), dynamics sensitivity ($L_2$), Q-network curvature ($L_3$), and temporal-difference (TD) target stability ($L_4$), where errors compound multiplicatively along the learning pipeline. Our analysis identifies a discrete-continuous mismatch as the root cause: value functions trained from sparse point samples must generalize over continuous manifolds, leading to multiplicative error amplification along the pathway. To address this, we introduce Action Manifold Smoothing (AMS), which replaces point-wise TD targets with orthogonally-sampled neighborhood averages, jointly regularizing $L_3$ (via implicit Laplacian smoothing) and $L_4$ (via local manifold supervision). We further characterize when Lipschitz-constrained Q-networks and geometric action priors are beneficial based on task structure. Empirically, AMS enables both TD3 and SAC to achieve over 400 reward on the 38-D Dog Run task within 1M steps, where baselines fail. These results validate the Lipschitz pathway as a principled framework for diagnosing and solving stability bottlenecks in high-dimensional control.