arXiv:2610.23053v1 Announce Type: new Abstract: Vision-language-action (VLA) models and world-action models (WAMs) have emerged as promising approaches to robot learning and control. These models integrate multimodal perception and action generation, but differ in their architectures, output representations, and learning objectives. In this paper, we present a unified, architecture-agnostic probabilistic formulation for embodied foundation models based on stochastic kernels. Our formulation has two key components: (i) we characterize embodied foundation models through conditional output distributions and action-realization kernels, which together induce task-conditioned robot policies; and (ii) we connect pretraining and post-training to policy deployment, accommodating model-specific pretraining objectives and offline or online post-training strategies. We instantiate the formulation for OpenVLA and DreamZero, covering autoregressive action-token prediction and joint video-action generation.