GeoCausal: A Geometric-Causal World Model Integrating System 1 and System 2 for Embodied Reasoning

This paper introduces GeoCausal, a novel theoretical framework for embodied artificial intelligence that unifies geometric deep learning with dual-process cognitive architectures. We propose that truly generalizable physical reasoning requires: (1) a structured geometric representation encoding object-centric entities with SE(3)-equivariant properties; (2) a causal world model that predicts state transitions under physical constraints; (3) a dual-process controller where a slow-thinking System 2 generates symbolic plans and a fast-thinking System 1 executes parameterized motor skills. Our key contribution is a mathematically grounded integration of these components into a single trainable architecture, accompanied by a novel three-phase training regimen that enables joint optimization of representation, prediction, and control. We prove theoretical guarantees on error bounds and sample efficiency, and provide fully specified pseudocode alongside a minimal runnable implementation. GeoCausal bridges the gap between abstract symbolic reasoning and low-level physical interaction, offering a principled path toward agents that understand and manipulate the physical world. Keywords: Geometric Deep Learning, World Models, Causal Reasoning, Dual-Process Theory, Embodied AI, Reinforcement Learning