Gnomon4D: Learning Where and When to Focus for Geometric Motion-Grounded World-Action Modeling

World-action models (WAMs) use video-generation priors to jointly predict future observations and robot actions. Yet RGB futures entangle physical state changes with appearance, while frame-wise depth and point maps lack temporal correspondence. Existing 4D approaches either keep motion implicit in visual features or translate predicted traces into actions through a separate interface. We present Gnomon4D, a WAM that jointly generates future video, robot actions, and GeoDelta, a video-aligned 4D geometric motion field encoding the cross-frame image-plane displacement and depth change of embodiment and object points. By placing embodiment motion and object responses in a shared space, GeoDelta exposes their motion relations to the action expert at inference. Interaction-Salient Attention (InSAttn) uses these relations to focus spatial readout on task-relevant interaction regions, while Transition-Anchored Aggregation (TransAgg) aggregates evidence around changes in their relative motion, such as grasp and release. GeoDelta describes surface motion independently of embodiment-specific controls, providing a shared human--robot quasi-action for action-free pretraining on EgoManip-5000H followed by robot mid-training on Robot-30H. Gnomon4D leads on RoboTwin~2.0, VLABench, and RoboDojo, and outperforms the strongest real-robot baseline by 7.0 and 16.0 percentage points under in-distribution and out-of-distribution settings, respectively.