Geometry-Centered 3D Latent World Models for Growing Surfaces

Many physical systems do not merely move or deform; they grow, adding material and changing the geometry that a world model must represent. Existing world models are typically optimized for pixel prediction, reward prediction, or fixed-support physical dynamics, leaving open how to model systems whose geometric state expands over time and whose future morphology depends on hidden material response. We introduce FOLIAGE, a geometry-centered latent world model for growing surfaces. FOLIAGE represents mature regions as a compact scaffold while allocating higher-resolution state to active growth fronts, allowing the model to focus capacity where new material and near-future change occur. It further separates observation, action, and privileged physics: heterogeneous RGB, point-cloud, and mesh observations are fused into a deployable geometric state; material controls condition only the latent dynamics; and hidden physical energies guide training without being required at deployment. To evaluate this setting, we introduce SURF-GARDEN and SURF-BENCH, providing controlled counterfactual branches, dense cross-modal correspondences, hidden physical signals, and stress tests for growing-geometry state learning. FOLIAGE reduces inverse-material error by $\approx 40\%$ and 5-step mesh forecasting Chamfer error by $\approx 20\%$ relative to strong baselines, while improving cross-modal retrieval by $\approx 25\%$ mAP@100. Stress tests show graceful degradation under sensor loss and correspondence corruption. Transfer experiments on real plant and dynamic-3D data indicate that the learned geometry-centered state generalizes beyond the simulator. We will release code and data upon acceptance.