What does a world model learn from physical exploration, without any linguistic supervision? We argue the answer is organized by a single principle: the geometric structure of the physical world. Training a VAE-based world model on random embodied exploration, we find that its latent space develops spatial semantic structure that mirrors physical geometry -- direction accuracy 0.677+-0.029 versus 0.547 for a randomly initialized encoder, and position RSA 0.192+-0.047 versus 0.029 for random encoders (6.6x improvement), showing that training induces genuine structural organization beyond CNN inductive bias. Across 20 temporal checkpoints, prediction performance and semantic alignment co-improve (Spearman r=-0.61, p=0.004), consistent with the shared-driver account. We confirm this through a double knockout: standard KL regularization (beta=0.1) forces the encoder away from geometric structure, and both prediction performance and semantic alignment collapse simultaneously to near-chance by step 50,000 -- exactly as the shared-driver account predicts. Reducing beta to 0.001 restores geometric access and recovers both capabilities together. These findings establish physical world geometry as the organizing principle of world model representations, with direct implications for the design of semantically grounded embodied agents.