Cognitive Map World Model

World models have emerged as a promising paradigm for spatial intelligence, yet current approaches typically treat action-observation pairs as unstructured sequences, neglecting the intrinsic algebraic properties of spatial actions and leading to data inefficiency. We propose the Cognitive Map World Model (CMWM), which integrates the Abelian group structure of spatial actions directly into the latent representation. Inspired by grid cell phase coding, CMWM maps continuous actions to phase shifts across multi-scale periodic modules, constructing a globally cognitive map that eliminates recursive prediction errors and ensures stable spatial memory and imagination. Furthermore, this structured latent space redefines navigation from expensive iterative planning to a congruence solving problem, enabling constant-time trajectory generation. Experiments demonstrate that CMWM significantly outperforms autoregressive and MaskGIT baselines in prediction fidelity, convergence speed, and navigation efficiency.