Spectral Identifiability for World Models: Polynomial Projectors, Resolvent Stability, and a Krylov Bottleneck

World models are central to model-based reinforcement learning and planning, but their latent dimensions, architectural spectra, and downstream behavior remain poorly understood. In this work, we develop a spectral theory of linear world models in three settings. First, for shared multi-horizon prediction in linear dynamical systems, we identify the finite-horizon block-Krylov operator ${\mathcal{K}}_H$ as the exact latent bottleneck and give a perturbation condition for recovering its empirical rank elbow. Second, for stable diagonal state-space models, we prove a Rademacher complexity bound governed by the spectral memory energy ${\sum }_j{L}_j(T{)}^2$, rather than the parameter count alone. Third, for jointly trained diagonal latent dynamics, we show that disjoint factor spectra imply factor-aligned latent coordinates, with a Riesz-projector stability extension for approximate recovery and a local gradient-flow companion. The supporting results connect these spectral quantities to compositional OOD prediction, rollout envelopes, and prediction-to-control mismatch. Further, generated-system experiments verify the predicted spectral behavior and illustrate why pixel prediction and control returns can rank world models differently.