When can a world model control ?

World models enable agents to simulate and plan, but when do decisions evaluated inside a model remain valid during physical interaction? In this work, we present a normative theoretical framework establishing the conditions under which a world model supports downstream physical control. By formalizing the response map from feedback policies to interaction laws, we show that matching optimal outcome values alone leaves command-consequence correspondence undetermined: a model and a physical system can agree on achievable task scores yet respond differently to the same command. We prove that preserving this policy-specific correspondence tightly bounds downstream planning loss by model substitution error. Through an exact projection onto causal behavior, we decompose a target trajectory's requirements into initial preparation, pre-action information, and post-action physical response. For Gaussian systems, we solve physical realization analytically over all randomized controls, exposing fundamental joint limits on precision, correlation, and sensor timing: a sensor reduces the outcome variance limit only if its reading precedes corrective action. Furthermore, our framework explains a counterintuitive practical failure: why expanding skill libraries can require finer state distinctions and amplify model selection errors. We validate our theoretical predictions across exact finite processes, learned visual models, and continuous contact simulations by systematically varying feedback timing, response coverage, and skill availability. These results connect prediction objectives, latent representations, and execution interfaces directly to the downstream control decisions a world model must support.