Learning world models that infer environment dynamics from high-dimensional observations and predict outcomes under candidate actions is central to planning and control. Joint-Embedding Predictive Architectures (JEPAs) provide a compelling framework for learning such models in representation space. Recent action-conditioned extensions perform promisingly in visual control and latent-space planning, but leave a fundamental question unresolved: when does controlled latent prediction identify both the underlying state and the controlled dynamics? This is challenging under nonlinear observations and behavior policies with limited conditional action variation, where state-dependent evolution and action effects can be statistically confounded. We establish a joint identifiability theory for controlled world models with Gaussian latent states under state-dependent Gaussian behavior policies. We identify two policy-dependent conditions: spectral separation of the predictable signal governs representation identifiability, while non-degenerate conditional action variation governs transition identifiability. We prove that when both conditions hold, every global minimizer of the JEPA objective identifies the latent state and controlled transition up to an orthogonal transformation. We further derive quantitative bounds on representation and transition identifiability under approximate optimization. Finally, we construct predictor perturbations along weakly excited action directions whose counterfactual-to-on-policy error ratio is the inverse transition-identifiability margin, revealing the cost of limited action coverage. Experiments across nonlinear observation maps and behavior policies corroborate the theory and demonstrate implications for transition identifiability, counterfactual prediction, and goal-conditioned latent planning.

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