When does LeJEPA learn a World Model?

A representation that scrambles the true degrees of freedom of the world cannot support reliable planning or compositional generalization. We prove that LeJEPA is guaranteed to recover a linear representation of latent variables from nonlinear observations, a property known as linear identifiability. What kind of world and learning objective admit this guarantee? We consider World Models with three properties (independent, stationary, and additive-noise transitions) that generate positive pairs, and study representations trained with the LeJEPA objective. Our main result: if the latent distribution is Gaussian, the representation provably achieves linear identifiability. The key insight comes from a spectral decomposition of the representation with respect to the transition structure: each spectral component corresponds to a degree of nonlinearity, and higher degrees are strictly penalized by alignment, making the linear map the unique optimum. We then prove the converse: among all latent variable distributions satisfying these three properties, the Gaussian is the unique one that leads to linear identifiability. The guarantee degrades gracefully when the two objectives are only approximately satisfied, with an explicit bound validated empirically. Our theory turns an empirically successful recipe into a mathematical guarantee, providing the foundation for building World Models that provably recover the structure of the world.