Typically, a learned world model is evaluated based on its ability to reconstruct observations or predict rewards, almost as if model quality is a binary feature. However, what a task actually requires from a model is narrower: the few predictive coordinates its queries depend on, termed the closure. We demonstrate that the extent to which a latent represents this closure is not dictated by the model's capacity or its observations, but by the dimensionality of the objective it is trained against. We measure this directly on a DreamerV3 stack in a controlled environment with known ground-truth closure. An aligned scalar value signal—the core objective of value equivalence—installs only a one-dimensional projection of a closure that requires several dimensions: when read through a single linear probe, the recoverable structure rises from $R^2=0.10$ to $0.76$ as the scalar is replaced by the full objective. Sweeping the dimensionality of the objective from one to four installs exactly that many predictive directions through an auxiliary head, and the same staircase appears—at attenuated magnitude but the same rank—through the model's own value head, indicating that the dissociation is dimensional rather than an artifact of head form. Capacity-matched comparisons and in-situ pressure checks rule out obvious alternatives. This law governs a regime, and we measure its boundary: in a companion closed-loop task where the structure is observable frame by frame, reconstruction installs that structure, and the scalar objective suffices—the objective dictates what a latent represents precisely where cheaper training signals cannot recover it. Thus, value equivalence is not all-or-nothing but dimensional: the familiar single-reward objective is its rank-one corner, and a model installs as much of a task's structure as the objective it is asked to predict.
Blogger's Review: This paper delves into the intricate relationship between world models and tasks, introducing the concept of dimensional value equivalence, highlighting the significance of objective selection on model performance. This insight provides a new perspective for model design, especially in the applications of reinforcement learning and self-supervised learning, where researchers can optimize performance by adjusting objective dimensions.