Prediction Is Not Control: Equivariance Enables Planning in Weight-Space World Models

A world model lets an agent plan by imagining how its environment responds to actions. For reward-free, goal-conditioned planning, such models arrive by two routes: large models pretrained on web-scale data, which control well but are opaque, and from-scratch functional models whose state is the weights of an implicit neural representation, inspectable and pretraining-free but not yet shown to plan. We show the second route fails not on prediction but on controllability; over the same INR-weight state, non-equivariant and latent-code operators match an equivariant operator's rollout PSNR yet plan no better than doing nothing. We trace this to symmetry where only a permutation-equivariant operator learns an action map that survives the weight-space gauge and can be inverted by a planner. We distill this into an offline diagnostic, the action-response, that predicts controllability at r=0.91. Composed with a classical planner on occupancy read off its state, the equivariant model is competitive-to-ahead of a pretrained state-of-the-art world model in distribution and more than 2× ahead out of distribution, from scratch and without reward. Controllability, not prediction, is what makes a world model useful for planning.