The Connectivity Cost of Shared Coordinates in World Models

In world models, each action can be implemented with a concise connectivity pattern under its own optimal coordinate system, but the optimal coordinate systems for different actions are generally inconsistent. World models require all actions to share the same coordinate system in order to read and update a consistent representation, yet this causes an increase in the number of connections required for the actions. Existing work has primarily focused on coordinate alignment methods that reduce the number of connections, but a specific characterization of how the connectivity scale changes when actions move from their respective coordinates to shared coordinates is still lacking. This paper investigates this connectivity expansion induced by shared state coordinates. In the symmetric linear model analyzed in this paper, we prove that for families of actions lacking common structure, each action requires only a linear number of connections under its own optimal coordinates, whereas the minimum number of connections typically grows to quadratic order under shared coordinates. The reason is that the connectivity structures required by different actions generally cannot be sufficiently aligned and reused within a single coordinate system, and this gap persists even when approximate compression is allowed. We further show that if different actions share a fixed sparse connectivity pattern under the same coordinate system, the number of connections required for the shared state can still remain at a linear scale, and we present a structure that avoids this increase in asymptotic order.