Interaction Locality in Spatial Reasoning

Spatial reasoning models must combine location-bound computation with location-invariant structure. These roles are often read off state names, as when a recursive model's high-level state is called global and its low-level state local. However, a name does not fix how far an update's effect travels, and every task defines its own notion of nearby. To address this, we propose interaction locality, which reports the fraction of an intervention's causal effect, at a calibrated perturbation scale, that stays inside a relation declared in advance over the sites a task admits. We analyse HRM and TRM on Maze-Hard, Sudoku Extreme and ARC-AGI, compare them with a matched non-recursive Transformer, and extend the measurement to a 3D grounding model. In the recursive models, a perturbation of the low-level state lands more tightly in the high-level write it produces than in the next low-level call, an ordering that holds for every task-defined relation we test. Under the primary intervention, a size-matched control shows that ordering is propagation in four cells, retention at the perturbed site in one, and unresolved in the cell with the highest raw score. With added depth the non-recursive baseline and HRM keep spreading a perturbation, whereas the weight-tied recursive model plateaus. Because the score is also differentiable, using it as a fine-tuning regularizer raises measured locality without costing accuracy.