Causal Depth and Limits of Scalable Multi-Agent Systems

We introduce an operational notion of object complexity based on the minimal causal structure required to construct, control, or stabilize an object over time. The proposed measure separates the number of active degrees of freedom from causal loop depth, defined as the number of hierarchically nested feedback loops operating on separated timescales. We show that flat, homogeneous multi-agent systems exhibit a fundamental scaling limit: increasing the number of agents improves spatial resolution but cannot increase causal depth. As a consequence, objects whose dynamics require multi-level temporal control are unrealizable by non-hierarchical swarms regardless of size. We formalize this limitation using frequency-domain control constraints and illustrate it via a toy numerical model.