Width, Memory, and Delay: A Resource Accounting for the Limits of Flat Multi-Agent Systems
A recurring question in the design of scalable multi-agent systems -- from robot swarms to collectives of large-language-model (LLM) agents -- is whether adding more agents can, on its own, overcome performance limits, or whether a qualitatively \emph{deeper} organization is required. A recent preprint argues that flat, homogeneous multi-agent systems face an irreducible, population-independent ``causal floor'' on achievable error, removable only by hierarchical (nested-loop) organization. Using a controlled disturbance-rejection testbed with an exactly computable optimum, we show this conclusion is too strong and replace it with a quantitative resource model built on three resources: population \emph{width} $N$, per-agent internal-model \emph{memory} $d$, and prediction across the observation \emph{delay} $\tau$. We establish three claims. (i) The achievable floor is governed not by architectural hierarchy but by per-agent internal-model content: a flat, homogeneous swarm whose agents carry a matched internal model of the disturbance matches or beats a designed two-loop hierarchy at equal per-agent memory -- so temporal depth can be dynamical (recurrent memory), not architectural (nesting). (ii) The three resources are \emph{not mutually interchangeable}; we chart the exchange rates and the hard non-exchange boundaries on an explicit width$\times$memory map, including a strict equal-total-state-budget comparison. (iii) A residual floor is set by the observation delay and the environment's unpredictability over that horizon, which we verify against the optimal controller. We quantify the price of replacing oracle knowledge of the disturbance spectrum with online learning, provide a preliminary robustness check against a mild bounded nonlinearity and a spatially-extended plant, and distill four design rules for practitioners.