The Dimensional Ceiling

The intuition runs deep: more components, more possible behaviors. A system with a thousand types of agent should exhibit richer collective dynamics than one with ten. Complexity scales with diversity.

Tozzi (arXiv:2603.18127) finds a geometric constraint that breaks this intuition. The number of stable macroscopic organizational regimes scales polynomially with the dimensionality of the interaction space — not with the number or diversity of microscopic components. A million distinct agents interacting along two axes produce the same few collective states as ten agents interacting along two axes. The organizational repertoire is bounded by the geometry of how things interact, not by what things interact.

This is a ceiling, not a floor. You can have arbitrarily many types of agent, arbitrarily rich internal states, arbitrarily complex individual behavior — none of it expands the macroscopic organizational repertoire unless the dimensionality of the interaction space also increases. The bottleneck is geometric. Adding components within a fixed-dimensional interaction space increases microscopic complexity but leaves macroscopic complexity unchanged.

The implication runs against standard complex systems thinking, where emergence is treated as a consequence of scale and diversity. Here, emergence has a dimensional budget. A social system where people interact along three dimensions (say: economic, political, cultural) supports the same number of stable organizational forms regardless of population size or individual variation. To get new collective states, you need new dimensions of interaction — new axes along which agents can affect each other.

This reframes the relationship between micro and macro. Microscopic complexity is free; macroscopic complexity is expensive. The price is paid not in the number of parts but in the geometry of their coupling.

Tozzi, "A Geometric Scaling Between Collective Organizations and Interaction-Space Dimension," arXiv:2603.18127 (2026).