A system with many microscopic components can organize into a limited number of macroscopic states. How limited? The answer depends not on the number of components but on the dimensionality of their interaction space.
The number of stable macroscopic regimes grows polynomially with exponent equal to the intrinsic dimension of the interaction space. A system where agents interact along one effective dimension (say, distance) can support a few stable collective states. Increase the interaction space to two dimensions (distance and orientation), and the number of accessible regimes grows. But the growth is polynomial, not exponential — adding microscopic complexity (more agents, more individual states) does not by itself expand the repertoire of collective organizations. Only higher-dimensional interactions do.
This is a geometric packing argument. Macroscopic regimes are regions in parameter space that must be separated from each other by boundaries. In a low-dimensional space, there is only so much room for non-overlapping regions. The maximum number of robust regimes is constrained by the same geometric logic that limits how many spheres fit in a box. The answer depends on the box's dimension, not on how many spheres you have.
The implication reverses a common intuition about complexity. Making a system more complex at the component level — more neurons, more genes, more interacting agents — does not necessarily produce richer collective behavior. The collective repertoire is capped by how many dimensions of interaction exist between components. A system with a billion components interacting along one axis has the same ceiling of collective organizations as a system with ten components interacting along one axis. The way to expand collective behavior is not to add more parts but to add more ways for parts to interact.
(arXiv:2603.18127)