# Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures

arXiv: https://arxiv.org/abs/2507.07343 · HTML: https://mlsanity.com/paper/2507.07343/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 55 references that are in the map.
- First posted on arXiv 10 Jul 2025.

## Builds on (most cited first, 2 of 2)

- [Topology, Convergence, and Reconstruction of Predictive States](https://mlsanity.com/paper/2109.09203/) (2021)
- [Signatures of Infinity: Nonergodicity and Resource Scaling in Prediction, Complexity, and Learning](https://mlsanity.com/paper/1504.00386/) (2015)

## Abstract (arXiv)

We show that mixtures comprised of multicomponent systems typically are much more structurally complex than the sum of their parts; sometimes, infinitely more complex. We contrast this with the more familiar notion of statistical mixtures, demonstrating how statistical mixtures miss key aspects of emergent hierarchical organization. This leads us to identify a new kind of structural complexity inherent in multicomponent systems and to draw out broad consequences for system ergodicity.