Symmetries at the origin of hierarchical emergence

TL;DR

The paper addresses how hierarchical emergence arises from underlying symmetries in dynamical systems. It combines information theory, group theory, and statistical mechanics to show that dynamical equivariance yields a hierarchy of informationally closed macrostates that correspond to subgroup lattices, and that the same symmetries shape Bayesian abstractions. The authors illustrate the theory with Hopfield networks and Ehrenfest diffusion, deriving macroscopic order parameters as emergent levels and demonstrating efficient coarse-grained inference. The results offer a principled framework for understanding and exploiting abstraction in high-dimensional inference, with potential implications for neuroscience, AI, and complex systems.

Abstract

Many systems of interest exhibit nested emergent layers with their own rules and regularities, and our knowledge about them seems naturally organised around these levels. This paper proposes that this type of hierarchical emergence arises as a result of underlying symmetries. By combining principles from information theory, group theory, and statistical mechanics, one finds that dynamical processes that are equivariant with respect to a symmetry group give rise to emergent macroscopic levels organised into a hierarchy determined by the subgroups of the symmetry. The same symmetries happen to also shape Bayesian beliefs, yielding hierarchies of abstract belief states that can be updated autonomously at different levels of resolution. These results are illustrated in Hopfield networks and Ehrenfest diffusion, showing that familiar macroscopic quantities emerge naturally from their symmetries. Together, these results suggest that symmetries provide a fundamental mechanism for emergence and support a structural correspondence between objective and epistemic processes, making feasible inferential problems that would otherwise be computationally intractable.

Figures (5)

• Figure 1: Symmetry gives rise to simpler self-contained levels. (A) Dynamics of a Markov chain over 6 states, which respects symmetries equivalent to a triangle --- known as the dihedral group of order 3, $D_3$. (B) Correspondence between the subgroups of $D_3$ and the lattice of informationally closed levels of the Markov chain, which preserves the partial ordering.
• Figure 2: Equivariant symmetries in a Hopfield network. (A) A Hopfield network is a recurrent neural network trained to retrieve a given set of patterns. (B) The dot products between each candidate pattern and the current state of the network are equivariant to the neural dynamics, providing natural order parameters known as Mattis magnetisation. (C) Training the network over letters and all their rotations and reflections induce additional equivariances to the dihedral group $D_4$. (D) The subgroups of $D_4$ correspond to different informationally closed coarse-grainings. For example, factoring all symmetries out yields a representation that captures only letter identity, while factoring by the rotation subgroup ($\langle r\rangle$) preserves letter identity and whether it is reflected. (E) Numerical results demonstrate that these symmetry-derived coarse-grainings (solid lines) yield substantially better self-prediction than arbitrary coarse-grainings of the same size (dotted lines matched by color), confirming that symmetry-based emergent variables capture all dynamically relevant information.
• Figure 3: Hierarchical Bayesian belief updating. A process with equivariant symmetries has informationally closed levels (left). If measurements respect the same symmetries, then there is a correspondence between the various levels of the process and measurements of different degrees of resolution (centre). This hierarchical structure of processes and measurements gives rise to Bayesian beliefs that can be updated autonomously at various scales, without the need of accounting for the information from levels below (right).
• Figure 4: Abstract beliefs in the Ehrenfest diffusion model.(A) The model considers $n$ particles contained in two connected chambers. The micro-level state is a binary vector that specifies in which container is each particle. (B) The system's dynamics are equivariant with respect to permutations among the particles, giving rise to equivalence classes corresponding to the particle count per chamber. (C) Measurements are noisy observations of each particle’s location, obtained via a binary symmetric channel. (D) Updating of micro-level beliefs about the microstate and abstract beliefs about the particle count for $n=8$. The dimensionality of abstract beliefs grows linearly with $n$, while for micro-level beliefs it grows exponentially.
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Theorems & Definitions (12)

• Theorem 1
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• Corollary 1
• proof
• Corollary 2
• proof
• Theorem 2
• Corollary 3
• proof
• Corollary 4