Compositional Symmetry as Compression: Lie-Pseudogroup Structure in Algorithmic Agents NeurReps 2025

In the algorithmic (Kolmogorov) view, agents are programs that track and compress sensory streams using generative programs. We propose a framework where the relevant structural prior is simplicity (Solomonoff) as compositional symmetry, where natural streams are well described by (local) actions of finite‑parameter Lie pseudogroups on geometrically and topologically complex low‑dimensional configuration manifolds (latent spaces). Modeling the agent as a generic neural dynamical system coupled to such streams, we show that accurate world‑tracking imposes (i) structural constraints (equivariance of the agent system constitutive equations and readouts) and (ii) dynamical constraints: under static inputs, symmetry induces conserved quantities (Noether‑style labels) in agent dynamics and confines trajectories to reduced invariant manifolds; under slow drift, these manifolds move but remain low‑dimensional. This yields a hierarchy of reduced manifolds aligned with the compositional factorization of the pseudogroup---a geometric account of the ``blessing of compositionality'' in deep models. We connect these ideas, at a high level, to the Spencer formalism for Lie pseudogroups, and formulate a symmetry‑based, self‑contained version of predictive coding in which higher layers receive only coarse-grained residual transformations (prediction‑error coordinates) along symmetry directions unresolved at lower layers.