Compositional Symmetry as Compression: Lie‑Pseudogroup Structure in Algorithmic Agents

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 \emph{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) \emph{structural} constraints (equivariance of the agent system constitutive equations and readouts) and (ii) \emph{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 \emph{coarse-grained residual transformations} (prediction‑error coordinates) along symmetry directions unresolved at lower layers.