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.