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) understood as \emph{compositional symmetry}: 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's constitutive equations and readouts -- and (ii) \emph{dynamical constraints}: under static inputs, symmetry induces conserved quantities (Noether-style labels) in the 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, providing a geometric account of the ``blessing of compositionality'' in deep models. We connect these ideas 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.
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