Structured Dynamics in the Algorithmic Agent (ICMNS 2025 poster)

A poster-length argument that a brain-like agent tracking the world must, by mathematical necessity, inherit the symmetry structure of that world.

The core idea is elegant: if the world generates data through continuous, structured transformations — rotations, translations, deformations — then any agent that successfully tracks that data cannot be arbitrary. Its internal dynamics are forced to mirror those same transformations. This is not a design choice; it falls out of the math.

The formal machinery comes from two places. First, Kolmogorov complexity theory, which frames intelligent agents as systems that compress and internally simulate the world's generative process. Second, Lie pseudogroups — a way of describing continuous symmetries using group theory. The paper marries these two traditions, showing they are saying the same thing from different angles. Compression-based theories of representation and symmetry-based theories of invariance turn out to be formally equivalent.

The technical spine is what the authors call the world-tracking neurodynamic equations (WTNE). Model the agent as a generic neural ODE — a dynamical system whose state evolves continuously. Require that some read-out of the agent's state matches the world's inputs. That single requirement is enough to constrain the agent's parameters: they cannot be arbitrary, they must lie on a manifold defined by the world's symmetry group. The agent's dynamics are then confined to an invariant submanifold — a reduced-dimensional slice of its full state space. This is a concrete, mechanistic grounding for the manifold hypothesis, the empirical observation that high-dimensional natural data clusters on low-dimensional surfaces.

The payoff is a unified "Lie Generative Model" framework. Environmental symmetries shape both the static wiring of the agent (its constitutive parameters) and its moment-to-moment dynamics. The authors argue this also enforces hierarchical, compositional organization — which connects to how cognition seems to work recursively and in layers.

This is a poster companion to a fuller Entropy 2025 paper (WP0106), so the source is sparse and some derivations are only gestured at rather than shown. The intuition is crisp; the detailed proofs live elsewhere.