Geometric Prediction Error: Free Energy on Lie Groups

Prediction error, the core quantity of the Free Energy Principle, is reformulated using compact Lie groups with bi-invariant metrics. A system composes sequential inputs into a running product on a Lie group, and prediction error becomes the geodesic distance from this product to the group identity. This geometric prediction error propagates exactly—a perturbation of any size at any position produces the same magnitude in the error, with no distortion regardless of sequence length—and is uniformly sensitive to every input. The reformulation eliminates the need for a generative model, replacing variational inference with a single group multiplication per timestep.

Study at a glance

Abstract

I reformulate the core quantity of the Free Energy Principleprediction error in the language of compact Lie groups with bi-invariant metrics. A system receiv-ing sequential inputs composes them into a running product on a Lie group G, and prediction error becomes the geodesic distance from this product to the groupidentity: σ = d(Ct , ⊮). This geometric prediction error inherits two properties that the information-theoretic version (KL divergence) lacks: exact propagation(a perturbation of magnitude ε at any position produces exactly ε in σ, with no distortion regardless of sequence length) and uniform sensitivity (every input isequally detectable, with no blind spots). The reformulation eliminates the need for a generative model entirelythe model is the identity element ⊮, universal andunlearnedand replaces variational inference with a single group multiplication per timestep. I show that the resulting coupling dynamics dier from Kuramoto syn-chronization in a precise way: Kuramoto drives oscillators toward uniformity (all phases equal), while closure coupling drives them toward coherence (phases composeto identity), admitting a vastly richer space of stable congurations. Four testable predictions distinguish the geometric formulation from the standard Free EnergyPrinciple, all accessible with existing experimental methods. I further show that the running product provides a formal substrate for the self, with ego dissolution corre-sponding to the running product approaching identity (the unique point equidistant from all states), life review as chain reconstruction, and the geometric distinctionbetween therapeutic integration and pathological fragmentation mapping onto the clinical phenomenology of psychedelic-assisted therapy and schizophrenia.