The Geometry of Agent Skills: Non-Commutative Composition in Representation Space

LLM agents increasingly rely on reusable skills for multi-step reasoning, tool use, and decision making. Yet most existing approaches represent a skill as a prompt, module, or direction in representation space. This view can be insufficient for agentic settings: (1) the behaviour induced by a skill can depend on the current context, and (2) composing two skills can yield order-dependent rollouts. We formalise a skill realization as a local map from continuous intervention coordinates to a readout representation. Its differential defines the context-dependent controllable distribution, and the residence-space norm induces a metric on this distribution. Skill objectives then define local vector fields on these controllable directions. Their \emph{Lie bracket} captures the order-dependent component of skill composition, revealing behaviour that cannot be represented by a single context-independent direction. Our central measurement is a finite-difference skill commutator, which estimates the non-commutative component of two-skill composition from paired ordered rollouts. Unlike standard Lie-bracket estimation, the relevant vector fields are induced by skill objectives and observable ordered rollouts rather than given analytically. We therefore reconstruct the commutator from paired ordered rollouts, preserving the direction of the order-dependent change in representation space. We instantiate the framework on frozen LLM agents using activation-based and prefix-based skill realizations. Same-skill activation and prefix Jacobians are more aligned than cross-skill pairings (Cohen’s $d=-0.76/-0.26$), but they assign different control costs; separately, the measured two-skill interaction predicts ordered-composition gaps (Spearman $\rho_S=0.918/0.889$). High-commutator skill pairs also yield non-saturating reachability signals, with positive spectral-entropy gains and principal-angle novelty, while synthetic bracketed systems validate the finite-difference estimator. Together, these results show that skills form context-dependent geometric objects whose compositions leave measurable vector-valued non-commutative residuals.