Tangential Action Spaces: Geometry, Memory and Cost in Holonomic and Nonholonomic Agents

Abstract

Living systems balance energetic efficiency with the capacity for path-dependent effects. We introduce Tangential Action Spaces (TAS), a geometric framework that models embodied agents as hierarchies of manifolds linked by projections from physical states to cognitive representations and onward to intentions. Lifts from intentions back to actions may follow multiple routes that differ in energy cost and in whether they leave memory-like traces. Under explicit assumptions, we prove: (i) if the physical-to-cognitive map is locally invertible, there is a unique lift that minimises instantaneous energy and yields no path-dependent memory; any memory requires strictly positive excess energy. (ii) If multiple physical states map to a cognitive state (a fibration), the energy-minimising lift is the metric-weighted pseudoinverse of the projection. (iii) In systems with holonomy, excess energy grows quadratically with the size of the induced memory for sufficiently small loops, establishing a local cost-memory law. These results motivate a classification of embodied systems by the origin of path dependence: intrinsically conservative, conditionally conservative, geometrically nonconservative, and dynamically nonconservative. Numerical examples illustrate each case. We also present a reflective extension (rTAS) in which perception depends on a learnable model state; a block metric formalises an effort-learning trade-off, and cross-curvature terms couple physical and model holonomy. Simulations of single- and two-agent settings show role asymmetries and sensitivity to coupling. TAS provides a geometric language linking embodiment, memory, and energetic cost, yielding testable predictions and design guidelines for biological and robotic systems.