Attention Field Theory: Extending the Active Inference Equivalence to Action

Description This companion preprint extends Attention Field Theory (AFT) from the perception-only formulation to action in the active sensing sense, i.e. action as control of sampling at the evidence surface. The paper derives the corresponding action update from the same free-energy objective used in AFT, and shows how the standard continuous-time Gaussian (Laplace) Active Inference action update emerges when sensory input depends on action via an action-dependent observation mapping. Main contributions Action extension of the AFT–Active Inference equivalence: a theorem-level derivation of the active sensing action update as descent on free energy under the same analytic regime used for the perception-only equivalence. Coupled perception–action dynamics (continuous time, Gaussian/Laplace regime): presentation of the joint system in which recognition dynamics (posterior mean update) and action co-evolve under free-energy minimisation. Unified geometric framing: retains the AFT geometric machinery (free-energy/salience functional and natural-gradient dynamics) and shows how the Active Inference action term is recovered within that framework under standard assumptions. Scope and assumptions This work focuses on continuous-time dynamics in the Gaussian/Laplace regime and treats action primarily as active sensing (sampling control). Temporally deep planning, policy selection, and motor execution via explicit value or expected-free-energy objectives are discussed as forward directions rather than treated as the central result. Relationship to the original AFT preprint This record is a direct follow-up to the original AFT preprint: Attention Field Theory: Riemannian Geometry of Free-Energy-Driven Attention DynamicsCite all versions (concept DOI, always resolves to the latest AFT version): 10.5281/zenodo.17703956 Keywords Attention Field Theory, Active Inference, Free Energy Principle, active sensing, natural gradient, Laplace approximation, Gaussian variational inference, perception–action coupling, Bayesian brain