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Contextual Loop-Spaces for Active Inference

 We develop a mathematically rigorous theory of context‑decorated loop spaces based on the pullback–pushforward duality of differential forms.    Contexts are modelled as elements of a group \(G\) of isometries of a Riemannian manifold \(X\).    A geometric free energy functional for a path \(\gamma\) is defined by    \[        F(g,\gamma)=-\int_\gamma g^*\psi+\frac12\int_0^1\|g^*\psi(\gamma(t))\|^2\,dt,    \]    where \(\psi\) is a sensory 1‑form and \(g\) the internal context map.    We prove the central theorem: \(F(g,\gamma)=F(\mathrm{id},g\circ\gamma)\), establishing a geometric duality between changing internal beliefs and deforming the observed trajectory.    The construction extends to \(N\)-loops and multi‑context settings, where an \(N\)-loop is a smooth map from the \(N\)-torus to the product manifold \(X^N\); periodicity guarantees that the map descends to a toroidal mapping space, maintaining a clean topological interpretation.    This functional is motivated by a formal path‑integral representation of variational free energy in active inference; the push–pull duality then unifies perception and action as dual operations minimising the same functional.    Perceptual inference yields an Euler–Lagrange equation coupling the Lie derivative of the sensory field with a metric energy term.    Action selection is recast as the choice of a probe loop minimising expected free energy.    An explicit example with planar rotations is analysed in full detail, illustrating non‑trivial perceptual inference and verifying the Euler–Lagrange condition.    The framework provides a purely geometric foundation for context‑sensitive field theories aligned with the free energy principle, without requiring a rigorous probabilistic construction.    We conclude with a discussion of limitations and future directions.

Contextual Loop-Spaces for Active Inference

We develop a theory of context-decorated loop-spaces on a smooth manifold X. Each context a is represented by a smooth map ϕa : X → X, and a collection of such contexts decorates a wedge product of differential forms Ψ = i ψa i. For an N-loop γ (N) :

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