RyanThe revised version:
https://zenodo.org/records/20379195
#mathematics #math #physics #contextuality #activeinference #cognition
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.
Alexander Huber
Quantum Observer
Rafael Wagner