Holonomy Grid Codes for Generalisation Under Directed Actions

Abstract

Efficient prediction and planning in structured environments often relies on spectral decompositions of transition operators, yet existing grid-cell and successor-representation theories implicitly assume “flat” action geometry where translations commute and a single Fourier eigenbasis suffices. We show that this assumption breaks in the presence of path-dependent effects—e.g., circulation, rotational drift, or topological loops—whose defining signature is nontrivial holonomy. We introduce a theory of directed-action prediction on discrete tori based on twisted translation operators forming a projective representation of the underlying motion group, and prove that the resulting controlled Markov operators admit an exact block-diagonalisation under a twisted Fourier transform: actions share a universal harmonic basis while their effects appear as small matrix-valued spectra rather than scalar eigenvalues. This yields closed-form resolvent expressions for the successor representation, a gauge-invariant transfer principle characterising when two environments admit identical predictive structure, and a curvature-induced lower bound showing that nonzero holonomy provably necessitates internal representational dimension. Together, these results generalise Fourier/grid-based prediction from commutative to curved action geometries, providing a principled foundation for generalisation under directed actions without learning environment-specific eigenvectors.