Contextuality, Holonomy and Discrete Fiber Bundles in Group-Valued Boltzmann Machines

Abstract: We propose a geometric extension of restricted Boltzmann machines (RBMs) by allowing weights to take values in abstract groups such as ( \mathrm{GL}_n(\mathbb{R}) ), ( \mathrm{SU}(2) ), or even infinite-dimensional operator groups. This generalization enables the modeling of complex relational structures, including projective transformations, spinor dynamics, and functional symmetries, with direct applications to vision, language, and quantum learning. A central contribution of this work is the introduction of a \emph{contextuality index} based on group-valued holonomies computed along cycles in the RBM graph. This index quantifies the global inconsistency or "curvature" induced by local weights, generalizing classical notions of coherence, consistency, and geometric flatness. We establish links with sheaf-theoretic contextuality, gauge theory, and noncommutative geometry, and provide numerical and diagrammatic examples in both finite and infinite dimensions. This framework opens novel directions in AI, from curvature-aware learning architectures to topological regularization in uncertain or adversarial environments.