Cognitive Shell F: Fiber Bundle Theory of the Perception-Symbol Interface — Geometric Mapping from Continuous Manifolds to Discrete Representations

Abstract

Cognitive systems face the interface problem between continuous perceptualinputs and discrete symbolic representations. This paper models the perceptionsymbol interface as a fiber bundle structure, with the perceptual manifold as thebase space and the candidate symbol distribution simplex as the fiber, and specifiesthe transport rules of symbol distributions along perceptual paths via a connection. The core findings include: the connection curvature exactly corresponds tolocal sequential effects in simply connected neighborhoods; the full holonomy groupdecomposes into a curvature-generated restricted holonomy component and a monodromy component given by the fundamental group of the base space, the formerbeing continuously tunable and the latter topologically stable. The discontinuity ofthe symbolization projection on the distribution simplex constitutes a boundary defect independent of curvature. Accordingly, we propose a geometric regularizationobjective comprising four terms—curvature penalty, small-loop constraint, largeloop symbol-level penalty, and task loss—providing design principles for neurosymbolic systems based on the dual mechanism of curvature and monodromy control. This framework unifies perceptual ambiguity, sequential effects, memory pathdependence, and framework incommensurability in the language of differential geometry, and establishes structural correspondence with existing neuro-symbolicmethods via sheaf cohomology