Informational Holonomy Curvature and Its Discrete-to-Continuous Convergence

We introduce a notion of curvature based on informational holonomy. Let (M,g) be a smooth Riemannian manifold and let π:P→M be a bundle of state spaces equipped fibrewise with a smooth divergence Dx inducing an information metric gPx. Assuming a connection on P compatible with this fibrewise information geometry, we measure the deviation of holonomy around small geodesic triangles by transporting a reference state μx and comparing it to its image via the induced informational distance dx=2Dx. Normalizing the resulting distance defect by the geometric area yields a continuous informational holonomy (sectional) curvatureKholcont(x,Π). We prove that this limit exists for all (x,Π) and equals the norm of a vector Wx(Π;μx)∈TμxPx depending linearly on the curvature of the connection along Π. In geometric models induced from the Levi–Civita connection via an isometric representation, Kholcont becomes a scalar invariant of Rg|Π and, on spaces of constant sectional curvature, reduces to a constant multiple of |secg|. On the discrete side, we consider quasi-uniform sampling graphs whose edges carry channels approximating parallel transport. Discrete triangle holonomies define a curvature estimator, and under explicit sampling, area-approximation, and channel-consistency assumptions, we establish a discrete-to-continuum convergence theorem with a quantitative error bound controlled by the sampling scale.