Discrete Informational Curvature from the Jensen–Shannon Metric: a Conditional Regge-Type Convergence to the Scalar Curvature

We introduce a purely metric, discrete notion of curvature built from the (quan-tum) Jensen–Shannon distance dJS between local states attached to the vertices ofsampling graphs Gε that approximate a smooth Riemannian manifold (M, g) at res-olution ε. Angles in small geodesic triangles are reconstructed via the law of cosines,and Regge-type angular deficits define a vertex curvature in n = 2. For n ≥ 3 weemploy a controlled sectional averaging over 2-planes that links our vertex estimatorto hinge-based Regge curvature.Our analysis is conditional: we work under explicit hypotheses (H1)–(H9) encodingquasi-uniform sampling, shape-regular piecewise-flat approximations, a local metricequivalence dJS ≃ dg at small scales, and smooth variation of the state field. Underthese assumptions, the informational curvature measureμε = ∑iκ(ε)i δxiconverges weak-∗ to Rg dvolg , with quantitative bound∣∣⟨f, με⟩ −∫Mf Rg dvolg∣∣ ≤ C∥f ∥C2(r2ε + δε + ηε + 1{n≥3} qε).Here rε is the reconstruction scale; δε controls the local distortion between dJS and dg ;ηε measures residual anisotropy of edge directions; and qε is a sectional quadratureerror from the averaging stencil.Conceptually, this yields a conditional consistency theorem: whenever an infor-mational metric approximates geodesic distance in the sense of (H2), a Regge-typeconstruction based on that metric produces a discrete curvature measure that con-verges to the scalar curvature of (M, g). All metric-compatibility requirements arespelled out explicitly.