Differentials of Means as Dispersion Diagnostics: A Barycentric Framework, with Structure, Generalizations, and Statistical Theory

A mean is a point that balances a configuration; in the variational formulation of Fr´echetit is the barycenter Mρ = arg minmPi wiρ(ci,m) that minimizes average dissimilarity, andthe difference of two barycenters under two geometries, Δρ,σ = Mρ − Mσ, is a measure ofdispersion. We study this barycentric differential from the general object down to its mosttractable instance. The framework subsumes the quasi-arithmetic (Chisini) means as the pulledbackEuclidean case, the operator geometric mean as a Riemannian (Karcher) barycenter, andWasserstein barycenters of distributions; and it exposes the structural spine of the theory—agradient/integrability dichotomy. The information-geometric content (Bregman representation,relative entropy, influence function, concentration) is the privilege of the gradient case, in whichthe geometry derives from a single convex potential; comparison and curvature laws survivefar more generally, including for non-gradient deviation and Bajraktarevi´c means, where theBregman structure provably breaks. We therefore drill down to the gradient case: the Chisinidifferentials Δϕ,ψ = Mϕ−Mψ, and in full quantitative detail the canonical arithmetic–geometricdifferential Δ = A − G, a single scalar measuring the dispersion of a positive signal. Part Iestablishes its deterministic structure from one organizing principle: the generator Jensen gapof every Chisini mean is a Bregman information, and for ψ = ln the logarithmic gap MLD =ln(A/G) is at once the Atkinson and Theil indices, a Kullback–Leibler divergence from anabundance-tilted distribution, and the minimal average Bregman divergence to the arithmeticmean. The arithmetic–geometric inequality is the Chisini comparison theorem, the varianceapproximation is a generator-curvature law, and the additive subgroup decomposition holds forevery generator—decomposability is a property of the generator’s domain, with ln distinguishedonly because its gap is also a relative entropy. From this follow sharp two-sided variance bounds,an additive decomposition that is the chain rule for relative entropy, majorization monotonicity,an embedding in the power-mean/diversity family, divergence-comparison corollaries, a lift topositive-definite matrices with a sharp Frobenius stability bound, and an honest categoricalplacement. Part II develops the statistical theory that the algebra cannot supply. Becausethe differential is a Bregman information, the influence function of its plug-in estimator isexactly the centered pointwise Bregman divergence; this yields a central limit theorem withvariance equal to the variance of that divergence, a nonasymptotic Efron–Stein bound, and—via the nonnegativity of the Bregman remainder—a one-sided Bernstein bound whose Gaussiancore uses the true variance while the diagnostic’s bottom-sensitivity enters only logarithmically,locating the “fragility” in the tail scale rather than the variance. The same influence functiondecomposes block-by-block for grouped sampling and lifts to sampled positive-definite matricesthrough the matrix Bregman divergence; the differential is also a homogeneity test statistic1with a weighted-χ2 null and Pearson-equivalent local power. We close with the two frontiersthe framework opens: the non-gradient edge, and the conditional extension that makes thedispersion diagnostic dynamic and links it to risk measures. We claim no new inequalities;the contribution is the framework, several exact identities, and a statistical theory organizedthroughout by the Bregman structure. Every claim is verified by simulation.