Chart-Generated Probability Geometry and Kolmogorov Expectations

Abstract

This paper develops a geometric reinterpretation of probability in which a cumulative distribution function is not used merely as a passive coordinate label, but as a generator of metric geometry. An admissible probability chart $G:I\to(0,1)$ pulls the Euclidean distance of the probability interval back to value space. This creates a decisive distinction from ordinary coordinate invariance: if $G$ is replaced by another chart $H$ while the observations and their law are kept fixed, the induced metric-measure structure changes. Probability charts preserve ordinal structure while altering metric notions such as distance, dispersion, boundary proximity, and barycentric centrality. Averaging linearly in the chart coordinate and pulling the result back defines the Fr\'echet barycenter of $X$ under $d_G$. These functionals coincide with Kolmogorov-Nagumo means, but the chart-generated viewpoint explains their chart dependence geometrically: changing the chart changes the loss being minimized, rather than merely rewriting one fixed centre in new coordinates. A rigidity result shows that, within normalized probability charts, preserving the barycenter for every law forces the chart itself to remain unchanged. Under the intrinsic chart the coordinates are uniform and the barycenter is the median; under an external benchmark chart, tail mismatch is represented by excess boundary occupation. Laws of large numbers, central limit theorems, and a non-asymptotic concentration inequality are established for the transformed functionals. Probability-coordinate moments exist for every law supported on the chart domain and determine the law uniquely, in contrast with classical moment non-existence and indeterminacy on unbounded value spaces.