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→(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 Fréchet barycenter of X under dG. 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.
Related
- On uniqueness of probability solutions to one-dimensional Fokker-Planck-Kolmogorov equations
- Reachable sets under Kolmogorov dynamics in the probability simplex
- Statistical Inference for Probability Barycenters and Kolmogorov Moments
- On the applicability of Kolmogorov's theory of probability to the description of quantum phenomena. Part II: Bell inequalities
- Fractional-order differential geometry of Gielis curves: chart regularization, Caputo curvature, and the fate of singularities
BibTeX
@misc{cojocea2026chartgenerated,
title = {{Chart-Generated Probability Geometry and Kolmogorov Expectations}},
author = {Manuela-Simona Cojocea},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.02104},
eprint = {2605.02104},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.02104},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Chart-Generated Probability Geometry and Kolmogorov Expectations
AU - Manuela-Simona Cojocea
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2605.02104
UR - https://arxiv.org/abs/2605.02104
AB - 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→(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 Fréchet 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.
ER -