arxiv: 2605.02104 · v1 · submitted 2026-05-04 · 🧮 math.PR

Recognition: 5 theorem links

· Lean Theorem

Probability Geometry and Kolmogorov Expectations via Coordinate Charts

Manuela-Simona Cojocea

Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:15 UTC · model grok-4.3

classification 🧮 math.PR

keywords probability geometryKolmogorov meanscumulative distribution functioncoordinate chartsgeneralized expectationheavy-tailed distributionslaws of large numberscentral limit theorem

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The pith

By treating cumulative distribution functions as coordinate charts, random variables are mapped to the unit interval where averaging is linear and the result pulled back coincides with Kolmogorov means.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reinterprets expectation by using cumulative distribution functions to transport a random variable into the unit interval, turning the averaging operation into a linear one in these probability coordinates. The averaged value is then mapped back to the original value space. This construction produces quantities identical to Kolmogorov means from the classical theory of generalized means. The approach removes the need for classical integrability, so heavy-tailed cases are handled by viewing divergence as boundary concentration in the probability interval. Laws of large numbers and central limit theorems are shown to hold after this coordinate change.

Core claim

Interpreting the cumulative distribution function as a coordinate chart transports a real-valued random variable into the unit interval, where averaging becomes linear in probability coordinates; pulling the result back to value space yields exactly the Kolmogorov means associated with that distribution. This geometric representation makes expectation well-defined even when the classical integral diverges, with heavy tails appearing as concentrations near the interval boundaries, and permits asymptotic laws to be recovered in the new coordinates.

What carries the argument

Cumulative distribution function viewed as a coordinate chart that linearizes averaging in probability space before pullback to value space.

If this is right

Expectation exists for heavy-tailed distributions whose classical moments diverge.
Laws of large numbers and central limit theorems hold after the change to probability coordinates.
Generalized means arise directly from the choice of probability representation rather than from algebraic axioms alone.
The same geometric construction extends to Rényi-type and other generalized-mean frameworks.
Expectation is not a single fixed functional but depends on the selected coordinate representation of probability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This coordinate view could supply a practical numerical route for approximating expectations of heavy-tailed variables by sampling uniformly in the unit interval and pulling back.
Similar chart changes might be applied to other functionals such as variance or entropy to obtain geometric versions that remain defined under weaker conditions.
Testing the method on distributions with known finite means would confirm it recovers the classical expectation as a special case.
The representation suggests examining whether other probability concepts, such as conditioning, also become simpler after the same CDF chart transformation.

Load-bearing premise

The cumulative distribution function acts as a valid coordinate chart that makes averaging strictly linear in probability space and whose pullback recovers a meaningful expectation without classical integrability or extra parameters.

What would settle it

For a Pareto random variable with shape parameter 0.5, compute the proposed coordinate-based average and compare it to the known Kolmogorov mean formula; mismatch or failure to remain finite while the classical integral diverges would disprove the claim.

Figures

Figures reproduced from arXiv: 2605.02104 by Manuela-Simona Cojocea.

**Figure 1.** Figure 1: Construction of the probability barycenter. Observations view at source ↗

**Figure 2.** Figure 2: Benchmark probability coordinate charts. A single random variable view at source ↗

**Figure 3.** Figure 3: Multivariate probability coordinates. A random vector view at source ↗

read the original abstract

This paper develops a geometric reinterpretation of probability in which expectation arises from averaging in probability coordinates rather than in value space. By interpreting the cumulative distribution functions as coordinate maps, a real-valued random variable is transported into the unit interval, where averaging becomes a linear operation in probability coordinates and is then pulled back to the value space. Within this representation, the resulting quantities coincide with Kolmogorov means, thereby linking the construction to the classical theory of generalized means associated with Kolmogorov, Nagumo, de Finetti, and Chisini. This connection clarifies that these means are not merely algebraic devices, but arise from a change of representation of probability. The framework provides a natural setting in which expectation exists beyond classical integrability assumptions. In particular, heavy-tailed phenomena are reinterpreted geometrically: divergence in value space corresponds to boundary concentration in probability coordinates. Laws of large numbers and central limit theorems are established in this coordinate system, showing that asymptotic behaviour can be recovered after this geometric representation. The perspective also connects to broader constructions based on generalized means, including R\'enyi-type formulations, and suggests that expectation should be viewed not as a fixed numerical functional, but as the outcome of a chosen probability representation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The CDF-as-coordinate-chart idea gives a geometric framing for Kolmogorov means but only works cleanly for continuous strictly monotonic cases and needs explicit handling of general RVs to support the broader claims.

read the letter

The paper's main move is to treat the CDF as a coordinate map that sends a random variable into the unit interval, perform linear averaging there, and pull the result back to the original scale, with the resulting objects matching Kolmogorov means. This is meant to let expectations exist for non-integrable variables by reinterpreting heavy tails as boundary piling in probability space, plus some LLNs and CLTs in the new coordinates. The geometric link to generalized means is the clearest part of the contribution and gives a readable picture for why certain averaging procedures appear naturally in probability. The boundary-concentration view for heavy tails is a reasonable intuition pump even if it does not immediately yield new theorems. The soft spot is exactly the one the stress test flags. For the construction to be a valid chart, the CDF must be continuous and strictly increasing so that it is bijective onto (0,1) and the inverse is well-defined. Discrete and mixed distributions produce jumps or flat regions, breaking the direct transport and the claimed linearity in probability coordinates. The abstract applies the framework to general real-valued random variables without showing generalized inverses or restricting the class, so the extension beyond integrability rests on an assumption that does not hold universally. If the full derivations simply recover the median or other known means by definition rather than deriving them from the geometry, the equivalence is not new in substance. This is for readers who already work with generalized means or foundational re-representations in probability. A specialist might find the coordinate perspective useful for thinking, but the technical gaps around scope and invertibility are real. I would send it to peer review so referees can check whether the authors supply the missing constructions and tighten the claims to cases where the chart is actually valid.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a geometric framework for probability by treating the cumulative distribution function (CDF) of a real-valued random variable as a coordinate chart that maps the variable into the unit interval [0,1]. In this probability coordinate system, averaging operations become linear, and the results are pulled back to the original value space using the inverse CDF. The paper claims that these pulled-back averages coincide with Kolmogorov means, providing a geometric interpretation of generalized means. It further asserts that this approach allows expectations to exist beyond classical integrability conditions, reinterprets heavy-tailed distributions geometrically as boundary concentrations in probability space, and establishes laws of large numbers and central limit theorems within the coordinate system.

Significance. If the construction is made rigorous and the equivalence to Kolmogorov means is derived rather than asserted, the work could offer a novel geometric perspective that unifies coordinate changes in probability space with the classical theory of generalized means (Kolmogorov, Nagumo, de Finetti, Chisini). It might provide tools for heavy-tailed phenomena by recasting divergence as boundary effects and could motivate representation-dependent views of expectation. The link to Rényi-type formulations is a potential strength if substantiated.

major comments (2)

[Abstract and coordinate chart construction] The central construction (abstract and the section introducing the coordinate chart) treats the CDF F as a diffeomorphic chart transporting X to U = F(X) ∈ [0,1], where averaging is linear before pullback by F^{-1}. This requires F to be continuous and strictly monotonic on the support, which fails for discrete RVs (jumps make F non-invertible) and distributions with flat regions. The abstract asserts the framework for general real-valued RVs and heavy tails without restrictions or generalized inverses; this is load-bearing for the claims of existence beyond integrability and the LLN/CLT results in coordinates.
[Section establishing coincidence with Kolmogorov means] The claim that the resulting quantities 'coincide with Kolmogorov means' (abstract) appears to follow by matching the pullback form to the generator f = F rather than being independently derived from the geometry. If the equivalence is definitional rather than a consequence of the transport, this introduces circularity that weakens the link to the classical theory.

minor comments (2)

[Notation and assumptions] Clarify whether the framework is restricted to absolutely continuous distributions or if a generalized inverse is used; add an explicit statement of the class of RVs for which the chart is valid.
[Asymptotic results] The abstract mentions 'Laws of large numbers and central limit theorems are established in this coordinate system' but provides no indication of the precise statements or proof sketches; include a brief outline or reference to the relevant theorem numbers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. The points raised regarding the assumptions in the coordinate chart and the derivation of the Kolmogorov means equivalence are well-taken. We have addressed both by clarifying the scope and strengthening the proofs in the revised version.

read point-by-point responses

Referee: [Abstract and coordinate chart construction] The central construction (abstract and the section introducing the coordinate chart) treats the CDF F as a diffeomorphic chart transporting X to U = F(X) ∈ [0,1], where averaging is linear before pullback by F^{-1}. This requires F to be continuous and strictly monotonic on the support, which fails for discrete RVs (jumps make F non-invertible) and distributions with flat regions. The abstract asserts the framework for general real-valued RVs and heavy tails without restrictions or generalized inverses; this is load-bearing for the claims of existence beyond integrability and the LLN/CLT results in coordinates.

Authors: We acknowledge the validity of this observation. The construction in the manuscript relies on the CDF being a diffeomorphism, which necessitates continuity and strict monotonicity. In the revised manuscript, we have updated the abstract and the relevant sections to explicitly state these assumptions on the random variable (continuous distributions with strictly increasing CDFs on their support). We also discuss the use of the quantile function for extension to more general cases, although the core geometric properties hold under the stated conditions. This ensures the claims regarding existence beyond integrability and the asymptotic results are appropriately scoped. We believe this resolves the concern without altering the main contributions. revision: yes
Referee: [Section establishing coincidence with Kolmogorov means] The claim that the resulting quantities 'coincide with Kolmogorov means' (abstract) appears to follow by matching the pullback form to the generator f = F rather than being independently derived from the geometry. If the equivalence is definitional rather than a consequence of the transport, this introduces circularity that weakens the link to the classical theory.

Authors: We appreciate the referee's insight into this potential issue of circularity. Upon review, the original presentation did match the form directly. In the revision, we have reorganized the section to first compute the expectation as the pullback of the linear average in probability coordinates, deriving its explicit form geometrically. We then independently show that this expression matches the Kolmogorov mean generated by the function f equal to the CDF. This establishes the coincidence as a derived result from the coordinate change. Additionally, we elaborate on the geometric interpretation this provides for the classical means, avoiding any definitional shortcut. revision: yes

Circularity Check

0 steps flagged

No circularity: geometric reinterpretation links to but does not reduce to Kolmogorov means by construction

full rationale

The provided abstract and description present the core construction as transporting via CDF coordinate charts to make averaging linear in probability space, then pulling back; the statement that resulting quantities 'coincide with Kolmogorov means' is framed as a clarifying link to existing theory rather than a claim that the new object is derived from or equivalent to the inputs by definition. No equations, self-citations, fitted parameters, or ansatzes are visible that would force the central claim to collapse into its own premises. The derivation remains self-contained as a change-of-representation perspective, with the Kolmogorov connection serving as external context rather than load-bearing justification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard properties of cumulative distribution functions and the existence of a linear averaging operation in the unit interval; no explicit free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Cumulative distribution functions are continuous and strictly increasing (or can be made so by suitable extension) so that they serve as coordinate charts.
Invoked when the paper states that CDFs transport the random variable into the unit interval.
domain assumption Averaging in the probability coordinate system is a linear operation that can be pulled back to value space.
Central to the claim that the pulled-back quantity coincides with Kolmogorov means.

pith-pipeline@v0.9.0 · 5501 in / 1511 out tokens · 59629 ms · 2026-05-08T19:15:48.199183+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

M_g(X) = g⁻¹(E[g(X)]) ... coincide with the class of generalized means introduced in the works of Kolmogorov, Nagumo, de Finetti, and Chisini.
IndisputableMonolith.Foundation.ArithmeticFromLogic embed (orbit transport via generator) echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

E_G(X) := G⁻¹(E[G(X)]) ... transporting the distribution to probability space, performing linear averaging there, and pulling back to value space.
IndisputableMonolith.Foundation.BranchSelection branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Two fundamentally distinct mechanisms generate notions of central tendency: (i) Optimization in value space ... (ii) Geometry in probability coordinates.
IndisputableMonolith.Foundation.AlphaCoordinateFixation alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the probability barycenter depends only on the affine structure of the probability scale (0,1) and is invariant under affine reparametrizations of probability coordinates.
IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel (RS forces uniqueness of J; paper explicitly does not force uniqueness of chart) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Different admissible charts may produce different barycenters for the same random variable X. In this sense, the construction does not define a unique notion of central tendency.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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