The Independence-Preserving Property and Planar Web Geometry
Source: arXiv:2607.20646 · Published 2026-07-22 · By Yoshihiro Gyotoku
TL;DR
This paper explores the characterization of probability distributions whose independence is preserved under certain real-analytic planar diffeomorphisms F : R² → R², by connecting this classical probability problem to the geometry of planar webs. Specifically, the map F(x, y) = (u, v) preserves independence of independent variables X and Y if and only if U = u(X, Y) and V = v(X, Y) remain independent. This strong restriction permits characterization of the distributions admitting such transformations. The main novelty is the introduction of planar web geometry and Abelian functional equations as a unifying framework. The author shows that the logarithmic densities of the distributions form vector spaces given by the Abelian relations of the 4-web defined by (x, y, u(x, y), v(x, y)). Bol’s rank bound caps the dimension of this space at three, revealing a natural maximal 3-parameter family of independent measures preserved by F. This geometric viewpoint recovers classical characterization theorems like Kac–Bernstein and Lukacs, as well as recent results for quadrirational Yang–Baxter maps, within a unified proof framework. Furthermore, the approach generates new families of independence-preserving maps, characterizes their preserved measures explicitly, and identifies the web as an invariant under coordinatewise reparametrization. The interplay between web geometry and independence-preservation suggests a pathway to a full classification of such maps along with their preserved measure classes.
Key findings
- For a real-analytic diffeomorphism F with nonsingular associated planar 4-web WF, F preserves independence of measures (µX, µY, µU, µV ) with positive continuous densities if and only if the logarithmic densities satisfy an Abelian functional equation, i.e., lie in the vector space of Abelian relations A(WF) (Theorem 6).
- Bol’s classical rank bound implies dim A(WF) ≤ 3 for nonsingular planar 4-webs, so at most a 3-parameter family of product measures can be preserved by such a map (Lemma 3).
- The framework recovers classical characterization theorems: Kac–Bernstein (normal laws), Lukacs (gamma laws), and Matsumoto–Yor properties as special cases where the Abelian relations correspond to known functional equations (Remark 14).
- Quadrirational Yang–Baxter maps HI, HII, and HIII restricted to (0, ∞)² are shown to be independence-preserving with explicit parametrizations of preserved measure families given by three parameters (Corollaries 11, 12, 13).
- A broader class of maps generalizing Koudou–Vallois transformations admits at least a two-parameter family of preserved product measures, extending beyond previously studied examples (Example 15).
- Independence-preserving property is invariant under coordinatewise one-variable analytic reparametrizations, which do not alter the associated web, making the planar web a natural geometric invariant (Remark 5).
- Changing the domain of a map can generate other independence-preserving maps sharing the same density structure, thus demonstrating the locality and flexibility within the framework (Example 17).
Methodology — deep read
Threat model & assumptions: The problem considers a real-analytic diffeomorphism F mapping from an open domain in R² (typically intervals IX×IY) to R², acting on two independent random variables X and Y. The primary interest is in characterizing those distributions µX, µY (and corresponding µU, µV under F) for which independence is preserved. The assumptions include positivity and continuity of densities relative to reference σ-finite Borel measures µ0T (often Lebesgue or weighted Lebesgue). The web WF associated with F is assumed nonsingular.
Data: The data corresponds to the distributions µT, with logarithmic densities φT = log(dµT/dµ0T), defined on intervals IT of R. The problem is formulated over open sets where F acts as a diffeomorphism. No empirical dataset is used; the work is theoretical.
Architecture / algorithm: The main mathematical object is the planar 4-web WF = (x, y, u(x,y), v(x,y)) induced by F. The space A(WF) of Abelian relations consists of 4-tuples of functions (ΦX, ΦY, ΦU, ΦV) satisfying a functional Abelian equation: ΦX(x) + ΦY(y) + ΦU(u) + ΦV(v) = constant. The key insight is that the logarithmic density functions satisfy such an equation if and only if independence is preserved under F. Bol’s rank bound limits dim A(WF) ≤ 3.
Training regime: Not applicable; the approach is fully analytical.
Evaluation protocol: The main evaluation is by proving the equivalence between independence preservation and Abelian relations (Theorem 6). Known characterizations of classical laws serve as baselines for verification, reproving Kac–Bernstein, Lukacs, and Matsumoto–Yor results. Further examples of quadrirational Yang–Baxter maps are reanalyzed and new examples constructed. Comparative study between transformed measures and their Abelian relations validates the framework.
Reproducibility: The paper is theoretical with explicit formulas and proofs; no code or datasets are used. The mathematical statements are concrete and grounded in classical differential geometry and probability theory. Known properties of planar webs and Abelian relations are cited from standard references (Blaschke & Bol 1938, Chern & Griffiths 1978, Bol 1932).
Concrete example end-to-end: For the quadrirational Yang–Baxter map H+I on (0, ∞)², the paper calculates the Jacobian determinant, verifies nonsingularity of WF, identifies a triple of linearly independent Abelian relations, and deduces that H+I preserves independence exactly for distributions with densities parametrized by (p,q,r) in specified ranges formulated in Corollary 11. This reproduces known characterization results within the Abelian relation framework and shows how the method unifies and extends these characterizations.
Technical innovations
- Connecting independence-preserving transformations of probability measures with the theory of planar web geometry and Abelian relations, providing a novel unifying framework.
- Demonstration that preservation of independence under an analytic map corresponds exactly to the logarithmic densities lying in the space of Abelian relations of the associated planar 4-web (Theorem 6).
- Use of Bol’s rank bound on the dimension of Abelian relations to limit the parameter space of preserved product measures to a maximum of three parameters.
- Identification of the planar web associated with a transformation as a natural geometric invariant under coordinatewise one-variable reparametrizations, linking geometric and probabilistic invariants.
Baselines vs proposed
- Kac–Bernstein theorem (classical): characterized Gaussian distributions via independence of X+Y and X−Y; recovered by Abelian relation framework using F(x,y)=(x+y,x−y).
- Lukacs theorem (classical): characterized gamma laws via independence of X+Y and X/Y; recovered by this framework using corresponding F and density parametrizations.
- Quadrirational Yang–Baxter maps H+I: preserves independence of product measures whose densities admit a 3-parameter family characterized by explicit positive density formulas (Corollary 11).
- Quadrirational Yang–Baxter maps H+II and HAIII similarly characterized with explicit 3-parameter density families (Corollaries 12 and 13).
Limitations
- The results require strong real-analyticity and positive continuous density assumptions, limiting applicability to some distributions with singularities or discrete support.
- The main classification is local and assumes nonsingularity of the associated planar 4-web; singular webs and discontinuous transformations are not covered.
- No adversarial or empirical robustness analysis—this is purely theoretical without practical testing on noisy or perturbed data.
- The full classification of all independence-preserving maps remains open and is conjectured rather than completed.
- The paper does not treat transformations involving more than two variables or higher-dimensional webs.
- The connection to discrete or algorithmic aspects of bot-detection or CAPTCHA-related independence transformations is not addressed.
Open questions / follow-ons
- Can the local algebraizability of nonsingular planar 4-webs of maximal rank be leveraged to yield a complete classification of all real-analytic maps preserving independence of three-parameter families of product measures?
- How can the framework extend to singular webs or to less regular maps and measures, e.g., with discontinuities or singularities in densities?
- Is there a higher-dimensional analogue connecting independence preservation for more variables with corresponding higher-rank webs or generalized Abelian relations?
- Can these geometric characterizations inform new probabilistic models or invariants useful in applied fields such as machine-learning-based bot detection or CAPTCHA analysis?
Why it matters for bot defense
From a bot-defense or CAPTCHA engineering perspective, the paper establishes fundamental structural constraints on transformations preserving independence of input distributions. In CAPTCHAs and bot detection, understanding independence properties of user interactions or challenge-responses under transformations could enable the design of tests that uniquely characterize human versus bot behavior distributions. Although highly theoretical, the identification of planar web invariants as natural descriptors suggests a novel geometric analytic toolkit in analyzing and classifying function transformations that preserve or alter independence. This might guide the construction of CAPTCHAs which resist bot strategy adaptations preserving independence. However, the paper itself does not address adversarial or approximate cases typical in bot detection, limiting direct practical impact without further adaptation.
Cite
@article{arxiv2607_20646,
title={ The Independence-Preserving Property and Planar Web Geometry },
author={ Yoshihiro Gyotoku },
journal={arXiv preprint arXiv:2607.20646},
year={ 2026 },
url={https://arxiv.org/abs/2607.20646}
}