Extending UPG to Power Laws

 Extension of UPG: From Gaussian Attractors to Power-Law Boundaries — q-Potential Functions and the Generalized Midpoint Extremum Theorem

Author: Zhang Suhang

Luoyang School of Mathematics

Abstract

This paper is an extension of the UPG framework. Papers 1–6 established a unified framework centered on the potential function h=-\log(d\mu/d\nu) and the Midpoint Extremum Theorem, whose cleanest manifestation is the Gaussian bell — a parabolic attractor under quadratic potential. Nevertheless, the Gaussian bell is only the expression of the kernel under quadratic potential, not the entirety of the framework. Following the path of "evolving naturally", this paper introduces the family of q-potential functions:

h_q(x)=\frac{1}{q-1}\log\left[1+(q-1)\frac{x^2}{2}\right].

It connects Gaussian distributions and power-law distributions as two ends of the same continuous deformation family. This paper defines the q-geometric measure space and proves the q-Midpoint Extremum Theorem: for q-quadratic potential functions, extremum points, curvature and Hessian structures vary continuously with q. As q\to 1, the theorem reduces to the static theorem in Paper 1; as q\to 2, the power-law boundary case is obtained. This paper further discusses the geometric correspondence of the q-heat equation, q-Central Limit Theorem and q-maximum entropy principle, and clearly marks newly constructed content, cited prior results and open research directions.

Keywords: UPG; q-potential function; q-Gaussian; power-law distribution; Generalized Midpoint Extremum Theorem; Tsallis entropy; fractional heat equation

§1 Introduction

1.1 Status and Limitations of the Gaussian Bell

The core of UPG reads:

h=-\log\frac{d\mu}{d\nu},\qquad \nabla h(\mu)=0 \iff \mu=\mathbb E[X] \iff \text{valley of the surface}.

When h is a quadratic form:

h(s)=\frac{s^2}{2}+\text{constant},

the probabilistic manifestation is the Gaussian bell, the geometric manifestation is a paraboloid, and the algebraic manifestation consists of linear gradient and constant Hessian. This is the minimal model and attractor of UPG.

However, the Gaussian bell is merely the form under quadratic potential. Real-world distributions often exhibit heavy tails, power laws and non-quadratic potential features. If UPG is to become a truly unified framework, it must account for these non-Gaussian cases.

1.2 Objectives of This Paper

The goal is not to "force power laws into Gaussians", but to find a continuous deformation path naturally transitioning from Gaussians to power laws.

Specific objectives:

1. Introduce the family of q-potential functions connecting quadratic potential and logarithmic potential;

2. Define the q-geometric measure space;

3. Prove the q-Midpoint Extremum Theorem, describing how extremum points, curvature and Hessian vary with q;

4. Discuss the geometric correspondence of the q-heat equation, q-Central Limit Theorem and q-maximum entropy;

5. Clearly label new constructions, citations and open directions.

1.3 Structure

- §2 Review of UPG core and the Gaussian case

- §3 q-potential functions and q-geometric measure spaces

- §4 q-Midpoint Extremum Theorem

- §5 q-heat equation and q-Central Limit Theorem

- §6 q-maximum entropy principle

- §7 Relations with power laws, stable distributions and Tsallis statistics

- §8 Conclusions and open directions

§2 Preliminaries: UPG Core and the Gaussian Case

2.1 UPG Core

Definition 2.1 (Geometric Measure Space)

A geometric measure space is a triple (M,\mathcal{B}(M),\mu), where M is a complete separable metric space, \mathcal{B}(M) is the Borel \sigma-algebra, and \mu is a probability measure absolutely continuous with respect to the reference measure \nu.

Definition 2.2 (Potential Function)

h(x)=-\log\frac{d\mu}{d\nu}(x).

Theorem 2.3 (Midpoint Extremum Theorem, Paper 1)

If h is strictly convex and C^2, the following are equivalent:

1. \nabla h(\mu)=0, \nabla^2 h(\mu)\succ 0;

2. \mu=\mathbb E[X], p(\mu)=\max p;

3. The surface z=h(x) attains its global valley at \mu.

2.2 Gaussian Case: Quadratic Potential

When

h(s)=\frac{s^2}{2}+\text{constant},

- Probabilistic side: p(s)\propto e^{-s^2/2}, Gaussian distribution;

- Geometric side: z=h(s) is a paraboloid;

- Algebraic side: \nabla h=s, \nabla^2 h=1, linear gradient and constant Hessian;

- Dynamics: fundamental solution of the heat equation \partial_t p=\frac12\Delta p;

- Limit: attractor of the Central Limit Theorem.

2.3 The Gaussian Bell Is a Manifestation, Not the Kernel

The kernel is: construct potential function via negative logarithm + Midpoint Extremum Theorem. The Gaussian bell is the canonical image of the kernel under quadratic potential. Changing to a non-quadratic potential preserves the kernel, while modifying the specific forms on probabilistic, geometric and algebraic sides.

§3 q-Potential Functions and q-Geometric Measure Spaces

3.1 From Quadratic Potential to Logarithmic Potential

Gaussian case:

h(s)\propto s^2.

Power-law case:

h(x)\propto \log x.

The natural transition between them is given by the q-logarithm.

Definition 3.1 (q-logarithm)

For q\in\mathbb R, define the q-logarithm:

\ln_q x=\frac{x^{1-q}-1}{1-q},\qquad x>0.

As q\to 1, \ln_q x\to\ln x.

Definition 3.2 (q-exponential)

\exp_q x=\left[1+(1-q)x\right]^{\frac{1}{1-q}}.

As q\to 1, \exp_q x\to e^x.

3.2 q-Potential Functions

Definition 3.3 (q-potential function)

h_q(x)=\frac{1}{q-1}\log\left[1+(q-1)\frac{x^2}{2}\right].

Properties:

- q\to 1: h_q(x)\to x^2/2, recovering quadratic potential;

- q=2: h_2(x)=\log(1+x^2/2), logarithmic-type potential corresponding to heavy tails;

- q>1: potential grows slower than quadratic, with heavier tails;

- q<1: potential diverges at finite x, compact support.

3.3 q-Geometric Measure Space

Definition 3.4 (q-Geometric Measure Space)

A q-geometric measure space is a quadruple (M,\mathcal{B}(M),\mu,q), where the potential function of \mu is h_q, i.e.:

\frac{d\mu}{d\nu}(x)\propto \exp_q\left(-\frac{x^2}{2}\right).

Remark 3.5: When q=1, \exp_q(-x^2/2)=e^{-x^2/2}, recovering the Gaussian case.

§4 q-Midpoint Extremum Theorem

4.1 Problem Statement

The original Midpoint Extremum Theorem requires h to be strictly convex and C^2. Under what conditions does the q-potential h_q satisfy these requirements? How do extremum points vary with q?

4.2 Convexity and Extrema

Lemma 4.1 (Convexity of q-potential)

h_q(x) is strictly convex for all x\in\mathbb R when q\ge 1. For q<1, it is strictly convex within |x|<\sqrt{2/(1-q)} and diverges at the boundary.

Proof: Compute second derivatives.
h_q'(x)=\frac{x}{1+(q-1)x^2/2},
h_q''(x)=\frac{1-(q-1)x^2/2}{\left[1+(q-1)x^2/2\right]^2}.

- q=1: h_1''=1>0, strictly convex;
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- q>1: h_q''>0 when x^2<2/(q-1); h_q''<0 for sufficiently large x. Thus for q>1, h_q is not globally convex, only convex on a finite interval and concave outside;
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- q<1: 1-(q-1)x^2/2=1+(1-q)x^2/2>0, and the denominator remains positive for |x|<\sqrt{2/(1-q)}, hence strictly convex.

This result is important: for q>1, the potential becomes concave far away, corresponding to heavy tails.

Theorem 4.2 (q-Midpoint Extremum Theorem)
Let h_q be a q-potential function. Then:

1. When q\le 1, h_q has a unique global minimum at x=0, with \nabla h_q(0)=0, \nabla^2 h_q(0)=1>0. The Midpoint Extremum Theorem holds.
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2. When q>1, h_q has a local minimum at x=0, with \nabla h_q(0)=0, \nabla^2 h_q(0)=1>0. However, h_q becomes concave for |x|>\sqrt{2/(q-1)}, so the minimum is no longer a global valley.
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3. As q\to 1, the theorem reduces to Theorem 2.3 for the Gaussian case.

Proof:
(1) By Lemma 4.1, h_q''>0 globally for q\le 1, so h_q is strictly convex. \nabla h_q(0)=0, hence x=0 is the unique global minimum. By definition, \mu=\mathbb E[X]=0, p(0)=\max p. The surface valley lies at x=0.

(2) For q>1, h_q''(0)=1>0, so x=0 is a local minimum. But h_q'' becomes negative when |x|>\sqrt{2/(q-1)}, so the function turns concave. Thus x=0 is not a global minimum. At this time the probability density p_q(x)\propto \exp_q(-x^2/2) has power-law tails, and expectation may diverge.

(3) Let q\to 1. Since \ln_q\to\ln, \exp_q\to\exp, h_q\to h_1=x^2/2, recovering Theorem 2.3.
∎

Remark 4.3 (Global vs Local)
The q-Midpoint Extremum Theorem reveals that the Gaussian case (q=1) is the watershed between global convexity and global minima. For q>1, extrema degenerate to local ones, and global structure is destroyed by heavy tails. This aligns with the intuition that power-law distributions may have divergent expectations, with extrema located at boundaries or infinity.

§5 q-Heat Equation and q-Central Limit Theorem

5.1 q-Heat Equation

The classical heat equation
\partial_t p=\frac12\Delta p
has Gaussian fundamental solutions.

The q-heat equation (Tsallis heat equation):
\partial_t p=\frac12\Delta p^q
admits q-Gaussian fundamental solutions:
p_q(x,t)\propto \exp_q\left(-\frac{x^2}{2t}\right).

Geometric interpretation: As q\to1, the q-heat equation recovers the heat equation, q-Gaussian recovers Gaussian, and the parabolic attractor is restored. When q>1, the q-heat equation describes anomalous diffusion, the attractor becomes q-Gaussian, and the geometric surface is no longer a paraboloid but a logarithmic or more general surface.

5.2 q-Central Limit Theorem

Classical CLT: i.i.d. variables with finite variance → Gaussian.

q-CLT (Tsallis Central Limit Theorem): under q-independence and q-convolution, attractors are q-Gaussians.

Geometric meaning: the attractor of classical CLT is a paraboloid; the attractor of q-CLT is a q-paraboloid. As q\to1, the q-paraboloid degenerates to a paraboloid.

5.3 Boundary: Stable Distributions

When variance diverges but the stable law holds, the attractor is not Gaussian but an \alpha-stable distribution, whose potential function reads:
h_\alpha(x)\propto |x|^\alpha,\qquad 0<\alpha<2.

Geometrically: |x|^\alpha is a paraboloid at \alpha=2, and a peaked surface for \alpha<2, with extremum at the origin but non-differentiable (\alpha<1) or singular derivative.

Relationship between q-potential and stable distributions: q-Gaussians have power-law tails for q>1, but their exact attractor property differs from stable distributions. q-Gaussians are attractors of the generalized q-CLT; stable distributions are attractors of classical i.i.d. variables with divergent variance. They are heavy-tail manifestations under different mechanisms.

§6 q-Maximum Entropy Principle

6.1 Classical Maximum Entropy

Under constraint \mathbb E[X^2]=\sigma^2, the distribution maximizing Shannon entropy is Gaussian.

Geometric meaning: quadratic constraint → quadratic potential → paraboloid → Gaussian.

6.2 q-Maximum Entropy

Under the constraint \mathbb E_q[X^2]=\sigma_q^2 (q-expectation), the distribution maximizing Tsallis entropy
S_q=-\int p^q\log_q p
is q-Gaussian.

Geometric meaning: q-quadratic constraint → q-quadratic potential → q-paraboloid → q-Gaussian.

As q\to1, Tsallis entropy reduces to Shannon entropy, and q-Gaussian reduces to Gaussian.

§7 Relations with Power Laws, Stable Distributions and Tsallis Statistics

7.1 Power-Law Distributions

Power-law distribution:
p(x)\propto x^{-\alpha},\qquad x\to\infty.
Potential function:
h(x)\propto \alpha\log x.

Within the q-potential framework, as q\to 2 and x\to\infty:
h_q(x)=\frac{1}{q-1}\log\left[1+(q-1)\frac{x^2}{2}\right]\sim \frac{2}{q-1}\log x.
Therefore, the q-potential naturally yields logarithmic potential in the tail region, corresponding to power-law tails. Power-law distributions are the asymptotic behavior of the q-potential framework for q>1 in the large-x regime.

7.2 Stable Distributions

Stable distributions are the generalization of classical CLT under divergent variance, with potential |x|^\alpha. The q-potential yields logarithmic potential when q\to2, while stable distributions yield fractional power potential for \alpha<2. They differ, yet both describe heavy tails.

7.3 Tsallis Statistics

Tsallis statistics take q-entropy, q-exponential and q-Gaussian as core components, serving as the direct source of the q-structure in this paper. The contribution of this work is embedding the q-structure of Tsallis statistics into the UPG potential-extremum theorem framework, providing geometric interpretation (q-paraboloid) and extremum conditions (q-Midpoint Extremum Theorem).

§8 Conclusions and Open Directions

8.1 Results of This Paper

1. Introduce the family of q-potential functions connecting quadratic potential (Gaussian) and logarithmic potential (power law);
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2. Define q-geometric measure spaces;
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3. Prove the q-Midpoint Extremum Theorem: global extrema hold for q\le1, while extrema degenerate to local ones for q>1, as heavy tails destroy global structure;
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4. Discuss the geometric correspondence of q-heat equation, q-Central Limit Theorem and q-maximum entropy;
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5. Clarify relations among q-Gaussians, power laws, stable distributions and Tsallis statistics.

8.2 Honest Positioning

- Cited prior results: q-logarithm, q-exponential, Tsallis entropy, q-Gaussian, q-heat equation, q-Central Limit Theorem;
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- New constructions: q-potential functions, q-geometric measure spaces, q-Midpoint Extremum Theorem;
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- Scope limitation: this paper restricts to one-dimensional case, real q, sufficiently smooth potential functions;
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- Open directions:
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- High-dimensional q-potential functions and q-geometric measure spaces;
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- Alternative extremum concepts after global extremum fails for q>1 (boundary extrema, constrained extrema, extrema at infinity);
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- Combination of q-Midpoint Extremum Theorem with the dynamic version in Paper 6;
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- Weighted versions of q-potential functions under ∞-topos framework;
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- Rigorous relations with stable distributions and Lévy flights.

8.3 One-Sentence Summary

The Gaussian bell is the canonical image of UPG under quadratic potential, and power laws are the boundary manifestation of UPG under logarithmic potential. The family of q-potential functions links them via a continuous deformation path, and the q-Midpoint Extremum Theorem describes the continuous change of extremum structure along this path. q=1 is the watershed for global convexity and global extrema. For q>1, heavy tails break global structure and extrema become local. This is the natural unification from Gaussian attractors to power-law boundaries.

References

Omitted