Paired Statistical Solitons and Torse-Forming Rigidity on Statistical Manifolds

Abstract

We study paired statistical $\eta$-Ricci–Bourguignon almost solitons on a statistical manifold endowed with two dual torsion-free statistical connections. The same potential field is required to satisfy compatible soliton equations for both connections. By adding and subtracting these equations, we obtain identities that separate the Levi–Civita contribution from the cubic tensor associated with the statistical difference tensor. We distinguish the connection-induced symmetric deformation tensor from the ordinary Lie derivative and derive sum and difference identities that isolate the cubic tensor associated with the statistical difference tensor. We prove a spanning-family criterion for full collapse of the dual connections and derive corresponding gradient $\eta$-Ricci–Bourguignon and gradient Schouten identities. Explicit affine and Fisher-geometric examples show non-vacuity and exhibit $\mathfrak{K}\ne 0$ together with $\mathfrak{K}(X,·)=0$. A Lorentzian perfect-fluid formulation and an exact FLRW-type realization illustrate how the geometric equations enter relativistic models. The mathematical conclusions are separated from model-dependent physical interpretations.

1. Introduction

Statistical geometry equips a manifold with a metric and a torsion-free affine connection whose covariant derivative of the metric is a symmetric cubic tensor. The metric determines a dual connection, so the geometry naturally contains two affine curvature tensors in addition to the Levi–Civita geometry. This dualistic framework originated in information geometry; foundational sources include Amari’s monograph and Lauritzen’s account of statistical manifolds [1,2]. The study of soliton equations provides a separate motivation: Ricci solitons model self-similar solutions of the Ricci flow [3]. The Ricci–Bourguignon flow modifies the Ricci-flow equation by a scalar-curvature term; its analytic properties and associated solitons were studied by Catino et al. [4], while Bourguignon’s work provides earlier geometric background on Ricci curvature and Einstein metrics [5].

Classical soliton theory includes Ricci, Ricci–Bourguignon, Schouten, Riemann, and related self-similar structures. Gradient solitons on statistical manifolds have been studied previously, while recent work on Lorentzian and semi-Riemannian solitons has produced rigidity results involving conformal, concircular, and torse-forming vector fields [6,7,8,9]. These developments suggest a natural question: what additional rigidity is forced when one and the same vector field generates compatible soliton structures for both members of a dual statistical pair?

Statistical solitons are not new as a general concept. Siddiqui, Chen, and Bahadir studied statistical solitons for Ricci-symmetric statistical warped products and used a common vector field in the equations associated with the dual connections [10]. Gradient solitons on statistical manifolds were subsequently investigated by Blaga and Chen [6]. A closely related, but geometrically different, paired construction was studied by Manev: the two almost-soliton equations there are associated with two related B-metrics on almost contact complex Riemannian manifolds, with a vertical torse-forming potential [11]. These works mean that neither the generic idea of a pair of soliton equations nor the use of torse-forming potentials can be claimed as new.

Recent works also show that torse-forming potentials and soliton-type equations remain active themes in differential geometry. Studies have considered Ricci solitons with torse-forming fields in modified-gravity settings, as well as potential vector fields of generalized soliton equations and soliton structures on Lorentzian manifolds [12]. These results provide useful geometric context, but they employ different structures from the present dual statistical formulation.

The narrower question addressed here is the following. Fix one metric g, dual statistical connections $\mathfrak{D}$ and ${\mathfrak{D}}^{*}$, a common 1-form $\eta$, a common potential field X, and common scalar parameters. What identities follow when the same $\eta$-Ricci–Bourguignon deformation equation is imposed for both connections? The key term is the symmetric cubic tensor

$$C(Y,Z,W)=g(\mathfrak{K}(Y,Z),W),\mathfrak{K}=\mathfrak{D}-\overline{\mathfrak{D}}=\frac{1}{2}(\mathfrak{D}-{\mathfrak{D}}^{*}),$$

where $\overline{\mathfrak{D}}$ is the Levi–Civita connection. Adding and subtracting the paired equations isolates this tensor explicitly.

The theorem-level distinction from earlier paired constructions is summarized in Table 1. Relative to the Ricci-operator equations in [10], we use a symmetric $(0,2)$ deformation equation with a Ricci–Bourguignon scalar term and a common $\eta \otimes \eta$ term, and we identify the cubic-tensor term produced by subtracting the two connection-induced deformation tensors. Relative to [11], the two connections here are metric-dual statistical connections for the same metric, rather than Levi–Civita connections of a pair of related B-metrics. We make no claim that this distinction alone establishes broad novelty; the results below are stated as explicit consequences of the dual identities and the additional common-torse hypothesis.

Table 1. Scope of the present equations compared with related paired constructions.

Work	Geometric Setting and Paired Data	Distinction Relevant to This Paper
Siddiqui et al. (2019) [10]	Statistical solitons on Ricci-symmetric statistical warped products; the same vector field enters equations for dual connections.	The present system is a symmetric $(0,2)$ $\eta$-Ricci–Bourguignon deformation equation and its difference identity displays the cubic tensor $C_X$.
Manev (2026) [11]	Associated $\eta$-Ricci–Bourguignon almost solitons for two related B-metrics on an almost contact complex Riemannian manifold; a vertical torse-forming potential is studied.	Here a single metric is paired with its dual torsion-free statistical connections; the directional conclusion is expressed through the statistical difference tensor.
Present paper	One metric, dual statistical connections, the same potential, the same $\eta$, and common parameters.	Sum/difference identities, the trace obstruction, directional annihilation under common torse data, and a spanning-family criterion are proved directly.

The main results are these. First, we derive the sum and difference identities for a paired statistical $\eta$-Ricci–Bourguignon almost soliton and record the scalar-curvature identity that holds for any dual torsion-free statistical connections. Second, we show that a common torse-forming field with identical torse data for the two connections satisfies $\mathfrak{K}(X,·)=0$; the difference equation then gives equality of the statistical Ricci tensors, with no restriction on the Bourguignon parameter. Third, we show that a local frame of such fields forces $\mathfrak{K}=0$. Fourth, we record the corresponding algebraic identities for gradient Ricci–Bourguignon and gradient Schouten systems; these formulations are treated as paired extensions of established one-connection equations, not as wholly new definitions. Finally, we provide explicit examples and a cautiously formulated Lorentzian specialization.

To place the new construction in context, it is useful to distinguish three levels of information. At the first level, a single statistical connection produces a soliton equation and the familiar trace, divergence, and conformal consequences. At the second level, a statistical manifold supplies a dual connection and hence a second curvature tensor. At the third level, requiring the same vector field to satisfy compatible soliton equations for both connections creates an overdetermined system. The present paper is concerned with this third level. In particular, the additional equation is not used merely to duplicate a known theorem: its role is to identify the part of the geometry that changes sign under the duality operation, namely the cubic tensor generated by $\mathfrak{K}$.

A second reason for retaining several classical consequences is methodological. Results concerning concircular fields, conformal potentials, perfect-fluid Ricci tensors, and gradient Schouten solitons form the natural one-connection boundary cases of the paired system. We therefore state them in a compact and explicitly labeled comparison form. This makes it possible to see which conclusions survive under one connection, which ones require both dual equations, and where the additional rigidity enters. In particular, every comparison result is cited to the corresponding literature, whereas the paired statements below are derived directly from the dual statistical identities.

We also emphasize that the term “statistical” is used structurally throughout the paper. No physical interpretation is invoked until the Lorentzian specialization in Section 7, and even there the geometric conclusions are separated from any model-dependent equation of state. This distinction prevents cosmological terminology from being mistaken for a mathematical consequence of the soliton equation itself.

The paper is organized as follows: Section 2 collects the statistical-geometric conventions. Section 3 introduces paired statistical Ricci–Bourguignon solitons and derives the fundamental sum and difference equations. Section 4 studies torse-forming and concircular rigidity. Section 5 develops paired gradient Ricci–Bourguignon and Schouten solitons. Section 6 gives rigidity and conditional warped-product classification under completeness and scalar-curvature assumptions. Section 7 presents explicit examples and the Lorentzian perfect-fluid form and an FLRW calculation in the Levi–Civita boundary case, and Section 8 concludes the paper.

2. Preliminaries

Let $(N^m,g)$ be a smooth semi-Riemannian manifold and let $\tau \left(N\right)$ denote its smooth vector fields. Throughout, all tensorial identities are understood on $\tau \left(N\right)$ unless otherwise stated.

Definition 1 

(Statistical structure and dual connection). A statistical structure on $(N,g)$ is a torsion-free affine connection $\mathfrak{D}$ for which

$$\left({\mathfrak{D}}_{X}g\right)(Y,Z)=\left({\mathfrak{D}}_{Y}g\right)(X,Z)$$

for all vector fields $X,Y,Z$. The dual connection ${\mathfrak{D}}^{*}$ is characterized by

$$Xg(Y,Z)=g({\mathfrak{D}}_{X}Y,Z)+g(Y,{\mathfrak{D}}_X^{*}Z).$$

(1)

For a statistical structure, this dual connection is also torsion-free and statistical; see [1,2].

Let $\overline{\mathfrak{D}}$ denote the Levi–Civita connection of g. Then

$$\overline{\mathfrak{D}}={\displaystyle \frac{1}{2}}(\mathfrak{D}+{\mathfrak{D}}^{*}).$$

(2)

The difference tensor is

$$\mathfrak{K}(X,Y)={\mathfrak{D}}_{X}Y-{\overline{\mathfrak{D}}}_{X}Y={\displaystyle \frac{1}{2}}({\mathfrak{D}}_{X}Y-{\mathfrak{D}}_X^{*}Y).$$

(3)

For a statistical manifold, $\mathfrak{K}(X,Y)=\mathfrak{K}(Y,X)$ and

$$g\left(\mathfrak{K}\right(X,Y),Z)=g(Y,\mathfrak{K}(X,Z\left)\right).$$

(4)

Consequently,

$$C(X,Y,Z):=g\left(\mathfrak{K}\right(X,Y),Z)$$

(5)

is a totally symmetric cubic tensor.

We use the curvature convention

$$\mathfrak{R}(X,Y)Z={\mathfrak{D}}_X{\mathfrak{D}}_{Y}Z-{\mathfrak{D}}_Y{\mathfrak{D}}_{X}Z-{\mathfrak{D}}_{[X,Y]}Z,$$

and define $\mathfrak{Ric}(Y,Z)=tr\{V\mapsto \mathfrak{R}(V,Y\left)Z\right\}$ and $r={tr}_g\mathfrak{Ric}$. Stars denote the corresponding quantities for ${\mathfrak{D}}^{*}$. A pseudo-orthonormal frame satisfies $g(e_i,e_j)={\epsilon }_i{\delta }_{ij}$ with ${\epsilon }_i\in \{-1,1\}$. We write $\mathfrak{Q},{\mathfrak{Q}}^{*}$ for the corresponding Ricci operators.

For $X\in \tau \left(N\right)$, define the connectionwise Lie derivatives

$$\left({\mathcal{L}}_X^{\mathfrak{D}}g\right)(Y,Z)=g({\mathfrak{D}}_{Y}X,Z)+g(Y,{\mathfrak{D}}_{Z}X),$$

(6)

$$\left({\mathcal{L}}_X^{{\mathfrak{D}}^{*}}g\right)(Y,Z)=g({\mathfrak{D}}_Y^{*}X,Z)+g(Y,{\mathfrak{D}}_Z^{*}X).$$

(7)

The Levi–Civita Lie derivative is denoted by ${\mathcal{L}}_X^{\overline{\mathfrak{D}}}g$. From (2) and (3),

$${\displaystyle \frac{1}{2}}\left({\mathcal{L}}_X^{\mathfrak{D}}g+{\mathcal{L}}_X^{{\mathfrak{D}}^{*}}g\right)={\mathcal{L}}_X^{\overline{\mathfrak{D}}}g,$$

(8)

and

$${\displaystyle \frac{1}{2}}\left({\mathcal{L}}_X^{\mathfrak{D}}g-{\mathcal{L}}_X^{{\mathfrak{D}}^{*}}g\right)(Y,Z)=2C(Y,Z,X).$$

(9)

Lemma 1 

(Equality of dual scalar curvatures). For dual torsion-free statistical connections, the scalar curvatures agree:

$$r=r^{*}.$$

(10)

Proof. 

The curvature duality identity is

$$g(\mathfrak{R}(X,Y)Z,W)=-g(Z,{\mathfrak{R}}^{*}(X,Y)W).$$

For a pseudo-orthonormal frame, contracting twice gives

$$r=\sum _{i,j}{\epsilon }_i{\epsilon }_{j}g(\mathfrak{R}(e_i,e_j)e_j,e_i)=-\sum _{i,j}{\epsilon }_i{\epsilon }_{j}g({\mathfrak{R}}^{*}(e_i,e_j)e_i,e_j).$$

Using skew-symmetry of curvature in the first two arguments and relabelling $i,j$ yields the contraction defining $r^{*}$. Hence $r=r^{*}$. □

A vector field X is called torse-forming with respect to $\mathfrak{D}$ if

$${\mathfrak{D}}_{Y}X=aY+\omega \left(Y\right)X,$$

(11)

for a smooth function a and a 1-form $\omega$. The analogous definition is used for ${\mathfrak{D}}^{*}$. It is called common torse-forming if the same pair $(a,\omega )$ occurs for both dual connections.

A vector field is called common concircular when

$${\mathfrak{D}}_{Y}X=aY,{\mathfrak{D}}_Y^{*}X=aY.$$

(12)

This is the special case $\omega =0$ of a common torse-forming field. Define the trace vector ${\tau }_{\mathfrak{K}}:={\sum }_i{\epsilon }_i\mathfrak{K}(e_i,e_i)$, so that

$${tr}_g{C}_X=g({\tau }_{\mathfrak{K}},X).$$

(13)

For later use, recall the statistical perfect-fluid form

$${\mathfrak{Ric}}^{*}(Y,Z)=Ag(Y,Z)+B\eta \left(Y\right)\eta \left(Z\right),$$

(14)

where $\eta \left(Y\right)=g(\sigma ,Y)$ and $g(\sigma ,\sigma )=-1$ in the Lorentzian case. The paired theory below will produce the same form with $\mathfrak{Ric}$ once the dual-rigidity condition is satisfied.

3. Paired Statistical Ricci–Bourguignon Solitons

Definition 2 

(Paired statistical $\eta$-Ricci–Bourguignon almost soliton). Let η be a fixed 1-form and let X be a vector field on N. We say that the data define a paired statistical η-Ricci–Bourguignon almost soliton if

$$\begin{array}{cc}\hfill \mathfrak{Ric}(Y,Z)+{\displaystyle \frac{1}{2}}\left({\mathcal{L}}_X^{\mathfrak{D}}g\right)(Y,Z)+\left(\lambda -\beta r\right)g(Y,Z)+\mu \eta \left(Y\right)\eta \left(Z\right)& =0,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\mathfrak{Ric}}^{*}(Y,Z)+{\displaystyle \frac{1}{2}}\left({\mathcal{L}}_X^{{\mathfrak{D}}^{*}}g\right)(Y,Z)+\left(\lambda -\beta r^{*}\right)g(Y,Z)+\mu \eta \left(Y\right)\eta \left(Z\right)& =0.\hfill \end{array}$$

(16)

Here λ is a smooth function and $\beta ,\mu$ are constants. If λ is constant as well, we call it a paired statistical η-Ricci–Bourguignon soliton. The same X, η, λ, β, and μ are used in both equations.

One-Connection Identities and the Paired Extension

Before studying the coupled system, we record a few identities that belong to the one-connection theory. They will later serve as comparison statements. Suppose that only the ${\mathfrak{D}}^{*}$-equation is imposed, namely

$${\mathfrak{Ric}}^{*}+{\displaystyle \frac{1}{2}}{\mathcal{L}}_X^{{\mathfrak{D}}^{*}}g+(\lambda -\beta r^{*})g+\mu \eta \otimes \eta =0.$$

(17)

Taking the metric trace gives the scalar balance

$$(1-m\beta )r^{*}+{tr}_g(\mu \eta \otimes \eta )+{div}^{{\mathfrak{D}}^{*}}\left(X\right)+m\lambda =0.$$

(18)

Thus, whenever $\eta$ is metric-dual to a unit vector field, the final term is simply $\mu g^{-1}(\eta ,\eta )$. Formula (18) is the one-connection analogue of the trace obstruction obtained below for the paired system.

The conformal case is equally instructive. If

$${\mathcal{L}}_X^{{\mathfrak{D}}^{*}}g=2\psi g,$$

(19)

then the soliton equation becomes

$${\mathfrak{Ric}}^{*}=\left(\beta r^{*}-\lambda -\psi \right)g-\mu \eta \otimes \eta .$$

(20)

In the Ricci case $(\mu =0)$, this is the familiar Einstein-type reduction in a conformal soliton. Such one-sided reductions are classical in spirit and are included here only to make the comparison with the genuinely dual theory transparent; see, for example, the statistical and semi-Riemannian soliton literature cited in [6,8,12].

Theorem 1 

(Paired decomposition identities). Every paired statistical η-Ricci–Bourguignon almost soliton satisfies

$${\displaystyle \frac{\mathfrak{Ric}+{\mathfrak{Ric}}^{*}}{2}}+{\displaystyle \frac{1}{2}}{\mathcal{L}}_X^{\overline{\mathfrak{D}}}g+\left(\lambda -\beta {\displaystyle \frac{r+r^{*}}{2}}\right)g+\mu \eta \otimes \eta =0,$$

(21)

and

$$\mathfrak{Ric}-{\mathfrak{Ric}}^{*}+2C_X=0,$$

(22)

Consequently,

$${tr}_g{C}_X=0,g({\tau }_{\mathfrak{K}},X)=0.$$

(23)

Proof. 

Adding (15) and (16), followed by (8), gives (21); the equality $r=r^{*}$ follows from Lemma 1. Subtracting (16) from (15) and using (9) gives (22). Taking the metric trace of (23) and again using $r=r^{*}$ yields $2{tr}_g{C}_X=0$. Equation (13) gives the final assertion. □

Remark 1. 

The equality $r=r^{*}$ is a property of dual statistical connections; it does not require the paired soliton equations. Thus no separate case at $\beta =1/m$ is needed for scalar-curvature comparison. A trace identity containing $(1-m\beta )(r-r^{*})$ should not be used to suggest otherwise: its scalar-curvature difference already vanishes in the statistical setting considered here.

Theorem 2 

(Directional rigidity under common torse data). Suppose the paired soliton of Definition 2 has a common torse-forming potential satisfying (11). Then

$$\mathfrak{K}(X,Y)=0\mathit{for}\mathit{all}Y,C_X=0,$$

(24)

and

$$\mathfrak{Ric}={\mathfrak{Ric}}^{*}.$$

(25)

No restriction on β is required for these conclusions.

Proof. 

Subtracting the two relations in (11) gives $2\mathfrak{K}(Y,X)=0$. Symmetry of $\mathfrak{K}$ yields $\mathfrak{K}(X,Y)=0$. By total symmetry of C, this is equivalent to $C_X(Y,Z)=C(X,Y,Z)=0$. Equation (22) therefore gives $\mathfrak{Ric}={\mathfrak{Ric}}^{*}$. The scalar-curvature equality was established independently in Lemma 1, so the argument includes the value $\beta =1/m$. □

4. Torse-Forming and Concircular Rigidity

The preceding theorem already shows that a common torse-forming potential annihilates the statistical difference tensor in the direction of the potential. We next derive the corresponding curvature identity.

Theorem 3 

(Common curvature identity). Let X be common torse-forming as in (11). Then

$$\mathfrak{R}(Y,Z)X=Y\left(a\right)Z-Z\left(a\right)Y+d\omega (Y,Z)X+a\omega \left(Z\right)Y-a\omega \left(Y\right)Z,$$

(26)

$${\mathfrak{R}}^{*}(Y,Z)X=Y\left(a\right)Z-Z\left(a\right)Y+d\omega (Y,Z)X+a\omega \left(Z\right)Y-a\omega \left(Y\right)Z.$$

(27)

Hence

$$\mathfrak{R}(Y,Z)X={\mathfrak{R}}^{*}(Y,Z)X.$$

(28)

Proof. 

From (11),

$${\mathfrak{D}}_Y{\mathfrak{D}}_{Z}X=Y\left(a\right)Z+a{\mathfrak{D}}_{Y}Z+Y\left(\omega \left(Z\right)\right)X+\omega \left(Z\right){\mathfrak{D}}_{Y}X.$$

Interchanging $Y,Z$, subtracting, and using torsion-freeness gives

$$\mathfrak{R}(Y,Z)X=Y\left(a\right)Z-Z\left(a\right)Y+d\omega (Y,Z)X+a\omega \left(Z\right)Y-a\omega \left(Y\right)Z.$$

The same calculation for ${\mathfrak{D}}^{*}$ gives the second identity. The factor a arises from the terms $a{\mathfrak{D}}_{Y}Z$ and $a{\mathfrak{D}}_{Z}Y$ in the expansion. □

Corollary 1 

(Common concircular rigidity). If X is common concircular, i.e., $\omega =0$, then

$$\mathfrak{R}(Y,Z)X={\mathfrak{R}}^{*}(Y,Z)X=Y\left(a\right)Z-Z\left(a\right)Y.$$

(29)

If, in addition, a is constant, then

$$\mathfrak{R}(Y,Z)X={\mathfrak{R}}^{*}(Y,Z)X=0.$$

(30)

Theorem 4 

(Rigidity of the difference tensor in the torse direction). Let X be a common torse-forming vector field on a statistical manifold. Then

$$\mathfrak{K}(X,Y)=0,\forall Y\in \tau \left(N\right),$$

(31)

and, consequently,

$$C(X,Y,Z)=0,\forall Y,Z\in \tau \left(N\right).$$

(32)

Thus the entire cubic statistical tensor vanishes whenever one of its arguments is the common torse-forming field.

Proof. 

Equation (31) is Theorem 2. Since $C(X,Y,Z)=g\left(\mathfrak{K}\right(X,Y),Z)$, (32) follows. □

A One-Connection Perfect-Fluid Comparison

The preceding rigidity statement can also be interpreted from the viewpoint of the perfect-fluid models that motivated several earlier statistical soliton papers. In four-dimensional Lorentzian signature, suppose a single statistical Ricci tensor admits the decomposition

$${\mathfrak{Ric}}^{*}=Ag+B\eta \otimes \eta ,g(\sigma ,\sigma )=-1,$$

(33)

where $\eta \left(Y\right)=g(\sigma ,Y)$. If $\sigma$ is statistical concircular, ${\mathfrak{D}}_Y^{*}\sigma =aY$, then

$${\mathfrak{R}}^{*}(Y,Z)\sigma =Y\left(a\right)Z-Z\left(a\right)Y,$$

(34)

and contraction against a pseudo-orthonormal frame gives a linear relation between $Y\left(a\right)$ and the coefficients $A,B$. In particular, the familiar conclusion that ${\mathfrak{Ric}}^{*}(Y,\sigma )$ is proportional to $\eta \left(Y\right)$ follows without requiring any paired hypothesis, and contraction against a pseudo-orthonormal frame gives a linear relation between $Y\left(a\right)$ and the coefficients $A,B$.

For the present problem, however, the paired assumption implies much more. If the same $\sigma$ is concircular for both dual connections, then $\mathfrak{K}(\sigma ,·)=0$, the two curvature tensors agree on $\sigma$, and the two Ricci tensors agree everywhere. Thus the perfect-fluid form can be transported from one side of the statistical structure to the other. This is the precise sense in which the paired formulation strengthens the earlier one-connection setting.

Theorem 5 

(Full difference-tensor rigidity under a spanning torse family). Let $U\subset N$ be an open set. Suppose there exist m common torse-forming vector fields $X_1,\dots ,X_m$ whose values form a local frame of $TN$ on U. Then

$${\mathfrak{K}|}_U=0.$$

(35)

Consequently,

$${\mathfrak{D}|}_U={\mathfrak{D}}^{*}{{|}_U=\overline{\mathfrak{D}}|}_U.$$

(36)

In particular, a paired statistical structure admitting a local spanning family of common torse-forming potentials is locally Riemannian.

Proof. 

By Theorem 4, every common torse-forming field $X_j$ satisfies $\mathfrak{K}(Y,X_j)=0$ for all Y. Since $X_1,\dots ,X_m$ form a local frame, bilinearity of $\mathfrak{K}$ in its second argument gives $\mathfrak{K}(Y,Z)=0$ for all $Y,Z$ on U. The identities $\mathfrak{D}=\overline{\mathfrak{D}}+\mathfrak{K}$ and ${\mathfrak{D}}^{*}=\overline{\mathfrak{D}}-\mathfrak{K}$ then yield (36). □

Remark 2 

(Sharpness). The explicit examples in Section 7 show that the spanning assumption cannot be omitted: a single common torse-forming potential may annihilate the difference tensor in its own direction while $\mathfrak{K}$ remains nonzero elsewhere.

5. Gradient Paired Solitons

We now assume that the common potential is a gradient. This introduces an additional statistical object: the pair of Hessians associated with the dual connections.

Let $X={grad}_{g}f$. Define

$${Hess}^{\mathfrak{D}}f(Y,Z)=g({\mathfrak{D}}_{Y}gradf,Z),$$

(37)

and

$${Hess}^{{\mathfrak{D}}^{*}}f(Y,Z)=g({\mathfrak{D}}_Y^{*}gradf,Z).$$

(38)

Then

$${Hess}^{\mathfrak{D}}f={Hess}^{\overline{\mathfrak{D}}}f+C(·,·,gradf),$$

(39)

$${Hess}^{{\mathfrak{D}}^{*}}f={Hess}^{\overline{\mathfrak{D}}}f-C(·,·,gradf).$$

(40)

Relation with One-Sided Gradient Statistical Solitons

Recent work on gradient Ricci solitons has also emphasized quantitative inequalities and equality cases in specialized geometric settings [13]. The present paired approach is different in emphasis: it uses dual statistical connections to identify a tensorial obstruction.

Gradient statistical solitons previously studied on statistical manifolds arise by imposing a single equation involving the Hessian associated with one statistical connection [6]. In the present notation, the one-sided gradient ${\mathfrak{D}}^{*}$-equation may be written as

$${\mathfrak{Ric}}^{*}+{Hess}^{{\mathfrak{D}}^{*}}f+(\lambda -\beta r^{*})g+\mu \eta \otimes \eta =0.$$

(41)

The importance of the paired formulation is that the two Hessians are not independent: their difference is exactly the cubic tensor term in (39) and (40). Consequently, an equality between the two gradient equations is equivalent to a rigidity statement for $\left(C\right(·,·,gradf\left)\right)$. This gives a concrete geometric meaning to the duality constraint and distinguishes the present construction from a parallel repetition of the one-sided theory.

Definition 3 

(Paired gradient Ricci–Bourguignon almost soliton). A statistical manifold admits a paired gradient η-Ricci–Bourguignon almost soliton with potential f if

$$\begin{array}{cc}\hfill \mathfrak{Ric}+{Hess}^{\mathfrak{D}}f+(\lambda -\beta r)g+\mu \eta \otimes \eta & =0,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\mathfrak{Ric}}^{*}+{Hess}^{{\mathfrak{D}}^{*}}f+(\lambda -\beta r^{*})g+\mu \eta \otimes \eta & =0.\hfill \end{array}$$

(43)

Theorem 6 

(Gradient sum-and-difference identities). A paired gradient η-Ricci–Bourguignon almost soliton satisfies

$${\displaystyle \frac{\mathfrak{Ric}+{\mathfrak{Ric}}^{*}}{2}}+{Hess}^{\overline{\mathfrak{D}}}f+\left(\lambda -\beta {\displaystyle \frac{r+r^{*}}{2}}\right)g+\mu \eta \otimes \eta =0,$$

(44)

and

$$\mathfrak{Ric}-{\mathfrak{Ric}}^{*}+2C(·,·,gradf)=0.$$

(45)

If $gradf$ is common torse-forming with the same torse data for $\mathfrak{D}$ and ${\mathfrak{D}}^{*}$, then

$$\mathfrak{K}(gradf,·)=0,C(·,·,gradf)=0,\mathfrak{Ric}={\mathfrak{Ric}}^{*}.$$

Proof. 

Add and subtract (42) and (43), then use (39) and (40). The common-torse conclusion follows exactly as in Theorem 2. □

Definition 4 

(Paired gradient Schouten almost soliton). For $m>1$, a statistical manifold admits a paired gradient Schouten almost soliton with potential f if

$$\begin{array}{cc}\hfill \mathfrak{Ric}+{Hess}^{\mathfrak{D}}f+\left(\lambda -{\displaystyle \frac{r}{2(m-1)}}\right)g& =0,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\mathfrak{Ric}}^{*}+{Hess}^{{\mathfrak{D}}^{*}}f+\left(\lambda -{\displaystyle \frac{r^{*}}{2(m-1)}}\right)g& =0.\hfill \end{array}$$

(47)

Theorem 7 

(Schouten difference identity). Every paired gradient Schouten almost soliton satisfies

$$\mathfrak{Ric}-{\mathfrak{Ric}}^{*}+2C(·,·,gradf)-{\displaystyle \frac{r-r^{*}}{2(m-1)}}g=0.$$

(48)

If $gradf$ is common torse-forming with identical data for the dual connections, then $\mathfrak{Ric}={\mathfrak{Ric}}^{*}$ and both Hessians equal ${Hess}^{\overline{\mathfrak{D}}}f$.

Proof. 

Subtract (47) from (46), and use (39) and (40). Under the common torse-forming assumption, the cubic term vanishes and the tensorial difference equation gives $\mathfrak{Ric}={\mathfrak{Ric}}^{*}$. Equations (39) and (40) then give equality of the Hessians with the Levi–Civita Hessian. □

6. Rigidity and Conditional Classification

We now investigate what happens when the paired gradient system is supplemented by completeness and conformality. The purpose of this section is to isolate a classification statement that is valid only after the statistical obstruction is removed.

Theorem 8 

(Conformal reduction in a paired gradient soliton). Let $(N^m,g,\mathfrak{D},{\mathfrak{D}}^{*})$ admit a paired gradient η-Ricci–Bourguignon almost soliton with potential f. Assume

$${\mathcal{L}}_{gradf}^{\overline{\mathfrak{D}}}g=2\psi g.$$

(49)

If $gradf$ is common torse-forming, then

$$\mathfrak{Ric}=\left(\beta r-\lambda -\psi \right)g-\mu \eta \otimes \eta .$$

(50)

In the Ricci case $\mu =0$, the statistical manifold is statistical Einstein.

Proof. 

By Theorem 2, $\mathfrak{Ric}={\mathfrak{Ric}}^{*}$ and the statistical difference tensor vanishes in the potential direction. Substitute (49) into (15) to obtain (50). □

Comparison with the Classical Complete Conformal Case

The classification problem is substantially more delicate for statistical manifolds than for ordinary Riemannian manifolds. In the latter setting, a complete manifold carrying a nontrivial closed conformal field is frequently forced into a warped-product or spherical model under suitable curvature assumptions. For the statistical problem, the same argument starts with the regular level sets of f, but an additional tensorial obstruction is present: the statistical Ricci tensor differs from the Levi–Civita Ricci tensor through terms involving $\overline{\mathfrak{D}}\mathfrak{K}$ and $\mathfrak{K}\circ \mathfrak{K}$.

This observation explains why the classification theorem below is deliberately formulated with an explicit curvature-coincidence assumption. Rather than silently applying a Riemannian theorem to a statistical manifold, we first eliminate the dual obstruction in the direction of the potential field and then impose the additional equality needed for the full Ricci tensor. The resulting statement is mathematically weaker than an unconditional classification, but it is also more defensible and makes the exact role of each hypothesis visible.

The comparison is particularly useful for novelty. Existing gradient Schouten classifications concern a single Levi–Civita connection or a single curvature tensor [7]. In the present setting, the new part is the preceding reduction from two dual statistical equations to one classical-type equation through $\mathfrak{K}(gradf,·)=0$ and $\mathfrak{Ric}={\mathfrak{Ric}}^{*}$.

Proposition 1 

(Conditional warped-product reduction). Assume that $(N^m,g)$ is complete and that a paired gradient Schouten almost soliton admits a nonconstant potential f whose gradient is common concircular. Assume further that

$$\mathfrak{Ric}={Ric}^{\overline{\mathfrak{D}}},\mathfrak{K}(gradf,·)=0,$$

(51)

and that the scalar curvature of the level hypersurfaces of f is constant. Then the underlying Riemannian manifold is locally a warped product

$$N\cong I{\times }_{\phi }{M}^{m-1},$$

(52)

where the fibers are Einstein. If, in addition, N is compact and the conformal field is nontrivial and closed, the standard closed-conformal rigidity theorem yields the spherical alternative under the corresponding positivity assumptions.

Proof. 

The common-concircular assumption gives ${\overline{\mathfrak{D}}}_{Y}gradf=aY$ because $\overline{\mathfrak{D}}=\frac{1}{2}(\mathfrak{D}+{\mathfrak{D}}^{*})$. Hence the regular level hypersurfaces of f are totally umbilical with second fundamental form proportional to the induced metric. Under the coincidence assumption (51), the paired Schouten equation reduces to the corresponding classical gradient Schouten equation for $\left(\overline{\mathfrak{D}}\right)$. The standard warped-product classification for complete gradient Schouten solitons with a nontrivial conformal potential then applies under the stated constant-fiber-scalar-curvature hypothesis; see the corresponding modern classification results in [7]. □

Remark 3. 

The additional hypothesis $\mathfrak{Ric}={Ric}^{\overline{\mathfrak{D}}}$ in Proposition 1 is essential. Without it, the statistical Ricci tensor contains information from $\mathfrak{K}$ that cannot be discarded merely from $\mathfrak{K}(gradf,·)=0$. Thus the theorem is a conditional transfer result rather than an unconditional statistical generalization of the classical classification theorem.

Corollary 2 

(Constant-scalar-curvature rigidity). Under the hypotheses of Proposition 1, if the scalar curvature of N is constant and the potential is common concircular with constant a, then

$$\mathfrak{R}(Y,Z)gradf={\mathfrak{R}}^{*}(Y,Z)gradf=0.$$

(53)

Thus the potential field belongs to the common curvature-nullity distribution of the two dual connections.

Proof. 

Use Corollary 1. □

7. Explicit Nontrivial Existence Examples and Lorentzian Statistical Perfect-Fluid

The preceding results are conditional on the existence of paired solitons. We now provide an explicit statistical structure for which the paired equations are satisfied and the difference tensor is nonzero. The example is elementary but nontrivial in the sense that the two affine connections differ from the Levi–Civita connection, while the common soliton potential lies in a direction annihilated by the difference tensor.

Example 1 

(A flat statistical model with nonzero difference tensor). Let $M={\mathbb{R}}^3$ with global coordinates $(x,y,z)$ and the Euclidean metric

$$g=d{x}^2+d{y}^2+d{z}^2.$$

Let $\overline{\mathfrak{D}}$ denote the Levi–Civita connection of g and put $X={\partial }_x$ and $\xi ={\partial }_y$. Define a symmetric $(1,2)$-tensor by

$$\mathfrak{K}(U,V)=g(U,\xi )g(V,\xi )\xi .$$

Then

$$\mathfrak{K}({\partial }_y,{\partial }_y)={\partial }_y,\mathfrak{K}({\partial }_i,{\partial }_j)=0\mathit{for}\mathit{all}\mathit{other}\mathit{coordinate}\mathit{pairs}.$$

The associated $(0,3)$-tensor is

$$C(U,V,W)=g\left(\mathfrak{K}\right(U,V),W)=g(U,\xi )g(V,\xi )g(W,\xi ),$$

which is totally symmetric. Hence the connections

$$\mathfrak{D}=\overline{\mathfrak{D}}+\mathfrak{K},{\mathfrak{D}}^{*}=\overline{\mathfrak{D}}-\mathfrak{K}$$

are torsion-free dual statistical connections. Their only nonzero Christoffel terms in the chosen coordinate frame are

$${\mathfrak{D}}_{{\partial }_y}{\partial }_y={\partial }_y,{\mathfrak{D}}_{{\partial }_y}^{*}{\partial }_y=-{\partial }_y.$$

Thus $\mathfrak{K}\ne 0$ and the statistical structure is not Levi–Civita.

Since $g(X,\xi )=0$, we have

$$\mathfrak{K}(X,Y)=0,{\mathfrak{D}}_{Y}X={\mathfrak{D}}_Y^{*}X=0$$

for every vector field Y. Consequently, X is common torse-forming with $a=0$ and $\omega =0$, hence common concircular. Moreover, the connection coefficients are constant and a direct curvature calculation gives

$$\mathfrak{R}=0,{\mathfrak{R}}^{*}=0,\mathfrak{Ric}={\mathfrak{Ric}}^{*}=0.$$

Because ${\mathfrak{D}}_{Y}X={\mathfrak{D}}_Y^{*}X=0$, both statistical Lie derivatives vanish:

$${\mathcal{L}}_X^{\mathfrak{D}}g=0,{\mathcal{L}}_X^{{\mathfrak{D}}^{*}}g=0.$$

Taking $\lambda =\beta =\mu =0$ therefore makes both equations in Definition 2 hold identically. We have thus obtained a nontrivial paired statistical soliton with

$$\mathfrak{K}\ne 0,\mathfrak{K}(X,·)=0.$$

The example shows that the rigidity conclusion of Theorem 2 is directional: it does not force the whole difference tensor to vanish.

Example 2 

(Fisher-geometric realization on a Gaussian family). Let $\mathcal{G}$ be the statistical manifold of univariate normal distributions $N(\mu ,{\sigma }^2)$, $\sigma >0$, equipped with the Fisher information metric

$$g_{\mathcal{G}}={\displaystyle \frac{1}{{\sigma }^2}}d{\mu }^2+{\displaystyle \frac{2}{{\sigma }^2}}d{\sigma }^2.$$

(54)

Let ${\mathfrak{D}}^{\left(1\right)}$ and ${\mathfrak{D}}^{(-1)}$ denote the exponential and mixture connections. They are dual and flat, while their difference tensor is nonzero; this is the standard dually flat realization of the normal exponential family in information geometry [1].

Consider the product

$$N=\mathcal{G}\times \mathbb{R},g=g_{\mathcal{G}}+d{t}^2,$$

and extend the two dual connections by the flat product connection on the t-factor. Let $X={\partial }_t$. Then

$${\mathfrak{R}}^{\left(1\right)}={\mathfrak{R}}^{(-1)}=0,{\mathfrak{Ric}}^{\left(1\right)}={\mathfrak{Ric}}^{(-1)}=0,{\mathfrak{D}}_Y^{\left(1\right)}X={\mathfrak{D}}_Y^{(-1)}X=0.$$

(55)

Moreover,

$$\mathfrak{K}\ne 0,\mathfrak{K}(X,·)=0.$$

(56)

Consequently, with $\lambda =\beta =\mu =0$, the product carries a paired statistical η-Ricci–Bourguignon soliton.

Unlike the elementary affine example above, this construction realizes the paired equations inside a genuine Fisher-information geometry associated with an exponential family. It therefore provides an information-geometric existence model rather than merely an abstract affine realization.

Example 3 

(A nonsteady constant-curvature boundary model). Let $(N^m,g)$ be an Einstein Riemannian manifold satisfying ${Ric}^{\overline{\mathfrak{D}}}=\kappa g$, and consider the boundary statistical case $\mathfrak{K}=0$, so that $\mathfrak{D}={\mathfrak{D}}^{*}=\overline{\mathfrak{D}}$. If X is Killing and $\mu =0$, then the paired equation reduces to

$$\kappa +(\lambda -\beta r)=0,$$

with $r=m\kappa$. Thus

$$\lambda =(m\beta -1)\kappa .$$

(57)

For $\kappa \ne 0$ and $\beta \ne 1/m$, this gives a nonzero soliton parameter on a nonflat Einstein background. This example is deliberately a boundary case: it shows that the formalism contains nonsteady/nonflat regimes, while the non-Riemannian content is supplied by the preceding statistical examples.

Proposition 2 

(Nontrivial existence of a paired statistical soliton). The preceding example proves that a non-Riemannian statistical manifold can carry a paired statistical η-Ricci–Bourguignon almost soliton with $\mathfrak{K}\ne 0$ while $\mathfrak{K}(X,·)=0$. Hence the class introduced in Definition 2 is nonempty.

Remark 4. 

The preceding example is an explicit affine statistical structure rather than a Fisher-information realization of a prescribed exponential family. Its purpose is to establish non-vacuity of the paired equations and to exhibit the coexistence of $\mathfrak{K}\ne 0$ with $\mathfrak{K}(X,·)=0$. It is not claimed here that this affine model is a Fisher metric or a particular exponential family; making the construction within a prescribed Fisher geometry is left as a separate model-dependent problem. Analogous constructions inside specific exponential families or Fisher geometries would provide a useful direction for further study.

7.1. An Exact Lorentzian Realization

A Lorentzian analog is obtained on Minkowski space $({\mathbb{R}}^4,g_{\eta })$ with

$$g_{\eta }=d{x}^2+d{y}^2+d{z}^2-d{t}^2,\xi ={\partial }_t,X={\partial }_x,$$

and

$$\mathfrak{K}(U,V)=g_{\eta }(U,\xi )g_{\eta }(V,\xi )\xi ,\mathfrak{D}=\overline{\mathfrak{D}}+\mathfrak{K},{\mathfrak{D}}^{*}=\overline{\mathfrak{D}}-\mathfrak{K}.$$

Again $\mathfrak{K}\ne 0$ while $\mathfrak{K}(X,·)=0$, and both statistical curvature tensors vanish. Thus the same construction gives an exact flat Lorentzian spacetime carrying a paired statistical soliton. Under the vacuum Einstein equations with zero cosmological constant this is the standard Minkowski vacuum. This provides a concrete spacetime realization for the Lorentzian discussion below without assigning additional cosmological dynamics to the soliton equation itself.

7.2. Sharpness of the Directional Rigidity

The explicit examples establish a useful distinction between directional and global rigidity. Theorem 4 gives $\mathfrak{K}(X,·)=0$ for a common torse-forming potential, but the affine and Fisher-geometric examples have $\mathfrak{K} /\equiv 0$. Hence the conclusion is sharp for a single potential field. Full vanishing of the difference tensor requires an additional spanning hypothesis, as established in Theorem 5.

Likewise, equality $\mathfrak{Ric}={\mathfrak{Ric}}^{*}$ is a curvature-level rigidity statement and should not be confused with connection-level equality. The paired framework therefore distinguishes three levels: directional annihilation of $\mathfrak{K}$, equality of the dual Ricci tensors, and full collapse $\mathfrak{K}=0$. The examples show that these levels are genuinely different.

7.3. Lorentzian Statistical Perfect-Fluid Consequences

We conclude the geometric development with a Lorentzian specialization. The perfect-fluid specialization is stated as a geometric consequence rather than as a full cosmological model. An exact flat Lorentzian realization is given in Section 7; the present section then explains how the common statistical Ricci tensor enters a perfect-fluid field equation when such a dynamical model is imposed. It also allows a direct comparison with earlier one-sided statistical perfect-fluid models, in which the Ricci tensor is prescribed in the form $(Ag+B\eta \otimes \eta )$. The paired construction shows that, under common torse-forming rigidity, this decomposition is shared by both statistical Ricci tensors.

Let $(N^m,g)$ have Lorentzian signature and let $\sigma$ be a unit timelike vector field with $g(\sigma ,\sigma )=-1$. Put $\eta \left(Y\right)=g(\sigma ,Y)$.

Theorem 9 

(Paired solitons and perfect-fluid Ricci form). Let a Lorentzian statistical manifold admit a paired statistical η-Ricci–Bourguignon almost soliton with soliton field X. Assume that X is common torse-forming, and

$${\mathcal{L}}_X^{\overline{\mathfrak{D}}}g=2\psi g.$$

(58)

Then the common statistical Ricci tensor has the perfect-fluid form

$$\mathfrak{Ric}={\mathfrak{Ric}}^{*}=\left(\beta r-\lambda -\psi \right)g-\mu \eta \otimes \eta .$$

(59)

In particular, the paired soliton automatically determines a statistical perfect-fluid decomposition with isotropic part $\beta r-\lambda -\psi$ and anisotropic coefficient $-\mu$.

Proof. 

Theorem 2 gives $\mathfrak{Ric}={\mathfrak{Ric}}^{*}$ and $r=r^{*}$. Substituting (58) into the reduced soliton equation gives (59). □

7.4. A Spatially Flat FLRW-Type Realization

We give an explicit Lorentzian realization that translates the paired soliton equation into ordinary differential equations for a cosmological scale factor. The purpose is to provide a concrete test geometry rather than to claim a complete cosmological model.

Let

$$g=-d{t}^2+a{\left(t\right)}^2{\displaystyle \sum _{i=1}^{m-1}}{\left(d{x}^i\right)}^2,X=c{\partial }_t,\eta =dt,$$

where c is constant. Consider first the boundary statistical case $\mathfrak{K}=0$, so that $\mathfrak{D}={\mathfrak{D}}^{*}=\overline{\mathfrak{D}}$. Put $n=m-1$. Then

$${\mathfrak{Ric}}_{tt}=-n{\displaystyle \frac{a^{″}}{a}},{\mathfrak{Ric}}_{ij}=\left({\displaystyle \frac{a{a}^{″}+(n-1){\left(a^{\prime }\right)}^2}{a^2}}\right)g_{ij},$$

(60)

and

$$r=2n{\displaystyle \frac{a^{″}}{a}}+n(n-1){\left({\displaystyle \frac{a^{\prime }}{a}}\right)}^2.$$

(61)

Moreover,

$${\displaystyle \frac{1}{2}}{\mathcal{L}}_X{g}_{tt}=0,{\displaystyle \frac{1}{2}}{\mathcal{L}}_X{g}_{ij}=c{\displaystyle \frac{a^{\prime }}{a}}{g}_{ij}.$$

Therefore the paired $\eta$-Ricci–Bourguignon equation is equivalent to

$$-n{\displaystyle \frac{a^{″}}{a}}-(\lambda -\beta r)+\mu =0,{\displaystyle \frac{a{a}^{″}+(n-1){\left(a^{\prime }\right)}^2}{a^2}}+c{\displaystyle \frac{a^{\prime }}{a}}+(\lambda -\beta r)=0.$$

(62)

For the de Sitter-type scale factor $a\left(t\right)=e^{Ht}$, $H>0$, one has $r=m(m-1)H^2$, and the equations hold for

$$\lambda =\beta m(m-1)H^2-(m-1)H^2-cH,\mu =-cH.$$

(63)

This is an explicit nonsteady paired soliton on a nonflat Lorentzian background in the boundary case $\mathfrak{K}=0$. It demonstrates how the abstract tensor equation can be tested against an exact cosmological geometry. If one seeks a genuinely statistical FLRW model with $\mathfrak{K}\ne 0$, an additional choice of statistical connection compatible with the cosmological ansatz is required.

8. Conclusions

We have developed a focused theory of paired statistical Ricci–Bourguignon solitons on a statistical manifold equipped with dual connections. The generic idea of statistical solitons involving dual connections is known; the contribution here is the analysis of a common-potential $\eta$-Ricci–Bourguignon system and the rigidity consequences obtained from its coupling with the statistical difference tensor. The central idea is to impose the same Ricci–Bourguignon soliton structure simultaneously with respect to $\mathfrak{D}$ and ${\mathfrak{D}}^{*}$. The sum of the two equations is governed by the Levi–Civita Lie derivative, while the difference detects the cubic statistical tensor generated by the difference tensor $\mathfrak{K}$.

The main rigidity result is that a common torse-forming potential field satisfies

$$\mathfrak{K}(X,·)=0.$$

Consequently, the paired equations force

$$\mathfrak{Ric}={\mathfrak{Ric}}^{*}.$$

Thus the dual statistical connections become indistinguishable at the level of Ricci curvature, although the full difference tensor need not vanish away from the direction of the potential field. We also established a common curvature identity for torse-forming fields and showed that the same mechanism governs the gradient Ricci–Bourguignon and gradient Schouten formulations.

Under completeness, conformality, and additional coincidence of the statistical and Levi–Civita Ricci tensors, the paired problem reduces to the classical gradient Schouten setting and yields a conditional warped-product classification. This qualification is important: without the extra curvature-coincidence hypothesis, the presence of the statistical difference tensor prevents a direct transfer of classical rigidity theorems.

For Lorentzian statistical perfect-fluid spacetimes, the paired theory yields an explicit perfect-fluid Ricci decomposition. The corresponding physical interpretation depends on the chosen statistical field equations and should therefore be separated from the purely geometric conclusions.

The Fisher-geometric realization also motivates future work on explicit exponential families and information-geometric flows.

Future Computational Directions

An interesting direction for future research is the development of neural-network-based computational methods for statistical soliton equations on statistical manifolds. Recent advances in neural approaches for solving time-dependent matrix problems demonstrate the potential of neural architectures for obtaining rapidly convergent and noise-tolerant numerical solutions [14]. At the same time, high-order convergent numerical schemes for nonlinear evolution equations, such as the Rosenau–Burgers equation, provide useful benchmarks for validating the accuracy and convergence of such learning-based approaches [15]. The recent study of localized gap solitons in nonlinear lattice models also illustrates the importance of combining analytical characterization with numerical computation when investigating the existence, stability, and dynamics of soliton solutions [16].

Building on these developments, one may formulate the paired statistical Ricci–Bourguignon soliton equations developed in this paper as physics-informed or neural differential systems on statistical manifolds. In particular, neural networks could be trained to approximate the potential vector field, the statistical difference tensor, and the associated curvature quantities while enforcing the dual connection structure and the geometric constraints derived in this work. Such an approach would make it possible to investigate soliton solutions in statistical manifolds for which closed-form solutions are difficult to obtain. Furthermore, the presence of non-IID data is a natural issue in information-geometric applications, and recent developments in generalized active stratified sampling provide a potentially useful strategy for constructing representative training sets in such settings [17]. Thus, combining statistical geometry, paired soliton theory, neural differential solvers, and structure-aware data sampling may provide a promising computational framework for studying more general statistical soliton equations and their applications in geometric models of physical systems.