Implicit Transport Geometry and Directional Admissibility in Biparametric Pullback Spaces

1. Introduction

Weighted function spaces offer a robust analytical framework for investigating endpoint singularities, boundary layer concentrations, and localization structures in functional analysis [8,13]. In classical coordinate transport problems, changing variables via a global diffeomorphism introduces a standard Jacobian derivative term that maps the underlying measure space to an equivalent configuration [14]. However, advanced settings in operator theory, singular differential equations, and non-autonomous transport systems require formulations where an external, asymmetric weight function is structurally coupled to the pullback mapping [5,10].

The principal contribution of this paper is to establish a rigorous localization and admissibility theory for a class of non-uniform transport spaces where the coordinate chart is driven by a non-linear geometric engine. This framework provides a mathematically distinct environment in which the transport map itself is generated implicitly rather than being prescribed a priori [1,2]. Rather than defining $\varphi :\mathbb{R}\to (0,1)$ directly, the coordinate map emerges as the non-linear compactification of an implicit biparametric relation:

$$R^p+qR=e^v,R>0,p\in (0,1],q\ge 0$$

(1)

which structurally induces the compactified mapping coordinate via the non-linear ratio:

$$\varphi (v;p,q)={\displaystyle \frac{R(v;p,q)}{1+R(v;p,q)}}$$

(2)

This design creates a rigid analytical hierarchy:

$$R⟶\mathrm{Compactified}\mathrm{Transport}\mathrm{Map}\varphi ⟶\mathrm{Localization}\mathrm{Density}\rho ⟶\mathrm{Admissibility}\mathrm{Structure}{D}_{\varphi }^w$$

This framework carries significant mathematical advantages. First, because $R>0$ globally, the ratio map natively compactifies the infinite real line $\mathbb{R}$ onto the open unit interval $(0,1)$, ensuring that $R\to 0\Rightarrow \varphi \to 0$ and $R\to \infty \Rightarrow \varphi \to 1$. Second, directional asymmetry is generated intrinsically by the transport engine: the parameter p regulates the non-linear left-endpoint compression ($x\to 0^{+}$), while q governs the right-endpoint redistribution ($x\to 1^{-}$). Finally, the effective pullback density $\rho \left(x\right)$ is geometrically induced rather than being arbitrarily assigned, providing a distinct geometric mechanism relative to standard pullback formulations. This relates structurally to the theory of weighted composition operators on Lebesgue and Sobolev spaces [19,20].

The paper is organized as follows. Section 2 establishes the existence, uniqueness, and boundary asymptotics of the implicit transport geometry via the generating functional framework. Section 3 constructs the foundational pullback representations and proves the surjectivity of the defining isometric isomorphism. Section 4 introduces the directional spatial decomposition and establishes a global admissibility criterion for dual power-law singular profiles. Section 5 derives the parametric variations of the localized energy densities under smooth parameter transformations. Section 6 develops explicit structural applications by passing functions through the directional variation fields, analyzing stable singular configurations, non-admissible blow-ups, and discontinuous regular interfaces. Section 7 verifies the polynomial density properties within this unified framework.

2. The Implicit Deformation Framework

We formalize the structural origin of the geometry and establish the existence, uniqueness, differentiability, and boundary asymptotics of the compactified coordinate mapping engine.

To formalize the structural origin of the geometry, let $\mathcal{F}:(0,\infty )\times \mathbb{R}\times (0,1]\times [0,\infty )\to \mathbb{R}$ be the smooth generating functional defined by:

$$\mathcal{F}(R,v;p,q)=R^p+qR-e^v$$

(3)

Definition 1 

(Implicit Biparametric Deformator System). Let $\mathcal{P}=(0,1]\times [0,\infty )$ denote the parameter space. Following the functional foundations introduced by John [1,2] for deformation geometry and localized operators, the Implicit Biparametric Deformator System (IBDS) is defined as the coupled geometric system $({\varphi }_{(p,q)},R)$ mapping the real coordinate track $v\in \mathbb{R}$ onto the bounded interval $(0,1)$ via the simultaneous relation:

$$\mathcal{F}(R,v;p,q)=0,{\varphi }_{(p,q)}\left(v\right)={\displaystyle \frac{R(v;p,q)}{1+R(v;p,q)}}$$

(4)

The operator ${\varphi }_{(p,q)}$ acts as a geometric compactification mechanism, compactly mapping the unbounded, implicitly defined manifold branch $R\in (0,\infty )$ onto the bounded spatial domain $(0,1)$.

Theorem 1 

(Implicit Transport Geometry). For every $v\in \mathbb{R}$ and $(p,q)\in \mathcal{P}$, the non-linear deformation equation $\mathcal{F}(R,v;p,q)=0$ admits a unique positive solution $R(v;p,q)\in (0,\infty )$. The induced compactified map

$$\varphi (v;p,q)={\displaystyle \frac{R(v;p,q)}{1+R(v;p,q)}}$$

(5)

defines a smooth ($C^{\infty }$) strictly increasing orientation-preserving diffeomorphism $\varphi :\mathbb{R}\to (0,1)$. Moreover, the mapping satisfies the non-uniform boundary asymptotics:

$$\begin{array}{cc}\hfill \varphi (v;p,q)& \sim e^{v/p}\mathrm{as}v\to -\infty \hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill 1-\varphi (v;p,q)& \sim q{e}^{-v}\mathrm{as}v\to +\infty (\mathrm{for}q>0)\hfill \end{array}$$

(7)

For the symmetric linear limit where $q=0$, the right-endpoint asymptotic behaves as:

$$1-\varphi (v;p,0)\sim e^{-v/p}\mathrm{as}v\to +\infty$$

(8)

Proof. 

Let $g\left(R\right)=R^p+qR$ for $R\in (0,\infty )$. Because $p\in (0,1]$ and $q\ge 0$, the derivative satisfies $g^{\prime }\left(R\right)=p{R}^{p-1}+q>0$ for all $R>0$. Thus, g is strictly increasing, mapping $(0,\infty )$ bijectively onto $(0,\infty )$ with ${lim}_{R\to 0}g\left(R\right)=0$ and ${lim}_{R\to \infty }g\left(R\right)=\infty$. Since $e^v>0$ for all $v\in \mathbb{R}$, the equation $\mathcal{F}(R,v;p,q)=0\iff g\left(R\right)=e^v$ possesses a unique positive solution $R(v;p,q)=g^{-1}\left(e^v\right)$.

Because $R>0$, the ratio map $\varphi =R/(1+R)$ maps $(0,\infty )$ bijectively onto $(0,1)$. As $v\to -\infty$, $e^v\to 0$, forcing $R\to 0$ and $\varphi \to 0$. As $v\to +\infty$, $e^v\to \infty$, forcing $R\to \infty$ and $\varphi \to 1$. Since g is a smooth function with a non-vanishing derivative on $(0,\infty )$, the Inverse Function Theorem guarantees that $g^{-1}$ is smooth, making $\varphi (v;p,q)$ a smooth diffeomorphism.

We now derive the derivative ${\varphi }^{\prime }$ explicitly. Differentiating (5) with respect to v via the quotient rule yields:

$${\varphi }^{\prime }(v;p,q)={\displaystyle \frac{R^{\prime }(1+R)-R{R}^{\prime }}{{(1+R)}^2}}={\displaystyle \frac{R^{\prime }(v;p,q)}{{(1+R(v;p,q))}^2}}$$

(9)

Applying implicit differentiation to the structural relation $\mathcal{F}(R,v;p,q)=0$ with respect to v gives:

$$(p{R}^{p-1}+q)R^{\prime }=e^v\Rightarrow R^{\prime }(v;p,q)={\displaystyle \frac{e^v}{p{R}^{p-1}+q}}$$

Substituting this expression back into (9) yields the explicit derivative formula:

$${\varphi }^{\prime }(v;p,q)={\displaystyle \frac{e^v}{{(1+R)}^2(p{R}^{p-1}+q)}}>0$$

(10)

which proves that $\varphi$ is strictly increasing and orientation-preserving.

To establish the left-endpoint asymptotic relation (6), let $v\to -\infty$, which implies $R\to 0$. For $0<p<1$, the exponent satisfies $p-1<0$, which yields $R^{p-1}\to \infty$, while for $p=1$, the term remains constant (equal to 1). Consequently, for $0<p<1$, the non-linear term $p{R}^{p-1}$ completely dominates the linear redistribution term q near the left boundary. Balancing $\mathcal{F}(R,v;p,q)=0$ gives $R^p\sim e^v\Rightarrow R\sim e^{v/p}$. Because $\varphi =R/(1+R)\sim R$ as $R\to 0$, we find that $\varphi (v;p,q)\sim e^{v/p}$ as $v\to -\infty$.

To establish the right-endpoint asymptotic relation (7), let $v\to +\infty$, which implies $R\to \infty$. For $q>0$, the linear term dominates the sub-linear or linear term $R^p$ since $R^{1-p}\to \infty$ (or $1+q$ if $p=1$). Balancing $\mathcal{F}(R,v;p,q)=0$ yields $qR\sim e^v\Rightarrow R\sim q^{-1}{e}^v$. Evaluating the target compactification profile yields:

$$1-\varphi =1-{\displaystyle \frac{R}{1+R}}={\displaystyle \frac{1}{1+R}}\sim {\displaystyle \frac{1}{R}}\sim q{e}^{-v}\mathrm{as}v\to +\infty$$

If $q=0$, then the relation collapses to $R^p=e^v$, which gives $R=e^{v/p}$ identically. In this case, as $v\to +\infty$:

$$1-\varphi (v;p,0)={\displaystyle \frac{1}{1+e^{v/p}}}\sim e^{-v/p}$$

This completes the rigorous derivation of the boundary mechanics. □

Remark 1. 

Using $R(v;p,q)\sim e^{v/p}$ as $v\to -\infty$, we obtain $R^{p-1}(v;p,q)\sim e^{{\displaystyle \frac{p-1}{p}}v}$. Hence, for $0<p<1$, $R^{p-1}(v;p,q)\to \infty$ as $v\to -\infty$, while for $p=1$ the expression remains identically equal to 1. Consequently, for any $0<p<1$, the non-linear compression term $p{R}^{p-1}$ completely dominates the linear redistribution term q near the left boundary, explaining the robust asymptotic behavior of the implicit transport geometry near $x=0$.

Example 1. 

Consider the coordinate system governed by the classic standard logistic map on $\mathbb{R}$:

$${\varphi }_0\left(v\right)={\displaystyle \frac{e^v}{1+e^v}}$$

(11)

which satisfies the boundary conditions ${lim}_{v\to -\infty }{\varphi }_0\left(v\right)=0$ and ${lim}_{v\to \infty }{\varphi }_0\left(v\right)=1$, yielding symmetric exponential scale parameters $\alpha =\beta =1$. Let the redistribution weight be specified globally as:

$$w_0\left(v\right)=e^{(\sigma +1)v}{(1+e^v)}^{-(\sigma +\tau +2)}$$

(12)

Evaluating the inverse map explicitly yields $e^v={\displaystyle \frac{x}{1-x}}$. By executing the substitution into our effective density representation, the reference effective density is recovered exactly for all $x\in (0,1)$:

$${\rho }_0\left(x\right)={\displaystyle \frac{w_0\left({\varphi }_0^{-1}\left(x\right)\right)}{{\varphi }_0^{\prime }\left({\varphi }_0^{-1}\left(x\right)\right)}}={\displaystyle \frac{{\left(\frac{x}{1-x}\right)}^{\sigma +1}{\left(1+\frac{x}{1-x}\right)}^{-(\sigma +\tau +2)}}{\left(\frac{x}{1-x}\right){\left(1+\frac{x}{1-x}\right)}^{-2}}}=x^{\sigma }{(1-x)}^{\tau }$$

(13)

The case $p=1,q=0$ constitutes a degenerate symmetric limit where the right-end redistribution mechanism disappears and the right-hand boundary asymptotic condition collapses because $1-{\varphi }_0\left(v\right)=e^{-v}/(1+e^{-v})\sim e^{-v}$ instead of scaling via q. This baseline model provides a uniform anchor from which non-uniform parameter configurations diverge when $q>0$.

3. Pullback Representations and Induced Hilbert Structures

Let $w:\mathbb{R}\to (0,\infty )$ be a measurable weight function matched to the parameters $\sigma ,\tau >-1$:

$$w\left(v\right)\sim e^{(\sigma +1)v/p}\mathrm{as}v\to -\infty \mathrm{and}w\left(v\right)\sim e^{-(\tau +1)v}\mathrm{as}v\to +\infty$$

(14)

To ensure the analytical consistency of the parametric variations throughout this study, we establish a formal differentiability framework for the weight family:

Assumption A1. 

The family of weights $w(v;p,q)$ is strictly positive, measurable in $v\in \mathbb{R}$, and continuously differentiable ($C^1$) with respect to the deformation parameters $(p,q)\in \mathcal{P}$ for each fixed v. Furthermore, its partial derivatives ${\partial }_{p}w(v;p,q)$ and ${\partial }_{q}w(v;p,q)$ satisfy compatible asymptotic decay rates preserving the boundary behaviors of (14) as $v\to \pm \infty$.

Definition 2. 

The implicit biparametric pullback space $D_{\varphi }^w$ is defined as the linear space of Lebesgue measurable functions $f:(0,1)\to \mathbb{C}$ such that $f\circ \varphi \in L^2(\mathbb{R},w\left(v\right)dv)$, equipped with the inner product

$${\langle f,g\rangle }_{D_{\varphi }^w}={\int }_{-\infty }^{\infty }f\left(\varphi \left(v\right)\right)\overline{g\left(\varphi \right(v\left)\right)}w\left(v\right)dv$$

(15)

and the naturally induced norm ${\parallel f\parallel }_{D_{\varphi }^w}=\sqrt{{\langle f,f\rangle }_{D_{\varphi }^w}}$. Functions that coincide almost everywhere with respect to the pulled-back Lebesgue measure are identified as the same equivalence class.

We introduce the linear transport operator $U:D_{\varphi }^w\to L^2\left(\mathbb{R}\right)$ defined by the explicit relational mapping $\left(Uf\right)\left(v\right)=f\left(\varphi \left(v\right)\right)\sqrt{w\left(v\right)}$.

Theorem 2. 

The implicit biparametric pullback space $D_{\varphi }^w$ is isometrically isomorphic to the classical weighted space $L^2((0,1),\rho \left(x\right)dx)$ via the identity mapping, where the induced effective density $\rho :(0,1)\to (0,\infty )$ is given by

$$\rho \left(x\right)={\displaystyle \frac{w\left({\varphi }^{-1}\left(x\right)\right)}{{\varphi }^{\prime }\left({\varphi }^{-1}\left(x\right)\right)}}$$

(16)

Consequently, $D_{\varphi }^w$ inherits completeness, separability, and reflexivity from $L^2((0,1),\rho \left(x\right)dx)$.

Proof. 

Let $f\in D_{\varphi }^w$, and apply the change of variables $x=\varphi \left(v\right)$. Because $\varphi$ is an orientation-preserving smooth diffeomorphism, its inverse $v={\varphi }^{-1}\left(x\right)$ is well-defined and continuously differentiable on $(0,1)$. The differential maps as $dx={\varphi }^{\prime }\left(v\right)dv$, which yields $dv={\displaystyle \frac{dx}{{\varphi }^{\prime }\left({\varphi }^{-1}\left(x\right)\right)}}$. Substituting these expressions directly into the norm definition yields:

$${\parallel f\parallel }_{D_{\varphi }^w}^2={\int }_{-\infty }^{\infty }{\left|f\left(\varphi \left(v\right)\right)\right|}^{2}w\left(v\right)dv={\int }_0^1{\left|f\left(x\right)\right|}^2{\displaystyle \frac{w\left({\varphi }^{-1}\left(x\right)\right)}{{\varphi }^{\prime }\left({\varphi }^{-1}\left(x\right)\right)}}dx={\int }_0^1{\left|f\left(x\right)\right|}^2\rho \left(x\right)dx={\parallel f\parallel }_{L^2((0,1),\rho \left(x\right)dx)}^2$$

(17)

This confirms that the mapping preserves the norm.

To prove that this relation constitutes a full structural isomorphism, we construct the inverse linear operator $V:L^2((0,1),\rho \left(x\right)dx)\to D_{\varphi }^w$ defined by $Vg=g$. For any $g\in L^2((0,1),\rho \left(x\right)dx)$, we evaluate its norm under the coordinate change $v={\varphi }^{-1}\left(x\right)$:

$${\parallel Vg\parallel }_{D_{\varphi }^w}^2={\int }_{-\infty }^{\infty }{\left|g\left(\varphi \left(v\right)\right)\right|}^{2}w\left(v\right)dv={\int }_0^1{\left|g\left(x\right)\right|}^2{\displaystyle \frac{w\left({\varphi }^{-1}\left(x\right)\right)}{{\varphi }^{\prime }\left({\varphi }^{-1}\left(x\right)\right)}}dx={\int }_0^1{\left|g\left(x\right)\right|}^2\rho \left(x\right)dx={\parallel g\parallel }_{L^2((0,1),\rho \left(x\right)dx)}^2<\infty$$

(18)

As a result, V maps every element of $L^2((0,1),\rho \left(x\right)dx)$ into a well-defined element of $D_{\varphi }^w$. Since V is the explicit two-sided inverse of the identity mapping, surjectivity is established. Completeness, separability, and reflexivity are preserved structurally under isometric isomorphism from $L^2((0,1),\rho \left(x\right)dx)$ [7]. □

Proposition 1. 

Under the structural asymptotics of Theorem 1 and equation (14), the geometrically induced effective density $\rho \left(x\right)$ satisfies:

$$\rho \left(x\right)\sim p{x}^{\sigma }\mathrm{as}x\to 0^{+}\mathrm{and}\rho \left(x\right)\sim C_q{(1-x)}^{\tau }\mathrm{as}x\to 1^{-}(\mathrm{for}q>0)$$

(19)

where the right-endpoint coefficient is given by $C_q=q^{-(\tau +2)}$.

Proof. 

We verify the left-endpoint limit. As $x\to 0^{+}$, ${\varphi }^{-1}\left(x\right)=v\to -\infty$. From Theorem 1, we have $\varphi \left(v\right)\sim e^{v/p}$ and ${\varphi }^{\prime }\left(v\right)\sim p^{-1}{e}^{v/p}\sim p^{-1}x$. The weight satisfies $w\left(v\right)\sim e^{(\sigma +1)v/p}\sim x^{\sigma +1}$. Substituting these behaviors into (16) gives:

$$\rho \left(x\right)={\displaystyle \frac{w\left(v\right)}{{\varphi }^{\prime }\left(v\right)}}\sim {\displaystyle \frac{x^{\sigma +1}}{\frac{1}{p}x}}=p{x}^{\sigma }\mathrm{as}x\to 0^{+}$$

At the right boundary, as $x\to 1^{-}$, ${\varphi }^{-1}\left(x\right)=v\to +\infty$. From Theorem 1, $1-\varphi \left(v\right)\sim q{e}^{-v}$ and $R\sim q^{-1}{e}^v\to \infty$. The derivative formula (10) satisfies ${\varphi }^{\prime }\left(v\right)\sim R^{-2}·q^{-1}{e}^v\sim q^2{e}^{-2v}·q^{-1}{e}^v=q{e}^{-v}\sim q^{-1}(1-x)$. The weight behaves as $w\left(v\right)\sim e^{-(\tau +1)v}\sim q^{-(\tau +1)}{(1-x)}^{\tau +1}$. Evaluating the ratio yields:

$$\rho \left(x\right)={\displaystyle \frac{w\left(v\right)}{{\varphi }^{\prime }\left(v\right)}}\sim {\displaystyle \frac{q^{-(\tau +1)}{(1-x)}^{\tau +1}}{\frac{1}{q}(1-x)}}=q^{-(\tau +2)}{(1-x)}^{\tau }=C_q{(1-x)}^{\tau }\mathrm{as}x\to 1^{-}$$

This establishes the consistency of the boundary density scaling coefficients. □

4. Spatial Decompositions and Global Admissibility Criteria

When modeling structural concentrations that are inherently directional, a singular function can remain stable at one endpoint while exhibiting a non-integrable blow-up at the opposite boundary. To isolate these behaviors, we introduce a spatial decomposition of the transported field $F\left(v\right)=\left(Uf\right)\left(v\right)$ based on the real line’s natural partition. Let $H\left(v\right)$ denote the standard Heaviside step function, and define the left- and right-directed fields by:

$$F_{-}\left(v\right)=F\left(v\right)·H(-v)\mathrm{and}{F}_{+}\left(v\right)=F\left(v\right)·H\left(v\right)$$

(20)

Proposition 2. 

The components $F_{-}$ and $F_{+}$ form an orthogonal decomposition of F in $L^2\left(\mathbb{R}\right)$, satisfying:

$${\langle F_{-},F_{+}\rangle }_{L^2\left(\mathbb{R}\right)}={0\mathrm{and}\parallel F\parallel }_{L^2\left(\mathbb{R}\right)}^2=\parallel F_{-}{\parallel }_{L^2\left(\mathbb{R}\right)}^2+{\parallel F_{+}\parallel }_{L^2\left(\mathbb{R}\right)}^2$$

(21)

With this split, the total norm of the space satisfies:

$${\parallel f\parallel }_{D_{\varphi }^w}^2={\int }_{-\infty }^0|F_{-}{\left(v\right)|}^{2}dv+{\int }_0^{\infty }{\left|F_{+}\left(v\right)\right|}^{2}dv.$$

Proof. 

We compute the inner product in $L^2\left(\mathbb{R}\right)$ directly from the definition of the components:

$${\langle F_{-},F_{+}\rangle }_{L^2\left(\mathbb{R}\right)}={\int }_{-\infty }^{\infty }F\left(v\right)H(-v)\overline{F\left(v\right)H\left(v\right)}dv={\int }_{-\infty }^{\infty }{\left|F\left(v\right)\right|}^{2}H(-v)H\left(v\right)dv$$

Because the product of the indicator functions satisfies $H(-v)H\left(v\right)=0$ for all $v\in \mathbb{R}\setminus \left\{0\right\}$, the integrand vanishes almost everywhere, establishing orthogonality. The norm identity follows immediately:

$${∥F∥}_{L^2\left(\mathbb{R}\right)}^2={∥F_{-}∥}_{L^2\left(\mathbb{R}\right)}^2+{∥F_{+}∥}_{L^2\left(\mathbb{R}\right)}^2$$

because orthogonality eliminates the cross term. □

Using this orthogonal spatial decomposition, we formulate a generalized structural classification theorem connecting boundary singularities, geometric parameters, and full space membership. Rather than analyzing only a single prototype profile, we characterize admissibility for any measurable function bounded by dual power-law profiles.

Theorem 3 

(General Directional Admissibility). Let $\sigma ,\tau >-1$, and let $D_{\varphi }^w\cong L^2((0,1),\rho \left(x\right)dx)$ be the implicit pullback space governed by the boundary density profiles of Proposition 1. Let $f:(0,1)\to \mathbb{C}$ be a measurable function.

(1) 

Upper Bound Sufficiency: If there exists a constant $M>0$ such that $\left|f\left(x\right)\right|\le M{x}^{-\lambda }{(1-x)}^{-\mu }$ almost everywhere on $(0,1)$, then the parameter conditions $\sigma -2\lambda >-1$ and $\tau -2\mu >-1$ imply that $f\in D_{\varphi }^w$.

(2) 

Lower Bound Necessity: Conversely, if there exist constants $c>0$ and $\delta \in (0,1/2)$ such that $\left|f\left(x\right)\right|\ge c{x}^{-\lambda }{(1-x)}^{-\mu }$ almost everywhere on $(0,\delta )\cup (1-\delta ,1)$, then $f\in D_{\varphi }^w$ implies that $\sigma -2\lambda >-1$ and $\tau -2\mu >-1$.

Proof. 

We prove both assertions by decomposing the global integral over $(0,1)$ via an internal indicator parameter $\delta \in (0,1/2)$:

$${\parallel f\parallel }_{D_{\varphi }^w}^2={\int }_0^{\delta }{\left|f\left(x\right)\right|}^2\rho \left(x\right)dx+{\int }_{\delta }^{1-\delta }{\left|f\left(x\right)\right|}^2\rho \left(x\right)dx+{\int }_{1-\delta }^1{\left|f\left(x\right)\right|}^2\rho \left(x\right)dx.$$

Because the effective density $\rho \left(x\right)$ is positive, smooth, and bounded away from zero on the compact interior domain $[\delta ,1-\delta ]$, the central integral ${\int }_{\delta }^{1-\delta }{\left|f\left(x\right)\right|}^2\rho \left(x\right)dx$ is finite for any locally square-integrable function. Thus, the global integrability is determined entirely by the boundary integrals.

To prove the first claim, we apply the upper bound assumption $\left|f\left(x\right)\right|\le M{x}^{-\lambda }{(1-x)}^{-\mu }$. The left-boundary integral is bounded by:

$${\int }_0^{\delta }{\left|f\left(x\right)\right|}^2\rho \left(x\right)dx\le M^2{\int }_0^{\delta }{x}^{-2\lambda }{(1-x)}^{-2\mu }\rho \left(x\right)dx$$

Using the left-endpoint equivalence $\rho \left(x\right)\sim p{x}^{\sigma }$ from Proposition 1, there exists a constant $K_1>0$ such that:

$${\int }_0^{\delta }{\left|f\left(x\right)\right|}^2\rho \left(x\right)dx\le K_1{\int }_0^{\delta }{x}^{\sigma -2\lambda }dx$$

This converges if and only if $\sigma -2\lambda >-1$. Similarly, using $\rho \left(x\right)\sim C_q{(1-x)}^{\tau }$ at the right boundary, the third integral satisfies:

$${\int }_{1-\delta }^1{\left|f\left(x\right)\right|}^2\rho \left(x\right)dx\le K_2{\int }_{1-\delta }^1{(1-x)}^{\tau -2\mu }dx$$

which converges if and only if $\tau -2\mu >-1$.

To prove the second claim, we apply the lower bound assumption $\left|f\left(x\right)\right|\ge c{x}^{-\lambda }{(1-x)}^{-\mu }$ on the boundary intervals. Under the condition $f\in D_{\varphi }^w$, we have:

$$\infty >{\int }_0^{\delta }{\left|f\left(x\right)\right|}^2\rho \left(x\right)dx\ge c^2{\int }_0^{\delta }{x}^{-2\lambda }{(1-x)}^{-2\mu }\rho \left(x\right)dx\ge C{\int }_0^{\delta }{x}^{\sigma -2\lambda }dx$$

for a positive constant C. This immediately yields the necessity of $\sigma -2\lambda >-1$. An identical evaluation on the right boundary subinterval $(1-\delta ,1)$ yields:

$$\infty >{\int }_{1-\delta }^1{\left|f\left(x\right)\right|}^2\rho \left(x\right)dx\ge C^{\prime }{\int }_{1-\delta }^1{(1-x)}^{\tau -2\mu }dx$$

which establishes the necessity of $\tau -2\mu >-1$, completing the proof. □

5. Parametric Gradients and Variational Energy Deformations

To evaluate how directional concentrations adjust under structural variations, we introduce a parameter $ϵ\in (-\delta ,\delta )$ driving a smooth pathway $ϵ\mapsto \left(p\right(ϵ),q(ϵ\left)\right)$ across the parameter manifold. This induces parameterized directional energy densities on their respective semi-infinite intervals:

$$L_{f,-}(v,ϵ)=|F_{-}{(v,ϵ)|}^2\mathrm{and}{L}_{f,+}(v,ϵ)={\left|F_{+}(v,ϵ)\right|}^2$$

(22)

The continuous change of the local density profiles under parameter variation is governed by the structural inner product identities derived below.

Lemma 1. 

Let $F(v,ϵ)$ be a parameter-dependent field that is differentiable with respect to ϵ. The continuous variations of the directional localized energy densities satisfy the algebraic identities:

$$\begin{array}{cc}\hfill {\partial }_{ϵ}{L}_{f,-}& =2Re\left(\overline{F_{-}}{\partial }_{ϵ}{F}_{-}\right)\mathrm{on}(-\infty ,0]\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\partial }_{ϵ}{L}_{f,+}& =2Re\left(\overline{F_{+}}{\partial }_{ϵ}{F}_{+}\right)\mathrm{on}[0,\infty )\hfill \end{array}$$

(24)

Proof. 

We compute the derivative of the product $L_{f,-}=F_{-}\overline{F_{-}}$ with respect to $ϵ$ on the left-hand interval $(-\infty ,0]$ via the standard product rule:

$${\partial }_{ϵ}{L}_{f,-}={\partial }_{ϵ}\left(F_{-}\overline{F_{-}}\right)=\left({\partial }_{ϵ}{F}_{-}\right)\overline{F_{-}}+F_{-}\left({\partial }_{ϵ}\overline{F_{-}}\right)$$

Because the second term is the complex conjugate of the first, adding them isolates exactly twice the real part of the expression ($z+\overline{z}=2Re\left(z\right)$). This directly yields ${\partial }_{ϵ}{L}_{f,-}=2Re\left(\overline{F_{-}}{\partial }_{ϵ}{F}_{-}\right)$. Applying this exact product expansion to $L_{f,+}=F_{+}\overline{F_{+}}$ over the positive ray $[0,\infty )$ establishes the formula for ${\partial }_{ϵ}{L}_{f,+}$. □

By evaluating the field component expansion ${\partial }_{ϵ}F$ via the multi-variable chain rule, the internal parameter variation breaks down into two operational contributions:

$${\partial }_{ϵ}F=f^{\prime }\left({\varphi }_{ϵ}\left(v\right)\right)\left({\partial }_{ϵ}{\varphi }_{ϵ}\left(v\right)\right)\sqrt{w_{ϵ}\left(v\right)}+{\displaystyle \frac{1}{2}}f\left({\varphi }_{ϵ}\left(v\right)\right){\displaystyle \frac{{\partial }_{ϵ}{w}_{ϵ}\left(v\right)}{\sqrt{w_{ϵ}\left(v\right)}}}$$

(25)

The transport contribution tracks the movement of functions along spatial coordinates, while the redistribution contribution manages the localized scaling of the energy profile.

6. Structural Applications: Parametric Field Component Deformations

We analyze the action of the directional parameter gradients on admissible transported fields. Let $F(v;p,q)=f\left(\varphi (v;p,q)\right)\sqrt{w\left(v\right)}$ for $f\in D_{\varphi }^w$. The identities derived below describe how the deformation parameters independently regulate left-endpoint compression and right-endpoint redistribution through the induced transport geometry.

Theorem 4 

(Directional Field Component Splitting). Assume that $f\in C^1\left((0,1)\right)$ and that Assumption A1 holds. The partial parameter gradients of the transported field $F(v;p,q)=f\left(\varphi (v;p,q)\right)\sqrt{w\left(v\right)}$ with respect to the left scaling parameter p and the right redistribution parameter q are given exactly by the structural identities:

$${\displaystyle \frac{\partial F}{\partial p}}=f^{\prime }\left(\varphi \right)\left[{\displaystyle \frac{-R^{p}lnR}{{(1+R)}^2(p{R}^{p-1}+q)}}\right]\sqrt{w\left(v\right)}+{\displaystyle \frac{1}{2}}f\left(\varphi \right){\displaystyle \frac{{\partial }_{p}w\left(v\right)}{\sqrt{w\left(v\right)}}}$$

(26)

$${\displaystyle \frac{\partial F}{\partial q}}=f^{\prime }\left(\varphi \right)\left[{\displaystyle \frac{-R}{{(1+R)}^2(p{R}^{p-1}+q)}}\right]\sqrt{w\left(v\right)}+{\displaystyle \frac{1}{2}}f\left(\varphi \right){\displaystyle \frac{{\partial }_{q}w\left(v\right)}{\sqrt{w\left(v\right)}}}$$

(27)

Proof. 

Differentiating $F(v;p,q)=f\left(\varphi (v;p,q)\right)\sqrt{w\left(v\right)}$ with respect to $\theta \in \{p,q\}$ yields:

$${\displaystyle \frac{\partial F}{\partial \theta }}=f^{\prime }\left(\varphi \right)\left({\displaystyle \frac{\partial \varphi }{\partial \theta }}\right)\sqrt{w\left(v\right)}+{\displaystyle \frac{1}{2}}f\left(\varphi \right){\displaystyle \frac{{\partial }_{\theta }w\left(v\right)}{\sqrt{w\left(v\right)}}}$$

Since $\varphi =R/(1+R)$, we have ${\partial }_{\theta }\varphi ={(1+R)}^{-2}{\partial }_{\theta }R$. Applying implicit differentiation to $R^p+qR=e^v$ gives:

$${\displaystyle \frac{\partial R}{\partial p}}={\displaystyle \frac{-R^{p}lnR}{p{R}^{p-1}+q}},{\displaystyle \frac{\partial R}{\partial q}}={\displaystyle \frac{-R}{p{R}^{p-1}+q}}$$

Substituting these relations into the chain-rule expansion yields formulas (26) and (27). □

We verify the stability of our model by tracking specific singular and discontinuous structures passing through these parametric directional derivative components.

6.1. Application to Singular Endpoint Scaling (p-Direction Action)

Let $f\left(x\right)=x^{-1/4}$, which satisfies the admissibility condition of Theorem 3 with $\lambda =1/4$. We analyze the left-endpoint behavior as $v\to -\infty$ for $0\le p<1$.

For illustrative purposes, we consider weight families satisfying the following asymptotic parameter derivatives near the left boundary:

$${\displaystyle \frac{{\partial }_{p}w\left(v\right)}{\sqrt{w\left(v\right)}}}\sim -{\displaystyle \frac{(\sigma +1)v}{p^2}}\sqrt{w\left(v\right)}$$

Using the geometric asymptotic relations $\varphi \sim R\sim e^{v/p}$, $lnR\sim {\displaystyle \frac{v}{p}}$, and $p{R}^{p-1}+q\sim p{R}^{p-1}$, the transport contribution in (26) satisfies:

$$f^{\prime }\left(\varphi \right)\left({\displaystyle \frac{\partial \varphi }{\partial p}}\right)\sqrt{w\left(v\right)}\sim \left({\displaystyle \frac{v}{4p^2}}\right)e^{-v/\left(4p\right)}\sqrt{w\left(v\right)}$$

The matching redistribution contribution behaves as:

$${\displaystyle \frac{1}{2}}f\left(\varphi \right){\displaystyle \frac{{\partial }_{p}w\left(v\right)}{\sqrt{w\left(v\right)}}}\sim -\left({\displaystyle \frac{(\sigma +1)v}{2p^2}}\right)e^{-v/\left(4p\right)}\sqrt{w\left(v\right)}$$

Combining both terms gives:

$${\displaystyle \frac{\partial F}{\partial p}}\sim \left({\displaystyle \frac{-2\sigma -1}{4p^2}}\right)v{e}^{-v/\left(4p\right)}\sqrt{w\left(v\right)}$$

Although the transport contribution contains polynomial growth in v, the combined weighted exponential factor generated by the asymptotic structure of $w\left(v\right)$ dominates sufficiently to preserve square integrability:

$${\int }_{-\infty }^0{\left|{\displaystyle \frac{\partial F}{\partial p}}\left(v\right)\right|}^{2}dv<\infty$$

provided $\sigma >-1/2$. Thus the induced deformation field remains square integrable under left-scaling variations.

6.2. Application to Domain Redistribution (q-Direction Action)

We now analyze the right-boundary behavior as $v\to +\infty$, where $R\to \infty$ and $1-\varphi \sim q{e}^{-v}$.

For $f\left(x\right)=x^{-1/4}$, the derivative satisfies $f^{\prime }\left(\varphi \right)\to -{\displaystyle \frac{1}{4}}$ as $\varphi \to 1$. For illustrative purposes, we assume a weight redistribution derivative profile of the form:

$${\displaystyle \frac{1}{2}}{\varphi }^{-1/4}{\displaystyle \frac{{\partial }_{q}w\left(v\right)}{\sqrt{w\left(v\right)}}}\sim -q^{-1}\sqrt{w\left(v\right)}$$

Using $1+R\sim R$ and $p{R}^{p-1}+q\sim q$, the transport contribution in (27) satisfies:

$$f^{\prime }\left(\varphi \right)\left({\displaystyle \frac{\partial \varphi }{\partial q}}\right)\sqrt{w\left(v\right)}\sim {\displaystyle \frac{1}{4}}{e}^{-v}\sqrt{w\left(v\right)}$$

Thus the transport contribution decays exponentially at the right boundary, while the total variation field satisfies

$${\displaystyle \frac{\partial F}{\partial q}}\sim -{\displaystyle \frac{1}{q}}\sqrt{w\left(v\right)}(v\to +\infty )$$

Therefore,

$${\int }_0^{\infty }{\left|{\displaystyle \frac{\partial F}{\partial q}}\left(v\right)\right|}^{2}dv<\infty$$

for every $\tau >-1$, confirming bounded right-endpoint variation under redistribution deformations.

7. Density of Polynomial Approximations

We verify that the non-linear modifications driven by our implicit transport geometry do not disrupt the fundamental approximation properties of the space.

Theorem 5. 

Let $D_{\varphi }^w\cong L^2((0,1),\rho \left(x\right)dx)$ be an implicit pullback space whose effective density satisfies the boundary asymptotic conditions $\rho \left(x\right)\sim p{x}^{\sigma }$ as $x\to 0^{+}$ and $\rho \left(x\right)\sim C_q{(1-x)}^{\tau }$ as $x\to 1^{-}$, with parameters $\sigma ,\tau >-1$. Then the set of classical algebraic polynomials $\mathcal{P}\left(\right[0,1\left]\right)$ is dense in $D_{\varphi }^w$.

Proof. 

Using the boundary asymptotics

$$\rho \left(x\right)\sim p{x}^{\sigma }(x\to 0^{+}),\rho \left(x\right)\sim C_q{(1-x)}^{\tau }(x\to 1^{-}),$$

with $\sigma ,\tau >-1$, choose $\delta \in (0,1/2)$ such that $\rho \left(x\right)\le 2p{x}^{\sigma }$ on $(0,\delta ]$, and $\rho \left(x\right)\le 2C_q{(1-x)}^{\tau }$ on $[1-\delta ,1)$. Splitting the integral over $(0,1)$ gives

$${\int }_0^1\rho \left(x\right)dx={\int }_0^{\delta }\rho \left(x\right)dx+{\int }_{\delta }^{1-\delta }\rho \left(x\right)dx+{\int }_{1-\delta }^1\rho \left(x\right)dx.$$

The boundary integrals are finite because $\sigma ,\tau >-1$, while the middle integral is finite since $\rho$ is bounded on compact subintervals of $(0,1)$. Hence

$${\int }_0^1\rho \left(x\right)dx<\infty ,$$

so $\rho \left(x\right)dx$ defines a finite regular Borel measure on $(0,1)$.

Therefore, $C\left(\right[0,1\left]\right)$ is dense in $L^2((0,1),\rho \left(x\right)dx)$, and by the Weierstrass Approximation Theorem, algebraic polynomials are uniformly dense in $C\left(\right[0,1\left]\right)$. Uniform convergence on a finite-measure space implies convergence in the weighted $L^2$-norm, proving that $\mathcal{P}\left(\right[0,1\left]\right)$ is dense in $D_{\varphi }^w\cong L^2((0,1),\rho \left(x\right)dx)$. □

8. Discussion and Concluding Remarks

The results established in this paper demonstrate that the structural properties of the weighted pullback space $D_{\varphi }^w\cong L^2((0,1),\rho \left(x\right)dx)$ are driven by an internal non-linear transport engine. By generating the coordinate mapping $\varphi (v;p,q)$ through the automatic compactification of the implicit relation $R^p+qR=e^v$, we establish a framework where directional boundary asymmetries are generated natively by the internal coordinates rather than being manually assigned. The directional variations computed in Lemma 1 and expanded in Theorem 4 elucidate the precise analytical balance regulating space membership, illustrating how the redistribution contribution compensates for the transport contribution to preserve integrability.

Passing functions through the explicit field gradients provides a robust variational method for tracing independent endpoint stability across the parameter manifold. This formulation introduces an implicitly generated transport geometry rather than a prescribed prior framework, establishing a solid foundation for future research into parameter-driven asymmetric localization manifolds and nonlinearly induced pullback operators.