Encoding-Relative Structural Diagnostics for Differential Operators

Abstract

Differential operators often admit multiple algebraically equivalent symbolic formulations, yet those formulations can differ in the organization of their internal structure prior to solution analysis. A reproducible symbolic framework is introduced to compare such formulations at the level of operator expressions. Within a declared symbolic specification consisting of a fixed grammar, an admissible weight class, canonical compression rules, and an admissible family of reformulations, we define four encoding-relative structural descriptors: structural strain $\tau$, structural curvature $\kappa$, compressibility $\sigma$, and the balance ratio $\Gamma =\kappa /\tau$. Structural strain compares an encoding to a designated reference representation, while compressibility measures reduction under canonical symbolic compression. These quantities are deterministic descriptors within the declared encoding class rather than coordinate-free invariants of the underlying operator. The structural length functional underlying these descriptors is developed, canonical compression is formalized, and finite symbolic comparison is distinguished from pathwise symbolic deformation. A robustness theorem shows that, away from the threshold surface $\Gamma =\sigma$, sufficiently small admissible perturbations preserve the induced diagnostic label. A supporting weight-robustness result further shows that qualitative labels persist across a local admissible family of weight choices under corresponding nondegeneracy conditions. The framework serves as a reproducible diagnostic for operator representations alongside Lyapunov, spectral, pseudospectral, and energy-based stability theories. Examples of representative ordinary and partial differential operators illustrate how the descriptors are computed and how they behave under admissible re-expression, while the appendices provide the technical backbone of the paper: formal definitions, reproducibility protocol, extended perturbation arguments, and explicit failure-mode analysis. Additional sensitivity checks regarding encoding, weights, and threshold variation clarify the method’s scope, and explicit failure modes delineate the boundary cases in which the descriptors cease to apply. The main contribution of this study is a formally delimited and reproducible symbolic framework for comparing differential operators under a fixed, declared specification, together with robustness results and worked examples that clarify the method’s scope.

1. Introduction

Differential equations are commonly studied in terms of their solution trajectories, spectral properties, or associated energy functionals. In practice, however, a given dynamical system or partial differential equation may permit multiple algebraically equivalent operator formulations. These formulations often arise from nondimensionalization, regrouping nonlinear terms, coefficient normalization, or alternative symbolic encodings used in analytical or numerical workflows. While such transformations preserve the underlying mathematical model, they can significantly alter the symbolic organization of the operator expression itself. Before any trajectory-based, spectral, or numerical analysis is undertaken, it is therefore natural to ask whether structurally equivalent formulations of a differential operator can be compared systematically at the level of their symbolic representation. The identification problem addressed here is whether algebraically equivalent operator formulations can be compared reproducibly within a fixed symbolic representation class before trajectory-based, spectral, or numerical analysis begins.

In many analytical and computational workflows, differential operators are manipulated symbolically prior to analysis, discretization, or implementation. However, there is currently no reproducible framework for comparing algebraically equivalent operator encodings at this pre-analytic stage. This gap becomes particularly relevant in settings where symbolic formulation affects conditioning, numerical stability, or implementation structure, even when the underlying differential equation remains unchanged.

Classical analysis of differential equations has developed a rich collection of tools for assessing stability and dynamical behavior, ranging from Lyapunov methods and nonlinear dynamical systems theory to operator-theoretic and spectral approaches. Foundational treatments of ordinary and partial differential equations emphasize trajectory-based and spectral perspectives on stability and long-time behavior [1,2,3,4]. In many contexts, particularly for linearized or nonnormal systems, spectral and pseudospectral analyses reveal transient growth mechanisms and sensitivity of operators to perturbations [5,6], while perturbation theory describes how the eigenstructure responds to structural changes in the governing operator [7,8], including structural relationships between eigenvalues and eigenvectors in linear operators [9]. For evolution equations generated by partial differential operators, semigroup theory provides a rigorous framework linking operator properties to solution dynamics [10,11], while nonlinear evolution equations introduce additional structural effects studied in semilinear PDE theory [12]. Simultaneously, numerical analysis has long recognized that the algebraic formulation of differential operators can affect stability, conditioning, and discretization behavior [13,14,15]. This paper instead addresses a logically prior question concerning the symbolic organization of equivalent operator expressions before such analyses begin.

Specifically, this study considers how algebraically equivalent formulations of a differential operator may differ in their symbolic structure under a declared representation. In many analytical and computational contexts, operators are manipulated symbolically before analysis or discretization. Such symbolic representations can be viewed as expressions generated by a fixed grammar comprising derivatives, coefficients, algebraic operations, and composition rules. Under this viewpoint, algebraically equivalent operators may nevertheless differ in structural length, nesting, or compressibility when written in explicit symbolic form. While these differences do not change the underlying equation, they may affect how the operator is interpreted, manipulated, or implemented in analytical and computational settings.

This analysis is also related to developments in symbolic computation and expression-tree representations, in which mathematical objects are encoded as structured symbolic graphs and manipulated by deterministic rewrite systems [16,17]. In numerical analysis, it is well known that algebraically equivalent formulations can exhibit different conditioning and stability properties after discretization [13,14,15,18]. The framework developed here connects to these perspectives by focusing explicitly on the representation level, providing a reproducible method for comparing operator encodings prior to analytical or numerical evaluation.

To formalize this idea, we introduce a reproducible symbolic framework that compares operator expressions against a declared specification comprising a fixed grammar, an admissible class of weights, canonical compression rules, and a family of admissible reformulations. Within this framework we define four encoding-relative structural descriptors: structural strain $\tau$, structural curvature $\kappa$, compressibility $\sigma$, and the balance ratio $\mathsf{\Gamma }=\kappa /\tau$. These quantities provide deterministic descriptors of symbolic organization within the declared encoding class rather than intrinsic invariants of the underlying operator. The central theoretical result of the paper, stated as Theorem 1, shows that, away from the threshold surface $\mathsf{\Gamma }=\sigma$, sufficiently small admissible perturbations of the symbolic specification preserve the induced diagnostic classification.

The scope of this study is deliberately restricted to representation-level diagnostics and does not attempt to address dynamical stability or long-time solution behavior. Instead of proposing a new theory of dynamical stability, we provide a reproducible symbolic diagnostic framework for comparing differential operators at the expression level. The derivational logic is straightforward: a declared symbolic grammar induces an expression-tree representation, the expression tree induces a structural length functional, and that functional, in turn, induces the descriptors and screening relation studied below. This approach is illustrated through worked examples involving representative ordinary and partial differential operators, together with computational demonstrations implemented using standard symbolic and numerical computing environments [16,19,20,21]. These examples clarify both the utility and the limitations of the proposed descriptors. The relationship of the framework to established stability theory is stated explicitly in Section 1.2. This provides a reproducible layer of analysis that sits between symbolic formulation and classical stability theory, enabling systematic comparison of operator encodings before downstream analytical or numerical methods are applied.

The appendices are part of the core argument rather than supplementary afterthoughts: they contain the formal descriptor definitions, explicit computational validation, extended perturbation derivations, and scope-delimiting counterexamples on which the main-text claims rely.

1.1. Scope and Terminology

Several clarifications are useful at the outset. The descriptors introduced here are encoding-relative quantities defined with respect to a declared symbolic specification consisting of a grammar, weight class, and canonical compression rules. They are best interpreted as encoding-relative descriptors within a declared symbolic specification rather than intrinsic or coordinate-free invariants in the sense of operator theory or dynamical systems.

The framework is designed as a symbolic pre-analytic diagnostic and sits alongside classical stability criteria. Approaches based on Lyapunov functions, spectral analysis, perturbation theory, and semigroup methods remain the appropriate tools for determining the stability or long-time behavior of solutions. The quantities introduced here instead describe symbolic organization prior to trajectory-based or spectral analysis. Within the declared symbolic framework, the screening relation $\Gamma \le \sigma$ functions as a diagnostic condition rather than a universal stability theorem. Its role is to classify operator expressions according to the balance between symbolic deformation and compressibility under admissible reformulations. The labels are therefore restricted to the symbolic comparison problem considered here.

The analysis is also restricted to a class of admissible symbolic transformations that preserve algebraic equivalence while maintaining the declared representation grammar. Transformations outside this class—including coordinate changes, asymptotic reductions, discretization effects, or model modifications—fall outside the scope of the structural comparisons developed in this paper.

1.2. Position Relative to Classical Stability Theory

The framework developed here is best understood as an operator-representation diagnostic that sits logically upstream of classical stability analysis. Its purpose is not to determine the asymptotic stability, instability, decay rates, invariant sets, or long-time behavior of solutions. Those questions remain the domain of established methods such as Lyapunov theory, spectral and pseudospectral analysis, perturbation theory, and semigroup methods [1,2,3,4,5,6,7,8,10,11], including classical Lyapunov stability formulations [22]. The role of this construction is narrower and logically prior: it compares algebraically equivalent symbolic formulations of a differential operator under a declared encoding specification and records how their symbolic organization differs before trajectory-based, spectral, or discretization-based analysis is undertaken.

This distinction is important because the objects being compared are different. Classical stability theories analyze solution behavior, spectra, or generated flows. This framework examines symbolic operator encodings under a fixed grammar, admissible weight scheme, and canonical compression procedure. The resulting quantities $\tau$, $\kappa$, $\sigma$, and $\mathsf{\Gamma }$ therefore describe features of representation rather than dynamical invariants of the underlying differential equation.

For convenience,

Table 1. Position of the present framework relative to classical stability theories.

Approach	Primary Object of Analysis	Typical Output	Role in the Present Paper
Lyapunov methods	trajectories and auxiliary functionals	stability, instability, attraction	Provides trajectory-based criteria; the present framework operates prior to such analysis at the representation level
Spectral/pseudospectral methods	linearized operators, spectra, resolvents	modal growth, transient amplification, sensitivity	Analyzes operator spectra; the present framework compares symbolic encodings before spectral evaluation
Semigroup methods	generated evolution operators	well-posedness, decay, long-time behavior	Characterizes generated flows; the present framework examines pre-analytic symbolic structure
Numerical stability analysis	discretized operators and schemes	conditioning, stability regions, error propagation	Studies discretized operators; the present framework compares symbolic formulations prior to discretization
Present symbolic framework	operator encodings under a declared symbolic specification	encoding-relative structural descriptors and diagnostic labels	Operates on symbolic representations to quantify encoding-dependent structural organization prior to analysis

summarizes the distinction between the present framework and several standard analytical approaches.

This study should therefore be read in a deliberately limited sense. Its purpose is to define reproducible structural descriptors for operator representations within a fixed declared encoding class and to establish the robustness of the induced diagnostic labels. Questions of Lyapunov stability, spectral stability, semigroup decay, and numerical stability remain with the corresponding classical theories. The technical development that follows, therefore, focuses on a narrower object: the symbolic organization of operator encodings under a fixed declared specification.

1.3. Contribution Summary

This study makes three main contributions. It introduces a formally specified symbolic encoding framework for differential operators, consisting of a fixed grammar, an admissible weight scheme, canonical compression rules, and an admissible class of operator reformulations. Within this representation, it defines four encoding-relative structural descriptors—structural strain τ, structural curvature κ, compressibility σ, and the balance ratio Γ = κ/τ—that quantify the symbolic organization of operator expressions. It then establishes a robustness property showing that, away from the threshold surface Γ = σ, sufficiently small admissible perturbations of the symbolic specification preserve the induced diagnostic classification. A deterministic computational workflow and worked ODE/PDE examples are included to show how the descriptors behave across representative operator encodings. Together, these elements provide a reproducible framework for comparing differential operator formulations at the symbolic representation level prior to trajectory-based, spectral, or numerical analysis.

This study therefore moves from declared symbolic structure to formal descriptors, from descriptors to robustness results, and from those results to reproducible worked examples.

1.4. Structure of the Paper

The remainder of this paper develops the framework outlined above. Section 2 introduces the symbolic encoding specification, including the grammar of admissible operator expressions, the structural length functional, and the canonical compression rules used to define symbolic complexity and compressibility. Section 3 defines the structural descriptors $\tau$, $\kappa$, $\sigma$, and $\mathsf{\Gamma }$ and establishes robustness properties for the induced diagnostic classifications under admissible perturbations. Section 4 describes the deterministic computational workflow used to evaluate the descriptors and discusses implementation considerations relevant to reproducibility. Section 5 presents worked examples involving representative ordinary and partial differential operators, illustrating how the descriptors behave across different operator structures. Section 6 presents compact procedural evaluations that connect the conceptual examples to the deterministic workflow. Section 7 examines sensitivity with respect to encoding choices and weight variations, identifies failure modes of the framework, and summarizes the principal conclusions. Appendix A records the formal descriptor definitions and declared symbolic specification; Appendix B contains explicit worked computations; Appendix C documents reproducibility and validation; Appendix D provides the extended perturbation and robustness derivations underlying Theorem 1 and Proposition 1; and Appendix E records counterexamples and diagnostic failure modes that delimit the scope of the framework.

2. Symbolic Encoding Framework, Structural Length, and Canonical Compression

The symbolic diagnostic framework developed in this paper operates on explicit representations of differential operators rather than on their solution trajectories or spectral properties. This section introduces the formal encoding specification used throughout the paper, including the grammar of admissible symbolic expressions, the structural length functional used to measure symbolic complexity, and the canonical compression procedure used to define compressibility. These elements together form the declared symbolic representation on which the structural descriptors of Section 3 are built.

The motivation for introducing an explicit encoding framework is straightforward. Algebraically equivalent differential operators may appear in multiple symbolic forms depending on normalization, regrouping, or representation conventions. Modern analytical and computational workflows also manipulate operators symbolically prior to numerical discretization or spectral analysis. Throughout this paper, all quantities are defined relative to the declared encoding specification introduced in this section. Once the grammar, admissible weight scheme, and compression rules are fixed, every descriptor and comparison rule that follows becomes a deterministic function of the symbolic operator expression.

2.1. Primitive Atoms and Expression Trees

Differential operators are represented as symbolic expressions constructed from a finite grammar of primitive atoms and algebraic operations.

Definition 1 (Symbolic Encoding Grammar). 

Let $G$denote a declared symbolic grammar consisting of derivative atoms, coefficient atoms, algebraic operation atoms, and composition or nesting operations. Derivative atoms include partial derivatives of order $k\ge 0$, including the identity operator $D^0$. Coefficient atoms represent scalar or functional coefficients multiplying derivative terms. Algebraic operation atoms include addition, multiplication, exponentiation, and scalar multiplication. Composition or nesting operations represent the symbolic composition of operators.

Under this grammar, a differential operator is represented as a finite expression tree whose nodes correspond to grammar atoms and whose edges encode algebraic composition. Each symbolic operator, therefore, admits a tree-based representation similar to those used in modern symbolic algebra systems [16]. This representation allows symbolic complexity to be measured directly from the structure of the expression tree.

All subsequent definitions and results operate on these expression-tree representations.

2.2. Symbolic Complexity and Structural Length

To compare symbolic operator forms, we introduce a functional measuring the structural complexity of a symbolic encoding.

Definition 2 (Structural Length). 

Let $T\left(O\right)$denote the expression tree corresponding to a symbolic operator $O$under the grammar $\mathcal{G}$. The structural length of $O$, denoted $L\left(O\right),$is defined as the weighted node count of $T\left(O\right)$under a declared weight assignment on primitive atoms.

Formally,

$$L\left(O\right)={\sum }_{v\in T\left(O\right)}w\left(v\right),$$

(1)

where $w\left(v\right)$ is the weight associated with the node $v$ according to the admissible weight scheme described in Section 2.3.

This definition captures the symbolic size of the operator expression better than any analytical property of the underlying differential equation.

Example 1 (Structural Length of Equivalent Forms). 

Consider two algebraically equivalent expressions:

$$L_1={\mathsf{\partial }}_x^{2}u+{\mathsf{\partial }}_y^{2}u$$

(2)

and

$$L_2=\mathsf{\Delta }u.$$

(3)

Under a symbolic grammar that treats $\mathsf{\Delta }$ as a primitive atom, the second expression has a shorter structural length. Under a grammar that expands the Laplacian into explicit derivatives, both expressions may have equal length. The structural comparison, therefore, depends on the declared encoding specification. This example shows that structural descriptors are encoding-relative quantities rather than intrinsic operator invariants.

2.3. Admissible Weight Classes

The structural length function depends on a weight assignment applied to grammar atoms. To prevent arbitrary choices from dominating the analysis, the admissible weights are restricted by structural axioms.

Definition 3 (Admissible Weight Scheme). 

A weight assignment $w$ is called admissible if it satisfies positivity, derivative monotonicity, nesting monotonicity, additivity, and scale invariance. Positivity requires $w\left(v\right)>0$for all grammar atoms. Derivative monotonicity requires higher-order derivatives to receive nondecreasing weight. Nesting monotonicity requires deeper compositions or expression-tree levels to receive nondecreasing weight. Additivity requires disjoint symbolic substructures to contribute additively to structural length. Scale invariance requires uniform rescaling of all weights by a positive constant without altering qualitative diagnostic classifications. These axioms ensure that the structural descriptors introduced later depend on broad structural properties rather than fine-tuned weight choices.

The robustness of the framework with respect to admissible weight variations will be analyzed formally in Section 3.

2.4. Canonical Compression

Symbolic expressions often contain redundancies that can be removed through deterministic rewriting. The framework therefore introduces a canonical compression operator.

Definition 4 (Canonical Compression). 

Let $O$ be a symbolic operator expression. The canonical compression operator $C\left(O\right)$is defined as the deterministic application of a fixed set of rewrite rules that reduce redundant symbolic structure while preserving algebraic equivalence.

The rewrite system includes constant folding, associative flattening of sums and products, common-subexpression elimination, inverse cancelation, and subtree sharing. These transformations correspond closely to normalization operations used in symbolic computing environments and compiler optimization systems [17].

The compressibility of an operator expression is then defined by the reduction in structural length under compression.

$$\sigma \left(O\right)=L\left(O\right)-L\left(C\left(O\right)\right).$$

(4)

Compressibility, therefore, measures the extent to which symbolic redundancy can be eliminated through canonical rewriting. This compression step provides the second structural ingredient needed for the descriptor system developed in Section 3.

2.5. Declared Versus Intrinsic Invariance

A key conceptual distinction in the framework concerns the difference between intrinsic invariance and invariance relative to a declared symbolic encoding.

Definition 5 (Declared Invariance). 

A property of symbolic operator expressions is declared invariant if it remains unchanged under admissible symbolic rewrites within a fixed encoding specification $E=\left(\mathcal{G},w,C,A\right).$

This notion differs from intrinsic invariance as encountered in classical operator theory, where invariants are defined independently of coordinate representation. The structural descriptors introduced here remain stable within the declared symbolic framework alone.

Making this distinction explicit avoids the common misunderstanding that symbolic diagnostics are intended to define coordinate-free operator invariants.

2.6. Admissible and Inadmissible Reformulations

Symbolic comparisons in this framework are restricted to transformations that preserve algebraic equivalence within the declared grammar. Admissible transformations include regrouping of algebraic terms, coefficient normalization, factorization or expansion that preserves algebraic equivalence, and canonical rewrite transformations. Discretization changes, truncation of asymptotic expansions, model modifications, elimination or insertion of physical terms, and coordinate transformations that alter the symbolic grammar lie outside of the admissible class.

Restricting the comparison class in this way ensures that the structural descriptors track symbolic organization while keeping the mathematical model fixed.

2.7. Running Example: Burgers Equation

To illustrate the symbolic framework introduced in this section, we introduce a simple nonlinear operator that serves as a running example throughout this paper.

Consider the one-dimensional viscous Burgers equation

$${\mathsf{\partial }}_{t}u+u{\mathsf{\partial }}_{x}u=\nu {\mathsf{\partial }}_x^{2}u,$$

(5)

where $u(x,t)$ is a scalar field, and $\nu >0$ denotes the viscosity coefficient.

Rewriting the equation in operator form yields

$$O\left(u\right)={\mathsf{\partial }}_{t}u+u {\mathsf{\partial }}_{x}u-\nu {\mathsf{\partial }}_x^{2}u.$$

(6)

Within the symbolic encoding framework of Section 2.1, this operator corresponds to an expression tree composed of derivative atoms ${\mathsf{\partial }}_t$ and ${\mathsf{\partial }}_x$, coefficient nodes $u$ and $\nu$, and algebraic operations, including addition and multiplication.

Alternative symbolic encodings of the same operator arise naturally from algebraic regrouping. For example,

$$O_1\left(u\right)={\mathsf{\partial }}_{t}u+u{\mathsf{\partial }}_{x}u-\nu {\mathsf{\partial }}_x^{2}u$$

(7)

and

$$O_2\left(u\right)={\mathsf{\partial }}_{t}u+{\mathsf{\partial }}_x\left({\displaystyle \frac{1}{2}}{u}^2\right)-\nu {\mathsf{\partial }}_x^{2}u$$

(8)

represent algebraically equivalent formulations of the nonlinear term. Under the symbolic grammar introduced earlier, these encodings produce different expression trees and therefore different structural descriptors. Burgers equation is used as a running example to illustrate how the symbolic descriptors $\tau$, $\kappa$, $\sigma$, and $\mathsf{\Gamma }$ behave across alternative operator encodings.

The symbolic encoding specification introduced in this section enables precise structural comparisons between equivalent operator forms. Section 3 builds on this framework by introducing the structural descriptors $\tau , \kappa , \sigma , \mathsf{\Gamma }$ and establishing robustness properties for the diagnostic classifications under admissible symbolic perturbations.

3. Structural Descriptors and Robust Diagnostic Classification

The symbolic encoding framework introduced in Section 2 provides a deterministic representation of differential operators as expression trees under a declared grammar. This section builds upon that representation by introducing a family of encoding-relative structural descriptors that quantify the symbolic organization of an operator expression under the declared specification. These descriptors make it possible to compare algebraically equivalent operator forms in a systematic and reproducible way within the declared symbolic specification.

The central quantities introduced here are structural strain $\tau$, structural curvature $\kappa$, compressibility $\sigma$, and the associated balance ratio:

$$\mathsf{\Gamma }={\displaystyle \frac{\kappa }{\tau }}.$$

(9)

These quantities should be interpreted as diagnostic descriptors of symbolic organization within the representation class fixed in Section 2. Their definitions depend explicitly on the symbolic grammar, weight scheme, and canonical compression operator introduced there.

This section defines the structural descriptors, clarifies how symbolic perturbations of operator expressions are modeled within the declared framework, and establishes a robustness theorem showing that the diagnostic classification is stable under sufficiently small admissible symbolic perturbations away from the threshold surface defined by the relation Γ = σ. The definitions are arranged so that each subsequent result depends only on previously declared symbolic data: grammar, weights, compression, reference encoding, and admissible perturbation class.

For readability, the main text states the principal results and includes only the amount of proof detail needed to orient the reader. The full technical development appears in the appendices, which establish the well-definedness of the descriptors, provide extended proofs of the robustness results, and document the perturbation analysis underlying the screening classification. Appendix A records the formal definitions and declared encoding specification, Appendix C documents reproducibility and validation, Appendix D contains the extended perturbation and robustness derivations supporting Theorem 1 and Proposition 1, and Appendix E presents boundary cases in which the diagnostics become undefined or lose discriminatory power.

3.1. Structural Strain

The first descriptor measures the symbolic complexity of the operator expression relative to the canonical compressed form introduced in Section 2.

Definition 6 (Structural Strain). 

Let $O$be a symbolic operator expression with structural length $L\left(O\right)$. Let $O_{\mathrm{r}\mathrm{e}\mathrm{f}}$ denote a reference encoding of the same operator within the declared symbolic specification. The structural strain of $O$ is defined as

$$\tau \left(O\right)=L\left(O\right)-L\left(O_{\mathrm{r}\mathrm{e}\mathrm{f}}\right).$$

(10)

Structural strain, therefore, measures the symbolic excess of a given encoding relative to a chosen reference representation of the operator.

Remark 1. 

Structural strain may be positive, zero, or negative, depending on the reference encoding. In practice, the reference representation $O_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is chosen as a canonical baseline within the declared encoding specification.

3.2. Structural Curvature

While structural strain measures symbolic excess, it does not capture how structural complexity is distributed across the expression tree. To quantify this aspect, we introduce a second descriptor based on the distribution of structural weights across the symbolic representation.

Definition 7 (Structural Curvature). 

Let $T\left(O\right)$ denote the expression tree associated with operator $O$. The structural curvature $\kappa \left(O\right)$ is defined as the weighted second moment of the node weights about their mean value:

$$\kappa \left(O\right)={\sum }_{v\in T\left(O\right)}w\left(v\right) \left(d\right(v)-\stackrel{-}{d}{)}^2,$$

(11)

where $d\left(v\right)$ denotes the depth of node $v$ in the expression tree, and $\stackrel{-}{d}$denotes the average node depth.

Structural curvature therefore measures the degree of symbolic concentration or dispersion of complexity across the expression tree. Low curvature corresponds to a relatively uniform symbolic structure, whereas high curvature indicates concentration of structural complexity in deeply nested substructures.

3.3. Compressibility

The third descriptor measures the potential reduction in symbolic complexity under canonical compression.

Definition 8 (Compressibility). 

The compressibility of operator expression $O$ is defined as

$$\sigma \left(O\right)=L\left(O\right)-L\left(C\left(O\right)\right).$$

(12)

Thus, compressibility records how much symbolic simplification canonical rewriting actually achieves. Unlike structural strain, which compares the operator encoding to a reference representation, compressibility remains internal to a single encoding.

Remark 2. 

In practice, compressibility can be evaluated deterministically using the canonical compression rules defined in Section 2.4. These rules correspond closely to normalization procedures implemented in symbolic algebra systems and rewrite frameworks [17].

3.4. Balance Ratio

To compare symbolic deformation with available compressibility, the framework introduces the balance ratio defined below.

Definition 9 (Balance Ratio). 

For operator $O$ with structural descriptors $\tau \left(O\right)$ and $\kappa \left(O\right)$, define

$$\mathsf{\Gamma }\left(O\right)={\displaystyle \frac{\kappa \left(O\right)}{\tau \left(O\right)}}$$

(13)

whenever $\tau \left(O\right)\ne 0$.

Example 2 (Worked Symbolic Operator Comparison). 

To illustrate the structural descriptors introduced above, consider two algebraically equivalent symbolic representations of the two-dimensional diffusion operator acting on a scalar field $u(x,y)$.

The first representation writes the Laplacian explicitly in terms of partial derivatives:

$$O_1\left(u\right)={\mathsf{\partial }}_x^{2}u+{\mathsf{\partial }}_y^{2}u.$$

(14)

An alternative representation introduces the Laplacian operator as a primitive symbol:

$$O_2\left(u\right)=\mathsf{\Delta }u.$$

(15)

Both expressions represent the same differential operator

$$\mathsf{\Delta }u={\mathsf{\partial }}_x^{2}u+{\mathsf{\partial }}_y^{2}u,$$

(16)

but their symbolic encodings differ depending on the grammar used to represent differential operators.

Under the symbolic encoding framework introduced in Section 2, each operator is represented as an expression tree whose nodes correspond to derivative atoms, algebraic operations, and coefficient nodes. Suppose the grammar treats the Laplacian symbol $\mathsf{\Delta }$ as a primitive atom with a weight comparable to a second-order derivative.

For the expanded representation $O_1$, the expression tree contains two second-order derivative nodes, one addition node, and the operand node $u$. In contrast, the compact representation $O_2$ contains one Laplacian atom and the operand node $u$.

Thus, the structural length of the compact encoding reduces to the weight assigned to the Laplacian atom,

$$L\left(O_2\right)=w\left(\mathsf{\Delta }\right).$$

(17)

If the grammar allows the Laplacian to be treated as a primitive atom, the canonical compression procedure defined in Section 2.4 maps the expanded form to the compact form:

$$C\left(O_1\right)=O_2.$$

Consequently, the structural strain and compressibility of $O_1$ are given by

$$\tau \left(O_1\right)=L\left(O_1\right)-L\left(O_{ref}\right)$$

(18)

$$\sigma \left(O_1\right)=L\left(O_1\right)-L\left(C\right(O_1\left)\right),$$

(19)

where $O_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is chosen as the canonical Laplacian encoding $O_2$. In this example the compact Laplacian representation $O_2$ naturally serves as the reference encoding $O_{\mathrm{r}\mathrm{e}\mathrm{f}}$.

In contrast, the compressed representation $O_2$ already lies in canonical form, so

$$\tau \left(O_2\right)=0,\sigma \left(O_2\right)=0.$$

(20)

Structural curvature depends on the distribution of node depths in the expression tree. The expanded representation $O_1$ exhibits greater symbolic dispersion because the derivative terms appear as separate branches of the tree, whereas $O_2$ concentrates symbolic complexity in a single operator node.

The balance ratio

$$\mathsf{\Gamma }(O)={\displaystyle \frac{\kappa \left(O\right)}{\tau (O)}}$$

differs between the two encodings, even though they represent the same mathematical operator.

The same symbolic comparison can be performed for nonlinear operators. For instance, the Burgers operator introduced in Section 2.7 admits multiple algebraically equivalent encodings of the nonlinear advection term. The structural descriptors defined above allow these encodings to be compared quantitatively within the declared symbolic framework. Section 5 provides a detailed analysis of this example.

3.5. Admissible Symbolic Perturbations

In the original manuscript version, symbolic variation was described using derivatives with respect to symbolic paths. Reviewers correctly noted that such derivatives are not naturally defined in discrete symbolic spaces. We therefore replace this notion with a perturbation framework based on admissible transformation sequences.

Definition 10 (Admissible Perturbation Sequence). 

Let $O_0$ be a symbolic operator expression. An admissible perturbation sequence

$$\left\{O_n{\}}_{n\ge 0}\right.$$

is a sequence of operator expressions generated by a finite sequence of admissible symbolic transformations defined in Section 2.6, such that each transformation preserves algebraic equivalence under the declared grammar.

The magnitude of a symbolic perturbation is measured by the change in structural length:

$${\mathsf{\Delta }}_n= \mid L\left(O_n\right)-L\left(O_0\right)\mid .$$

(21)

Small symbolic perturbations correspond to sequences with bounded ${\mathsf{\Delta }}_n$.

This formulation avoids introducing differential structures on symbolic spaces while preserving the intuitive notion of local structural variation.

3.6. Diagnostic Classification

The symbolic descriptors defined above induce a natural classification of operator encodings according to the relationship between symbolic deformation and compressibility.

Definition 11 (Diagnostic Screening Condition). 

An operator expression $O$ is said to satisfy the symbolic screening condition if

$$\mathsf{\Gamma }\left(O\right)\le \sigma \left(O\right).$$

(22)

Expressions satisfying this condition exhibit symbolic deformation that remains balanced relative to the compressibility available under canonical rewriting.

This condition provides a structural classification within the declared symbolic framework rather than a stability criterion for the underlying differential equation.

3.7. Robustness of the Diagnostic Classification

The relevant notion of robustness concerns preservation of the sign of the difference $\sigma - \Gamma$ under admissible perturbations rather than invariance of the descriptor values themselves. The descriptors introduced above, therefore, induce a structural classification of operator encodings whose stability under symbolic perturbation can be analyzed mathematically. The following theorem establishes that this screening classification remains stable under sufficiently small admissible perturbations of the symbolic representation.

Theorem 1 (Robust Diagnostic Classification Away from the Threshold Surface). 

Let $O_0$ be a symbolic operator expression under a fixed declared grammar, admissible weight scheme, and canonical compression rule set, for which the descriptors $\tau$, $\kappa$, $\sigma$, and $\mathsf{\Gamma }=\kappa /\tau$ are well defined, and assume

$$\mathsf{\Gamma }\left(O_0\right)<\sigma \left(O_0\right).$$

Define the threshold margin

$$\mathsf{\Delta }\left(O_0\right)=\sigma \left(O_0\right)-\mathsf{\Gamma }\left(O_0\right)>0.$$

Suppose $\left\{O_n{\}}_{n\ge 0}\right.$ is an admissible perturbation sequence converging symbolically to $O_0$ in the sense that

$$\mid \mathsf{\Gamma }\left(O_n\right)-\mathsf{\Gamma }\left(O_0\right)\mid +\mid \sigma \left(O_n\right)-\sigma \left(O_0\right)\mid \to 0.$$

(23)

Then, there exists $N\in \mathbb{N}$ such that for all $n\ge N$,

$$\mathsf{\Gamma }\left(O_n\right)<\sigma \left(O_n\right).$$

(24)

Equivalently, the diagnostic label induced by the screening relation is preserved under all sufficiently small admissible perturbations whose combined variation in $\Gamma$ and $\sigma$ is strictly smaller than the threshold margin $\mathsf{\Delta }\left(O_0\right)$.

Proof. 

The argument relies on the continuity of the descriptors with respect to admissible symbolic perturbations under the declared encoding specification introduced in Section 2. Because the descriptors are deterministic functions of the symbolic representation, admissible perturbation sequences induce corresponding variations in the quantities $\mathsf{\Gamma }$ and $\sigma$.

At the reference operator $O_0$, the strict inequality $\mathsf{\Gamma }\left(O_0\right)<\sigma \left(O_0\right)$ defines a positive threshold margin

$$\mathsf{\Delta }\left(O_0\right)=\sigma \left(O_0\right)-\mathsf{\Gamma }\left(O_0\right)>0.$$

(25)

if an admissible perturbation sequence satisfies

$$\mid \mathsf{\Gamma }\left(O_n\right)-\mathsf{\Gamma }\left(O_0\right)\mid +\mid \sigma \left(O_n\right)-\sigma \left(O_0\right)\mid <\mathsf{\Delta }\left(O_0\right),$$

(26)

Therefore, perturbations smaller than the threshold margin cannot reverse the sign of $\sigma \left(O_n\right)-\mathsf{\Gamma }\left(O_n\right)$. Consequently,

$$\mathsf{\Gamma }\left(O_n\right)<\sigma \left(O_n\right)$$

(27)

for all sufficiently large $n$.

The full derivation of the perturbation bounds, continuity assumptions, and margin-preservation argument is given in Appendix D, where Theorem 1 is proved in extended form together with the perturbative expansion of the balance ratio used to justify the main-text reasoning. □

3.8. Weight Robustness of the Screening Classification

The diagnostic classification defined by the screening relation also depends on the admissible weight scheme introduced in Section 2.3. Because the structural length function depends on the admissible weight assignment, it is natural to ask whether variations in the weight scheme can alter the screening classification. The next result shows that the classification remains locally stable under sufficiently small admissible weight perturbations.

Proposition 1 (Weight Robustness of Diagnostic Labels). 

Let $O$ be a symbolic operator expression, for which the descriptors $\tau \left(O\right)$, $\kappa \left(O\right)$, $\sigma \left(O\right)$, and $\mathsf{\Gamma }\left(O\right)=\kappa \left(O\right)/\tau \left(O\right)$ are well defined under an admissible weight scheme $w$. Assume that the screening inequality

$$\mathsf{\Gamma }\left(O\right)<\sigma \left(O\right)$$

holds with positive margin

$$\mathsf{\Delta }\left(O\right)=\sigma \left(O\right)-\mathsf{\Gamma }\left(O\right)>0.$$

Then, there exists a neighborhood of admissible weight schemes $w^{\prime }$ such that, when the descriptors are recomputed under $w^{\prime }$, the screening inequality

$${\mathsf{\Gamma }}_{w^{\prime }}\left(O\right)<{\sigma }_{w^{\prime }}\left(O\right)$$

(28)

continues to hold. Consequently, the diagnostic classification induced by the screening relation is locally stable under sufficiently small admissible perturbations of the weight scheme.

Proof. 

The descriptors $L\left(O\right)$, $\kappa \left(O\right)$, and $\sigma \left(O\right)$ are finite sums of node weights determined by the declared grammar introduced in Section 2. Small admissible perturbations of the weight scheme, therefore, induce corresponding variations in these quantities.

Because the screening inequality

$$\mathsf{\Gamma }\left(O\right)<\sigma \left(O\right)$$

holds with positive margin

$$\mathsf{\Delta }\left(O\right)=\sigma \left(O\right)-\mathsf{\Gamma }\left(O\right)>0,$$

sufficiently small admissible weight variations cannot reverse the sign of $\sigma \left(O\right)-\mathsf{\Gamma }\left(O\right)$. Consequently, the screening relation persists under all admissible weight schemes lying within a sufficiently small neighborhood of the original assignment.

The full argument establishing local persistence of the diagnostic label under admissible weight perturbations is given in Appendix D, which treats weight perturbations explicitly and places Proposition 1 alongside the extended proof machinery for Theorem 1. □

When the equality $\mathsf{\Gamma }=\sigma$ holds exactly, arbitrarily small admissible perturbations may alter the diagnostic classification. This threshold, therefore, represents a structurally sensitive region of symbolic space. Detailed examples illustrating this behavior are presented in Section 5 and Appendix A.

Together, Theorem 1 and Proposition 1 establish that the screening classification is locally stable with respect to both symbolic perturbations of operator encodings and admissible perturbations of the weight scheme.

4. Deterministic Computation and Reproducibility Protocol

The structural descriptors introduced in Section 3 are defined directly on symbolic representations of differential operators. Their evaluation therefore requires a deterministic symbolic workflow capable of parsing operator expressions, applying canonical compression rules, and computing the associated structural quantities. This section sets out the computational protocol used to implement the framework and the reproducibility assumptions under which the descriptors are evaluated. Appendix C records the reproducibility protocol, implementation logic, and encoding stability checks supporting the deterministic interpretation used throughout this section.

Symbolic operator manipulation is now a standard component of modern analytical and computational environments, where differential operators are represented internally as expression trees or syntax graphs [16,19]. Canonical rewriting and symbolic normalization techniques used in such systems draw from formal rewriting theory [17] and numerical analysis workflows that operate on structured symbolic representations of differential equations prior to discretization [14,18]. The framework explicitly adopts this paradigm: the symbolic descriptors are evaluated using a deterministic procedure defined relative to a declared encoding specification.

The goal of this section is practical rather than theoretical. Instead of introducing additional mathematical results, we provide an explicit evaluation protocol that allows the descriptors $\tau , \kappa , \sigma , \mathsf{\Gamma }$ to be computed reproducibly for any operator expression represented within the grammar defined in Section 2.

4.1. Applying the Deterministic Evaluation Procedure

The evaluation of the structural descriptors proceeds deterministically once an operator expression is specified within the declared grammar. The symbolic representation is first interpreted as an expression tree $T\left(O\right)$ whose nodes correspond to derivative atoms, coefficient nodes, and algebraic operations permitted by the grammar of Section 2. Each node of this tree receives a structural weight according to the admissible weight scheme defined in Section 2.3.

The structural length of the operator is then obtained as the weighted node count

$$L\left(O\right)={\sum }_{v\in T\left(O\right)}w\left(v\right).$$

(29)

Canonical compression is subsequently applied using the deterministic rewrite rules introduced in Section 2.4, producing a normalized expression $C\left(O\right)$ that removes redundant symbolic structure while preserving algebraic equivalence. The compressibility descriptor is therefore

$$\sigma \left(O\right)=L\left(O\right)-L\left(C\left(O\right)\right).$$

(30)

Structural strain is evaluated relative to the designated reference encoding $O_{\mathrm{r}\mathrm{e}\mathrm{f}}$,

$$\tau \left(O\right)=L\left(O\right)-L\left(O_{\mathrm{r}\mathrm{e}\mathrm{f}}\right),$$

(31)

and structural curvature is obtained from the weighted dispersion of node depths within the expression tree,

$$\kappa \left(O\right)={\sum }_{v\in T\left(O\right)}w\left(v\right)\left(d\right(v)-\stackrel{-}{d}{)}^2.$$

(32)

When $\tau \left(O\right)\ne 0$, the balance ratio

$$\mathsf{\Gamma }\left(O\right)={\displaystyle \frac{\kappa \left(O\right)}{\tau \left(O\right)}}$$

is defined and compared with $\sigma \left(O\right)$ to determine whether the screening relation $\Gamma \left(O\right)\le \sigma \left(O\right)$ holds. Because each part of this evaluation depends only on the declared encoding specification and the symbolic operator expression, the procedure defines unique mapping from symbolic operator expressions to the descriptor tuple (τ, κ, σ, Γ). The resulting descriptor values are therefore deterministic and reproducible across independent implementations. Spectral discretization frameworks frequently manipulate operators symbolically prior to discretization and matrix assembly [23]. Appendix C formalizes these reproducibility claims by recording the validation protocol for deterministic evaluation and encoding robustness.

4.2. Algorithmic Representation

The procedure of Section 4.1 may be viewed algorithmically as a deterministic map from a declared symbolic encoding to the descriptor tuple (τ, κ, σ, Γ). Its essential point is that the calculation depends only on the chosen grammar, weight scheme, and compression rules, and therefore requires neither solving the differential equation nor computing its spectrum.

4.3. Computational Complexity

The computational cost of the procedure depends primarily on the size of the symbolic expression tree.

Let $n$ denote the number of nodes in the expression tree $T\left(O\right)$. Parsing and tree construction require $O\left(n\right)$ time in typical symbolic algebra implementations.

Structural length evaluation and descriptor calculation also scale linearly with tree size. Canonical compression requires pattern matching and subtree rewriting; its complexity depends on the rewrite system, but typically scales between

$$O\left(n\right) \mathrm{a}\mathrm{n}\mathrm{d} O(nlog n)$$

(33)

for the small operator expressions considered in this paper.

Because the descriptors are evaluated on symbolic expressions rather than on discretized operators, the computational burden remains modest compared with numerical eigenvalue or time integration analyses [13,15].

4.4. Reproducibility Conditions

Specifically, the descriptor values are uniquely determined once the symbolic grammar, admissible weight scheme, canonical compression rules, and admissible class of operator reformulations are fixed. Under this declared specification, independent implementations produce identical descriptor values. However, alternative encoding specifications may produce different descriptor values. This is why the framework consistently refers to encoding-relative structural descriptors rather than intrinsic operator invariants.

These claims are formalized operationally in Appendix C, where the workflow is restated as a validation protocol and directly linked to the examples in Section 5 and Section 6.

4.5. Implementation in the Burgers Operator

To evaluate structural strain, one encoding is designated as the reference representation $O_{\mathrm{r}\mathrm{e}\mathrm{f}}$. In the examples below, the product form $u {\mathsf{\partial }}_{x}u$ is used as the reference encoding.

The deterministic workflow can be illustrated using the Burgers equation introduced in Section 2.7.

Consider the viscous Burgers operator

$$O\left(u\right)={\mathsf{\partial }}_{t}u+u{\mathsf{\partial }}_{x}u-\nu {\mathsf{\partial }}_x^{2}u.$$

(34)

Two algebraically equivalent symbolic encodings of the nonlinear advection term are

$$O_1\left(u\right)={\mathsf{\partial }}_{t}u+u{\mathsf{\partial }}_{x}u-\nu {\mathsf{\partial }}_x^{2}u$$

(35)

and

$$O_2\left(u\right)={\mathsf{\partial }}_{t}u+{\mathsf{\partial }}_x\left({\displaystyle \frac{1}{2}}{u}^2\right)-\nu {\mathsf{\partial }}_x^{2}u.$$

(36)

Applying the evaluation procedure reveals how the alternative encodings differ structurally. Both expressions are first interpreted as expression trees containing derivative nodes ${\mathsf{\partial }}_t$ and ${\mathsf{\partial }}_x$, together with coefficient nodes and algebraic operations linking coefficients and derivatives. Because the nonlinear term is represented either as a product $u {\mathsf{\partial }}_{x}u$ or as a derivative acting on a quadratic expression ${\mathsf{\partial }}_x\left({\displaystyle \frac{1}{2}}{u}^2\right)$, the resulting trees contain different node configurations and therefore different structural lengths. Canonical compression normalizes the symbolic representation by removing redundant subexpressions while preserving algebraic equivalence. Evaluating the structural descriptors on the resulting trees yields distinct values for $\tau \left(O_1\right),\kappa \left(O_1\right),\mathsf{\Gamma }\left(O_1\right)$ and $\tau \left(O_2\right),\kappa \left(O_2\right),\mathsf{\Gamma }\left(O_2\right)$. Although both encodings represent the same dynamical equation, their symbolic organizations differ under the declared grammar and therefore occupy different locations in the descriptor space.

This Burgers example previews how algebraically equivalent encodings of the same operator separate in descriptor space once the evaluation procedure has been fixed.

5. Conceptual Examples of Symbolic Operator Structure

The symbolic framework developed in the preceding sections provides a deterministic method for comparing algebraically equivalent operator encodings with respect to a declared symbolic specification. To show how the structural descriptors behave in practice, we examine several representative operators from both ordinary and partial differential equations. Each example highlights how symbolic organization changes when operators contain nonlinear terms, multiple differential mechanisms, or alternative algebraic encodings. The goal is to illustrate how the descriptors τ, κ, σ, and Γ characterize the structure of operator representations prior to spectral, Lyapunov, or semigroup analysis [1,2,4,10].

Section 6 presents compact procedural evaluations for representative ODE and PDE cases, while Appendix B contains the fuller worked calculations that make the example section reproducible rather than merely illustrative.

5.1. Harmonic Oscillator

The harmonic oscillator provides a simple linear benchmark for evaluating the symbolic descriptors. Consider the classical second-order equation

$${\displaystyle \frac{d^{2}u}{d{t}^2}}+{\omega }^{2}u=0.$$

(37)

In operator form,

$$O\left(u\right)={\mathsf{\partial }}_t^{2}u+{\omega }^{2}u.$$

(38)

Under the symbolic grammar defined in Section 2, the expression tree contains a second-order derivative atom, a multiplication node representing the coefficient term ${\omega }^{2}u$, and an addition node linking the two components.

Because the expression contains no redundant symbolic substructures, canonical compression leaves the representation unchanged: $C\left(O\right)=O.$

The structural descriptors therefore identify the operator as structurally simple with minimal symbolic curvature. In this case, the encoding already appears in canonical compressed form and serves as a reference configuration illustrating the behavior of the descriptors for a minimal linear system [3,24].

5.2. Nonlinear Oscillator

Nonlinear terms introduce an additional symbolic structure that alters the distribution of complexity within the expression tree. Consider the cubic oscillator

$${\displaystyle \frac{d^{2}u}{d{t}^2}}+\alpha u+\beta u^3=0.$$

(39)

The operator representation

$$O\left(u\right)={\mathsf{\partial }}_t^{2}u+\alpha u+\beta u^3$$

(40)

contains a power node representing the cubic nonlinearity in addition to the derivative and coefficient nodes present in the harmonic oscillator example.

The nonlinear term increases the depth and branching of the symbolic expression tree. Consequently, the curvature descriptor $\kappa \left(O\right)$ becomes larger than in the linear case, reflecting the increased concentration of symbolic structure.

From the perspective of classical dynamical systems theory, this equation produces nonlinear oscillatory behavior and amplitude-dependent dynamics [3,24]. The symbolic descriptors introduced here do not attempt to predict such dynamics; rather, they quantify how nonlinear structure appears within the operator encoding.

5.3. Diffusion Operator

We next consider a partial differential operator representing linear diffusion:

$${\mathsf{\partial }}_{t}u=\nu {\mathsf{\partial }}_x^{2}u.$$

(41)

In operator form,

$$O\left(u\right)={\mathsf{\partial }}_{t}u-\nu {\mathsf{\partial }}_x^{2}u.$$

(42)

The expression tree contains a first-order temporal derivative, a second-order spatial derivative, and a coefficient multiplication node.

Because the symbolic representation contains no repeated substructures, canonical compression again leaves the operator unchanged. The resulting structural descriptors indicate a relatively simple hierarchical operator structure with modest symbolic curvature.

In classical PDE theory, diffusion operators generate smoothing semigroups and dissipative dynamics under appropriate boundary conditions [1,10]. Within the symbolic framework, the operator primarily serves as an example of a structurally balanced PDE encoding.

5.4. Advection–Diffusion Operator

A richer symbolic structure emerges when multiple physical mechanisms operate simultaneously. Consider the advection–diffusion equation

$${\mathsf{\partial }}_{t}u+a{\mathsf{\partial }}_{x}u=\nu {\mathsf{\partial }}_x^{2}u.$$

(43)

The operator

$$O\left(u\right)={\mathsf{\partial }}_{t}u+a{\mathsf{\partial }}_{x}u-\nu {\mathsf{\partial }}_x^{2}u$$

(44)

contains three differential components corresponding to temporal evolution, advective transport, and diffusive smoothing.

The expression tree therefore contains multiple derivative atoms of different orders together with their associated coefficient nodes. This branching structure increases the dispersion of symbolic weights throughout the expression tree and yields a larger curvature value than the pure diffusion operator.

From a classical perspective, the equation represents competition between transport and diffusion processes [1,2]. The symbolic descriptors capture this difference not through physical modeling but through the structural organization of the operator representation.

5.5. Burgers Equation

Finally, we consider the viscous Burgers equation

$${\mathsf{\partial }}_{t}u+u{\mathsf{\partial }}_{x}u=\nu {\mathsf{\partial }}_x^{2}u.$$

(45)

Two algebraically equivalent symbolic encodings of the nonlinear term are

$$O_1\left(u\right)={\mathsf{\partial }}_{t}u+u{\mathsf{\partial }}_{x}u-\nu {\mathsf{\partial }}_x^{2}u$$

(46)

and

$$O_2\left(u\right)={\mathsf{\partial }}_{t}u+{\mathsf{\partial }}_x\left({\displaystyle \frac{1}{2}}{u}^2\right)-\nu {\mathsf{\partial }}_x^{2}u.$$

(47)

Although both encodings represent the same differential equation, the symbolic grammar produces different expression trees. In the first representation, the nonlinear term appears as a product node linking $u$ and ${\mathsf{\partial }}_{x}u$. In the second, it appears as a derivative acting on a quadratic expression.

These structural differences lead to distinct descriptor values $\tau$, $\kappa$, and $\mathsf{\Gamma }$. The example therefore illustrates the central idea of the framework: algebraically equivalent operators may occupy different locations in descriptor space when expressed under a fixed symbolic encoding.

5.6. Summary of Structural Regimes

The examples above illustrate several characteristic symbolic regimes. Linear operators such as the harmonic oscillator and diffusion equation exhibit relatively simple expression trees with minimal compressibility. Nonlinear operators introduce additional symbolic nesting, increasing structural curvature. PDE operators involving multiple physical mechanisms produce richer expression trees due to the presence of several derivative components.

Table 2. Qualitative descriptor regimes used for representative operator encodings.

Operator	Structural Features	Typical Descriptor Behavior
Harmonic oscillator	Linear ODE with minimal symbolic structure	τ ≈ 0; low κ; negligible σ
Nonlinear oscillator	Polynomial nonlinearity introduces deeper symbolic nesting	Increased κ; moderate Γ
Diffusion operator	Simple PDE with single spatial derivative mechanism	Low curvature; minimal compressibility
Advection–diffusion operator	Multiple differential mechanisms increase tree branching	Larger κ; moderate structural dispersion
Burgers operator (two encodings)	Alternative symbolic forms of nonlinear term	Differing τ, κ, and Γ across encodings

summarizes the qualitative descriptor regimes observed for the representative operators considered in this section.

To make the comparison between operator encodings explicit, we report representative numerical values of the structural descriptors under a fixed admissible weight scheme (

Table 3. Illustrative descriptor values for equivalent operator encodings under a fixed admissible weight scheme.

Operator	Encoding	τ	κ	σ	Γ
Diffusion	∂x2u + ∂γ2u	2	1.5	2	0.75
Diffusion	Δu	0	0.5	0	—
Burgers	u∂xu	3	2.2	1	0.73
Burgers	∂x(½u2)	1	1.1	1	1.10

), illustrating how algebraically equivalent formulations separate in descriptor space.

These values are illustrative and correspond to a fixed admissible weight scheme; their purpose is to demonstrate concretely how algebraically equivalent encodings can produce different structural descriptors under the same symbolic specification.

These examples show how the structural descriptors provide a reproducible means of comparing operator encodings within a declared symbolic grammar. Section 6 now turns from qualitative structural interpretation to compact procedural evaluations of the same examples, while Appendix B records the fuller worked calculations.

6. Procedural Evaluation Examples

The examples in Section 5 illustrate how the symbolic descriptors $\tau$, $\kappa$, $\sigma$, and $\mathsf{\Gamma }$ characterize differences between operator encodings. Those discussions emphasized structural interpretation. In this section, we demonstrate how the descriptors are obtained procedurally by executing the deterministic workflow introduced in Section 4.

Two representative operator evaluations are presented below, one drawn from an ordinary differential equation and the other from a partial differential equation. They are intentionally compact and are meant to display the logic of the symbolic evaluation procedure—construction of the operator expression tree, assignment of structural weights, canonical symbolic compression, and evaluation of the associated structural descriptors—without interrupting the main narrative. The corresponding expanded calculations, including explicit structural length arithmetic and alternative weight choices, are recorded in Appendix B.

6.1. Oscillator Evaluation

Consider the harmonic oscillator

$${\displaystyle \frac{d^{2}u}{d{t}^2}}+{\omega }^{2}u=0,$$

(48)

with the operator representation

$$O\left(u\right)={\mathsf{\partial }}_t^{2}u+{\omega }^{2}u.$$

(49)

Under the symbolic grammar introduced in Section 2, the expression tree contains a second-order derivative node, a multiplication node associated with ${\omega }^{2}u$, a coefficient node, and an addition node.

To illustrate the deterministic evaluation procedure, consider the admissible illustrative weight scheme

Symbolic Atom	Weight
${\mathsf{\partial }}_t^2$	3
Multiplication	2
Coefficient	1
Addition	1

The calculation proceeds directly from the declared grammar and illustrative weight assignment.

The symbolic representation produces an expression tree containing four principal nodes: the second-order derivative node, the multiplication node associated with ${\omega }^{2}u$, the coefficient node representing ${\omega }^2$, and the addition node combining the two terms. Under the illustrative weight assignment introduced above, the structural length becomes

$$L\left(O\right)=3+2+1+1=7.$$

(50)

No redundant symbolic substructure appears in this representation, so canonical compression leaves the expression unchanged, $C\left(O\right)=O$. Consequently, the compressibility vanishes, $\sigma \left(O\right)=0$, and the strain relative to the reference encoding is also zero when the canonical oscillator form is chosen as $O_{\mathrm{r}\mathrm{e}\mathrm{f}}$. Because the node depths are shallow and nearly uniform, the resulting curvature is negligible, and the balance ratio is correspondingly small.

This simple example shows explicitly how the descriptor values arise from the deterministic symbolic pipeline.

6.2. PDE Evaluation Example

A similar evaluation can be performed for partial differential operators. Consider the advection–diffusion equation

$${\mathsf{\partial }}_{t}u+a{\mathsf{\partial }}_{x}u=\nu {\mathsf{\partial }}_x^{2}u,$$

(51)

with operator

$$O\left(u\right)={\mathsf{\partial }}_{t}u+a{\mathsf{\partial }}_{x}u-\nu {\mathsf{\partial }}_x^{2}u.$$

(52)

The symbolic representation contains three derivative atoms, ${\mathsf{\partial }}_t$, ${\mathsf{\partial }}_x$, and ${\mathsf{\partial }}_x^2$, together with coefficient nodes $a$ and $\nu$ and the algebraic operations linking the terms. Under the admissible weight scheme introduced earlier, the structural length may therefore be written schematically as

$$L\left(O\right)=w\left({\mathsf{\partial }}_t\right)+w\left({\mathsf{\partial }}_x\right)+w\left({\mathsf{\partial }}_x^2\right)+w\left(a\right)+w\left(\nu \right)+w\left(+\right).$$

(53)

Because the operator contains no repeated symbolic substructures, canonical compression leaves the expression unchanged, so $C\left(O\right)=O$, and hence $\sigma \left(O\right)=0$. The simultaneous presence of three differential mechanisms produces greater dispersion in node depth than in the pure diffusion case, leading to a larger curvature value $\kappa \left(O\right)$. The resulting descriptors therefore register a richer symbolic organization while remaining tied entirely to the encoding rather than to the solution behavior of the PDE.

The evaluations above demonstrate that the symbolic descriptors can be computed directly from the operator representation using the deterministic procedure described in Section 4. Additional explicit calculations and alternative weight choices are documented in Appendix B to ensure full computational reproducibility.

With the procedural evaluations in place, the final section of this study focuses on sensitivity, limitations, and the precise scope of the symbolic diagnostics.

7. Sensitivity, Limitations, and Scope of the Symbolic Diagnostic Framework

The preceding sections developed the symbolic framework and illustrated its computation using representative examples of ordinary and partial differential operators. Here, τ measures deviation from a reference encoding of the operator, while σ measures reduction under canonical symbolic compression. The examples demonstrate that algebraically equivalent formulations of a differential equation can occupy distinct locations in the descriptor space when written under a fixed symbolic encoding specification.

The purpose of this final section is to clarify the scope of the framework, examine its sensitivity to encoding choices, and summarize how the proposed descriptors relate to established analytical perspectives in the differential equations literature. The technical counterpart to this section is Appendix E, which collects explicit counterexamples and diagnostic failure modes so that the limits discussed here are documented constructively, rather than only described in prose.

7.1. Encoding Dependence and Representation Sensitivity

A central feature of the framework is that the structural descriptors are encoding-relative quantities. Their values depend on the symbolic grammar, weight scheme, and canonical compression rules declared in Section 2. As a result, the descriptors should not be interpreted as intrinsic invariants of the underlying differential equation. Instead, they measure structural organization within a specified symbolic representation.

This distinction parallels well-known representation effects in numerical analysis and dynamical systems modeling. For example, equivalent operator formulations may exhibit different conditioning properties after discretization, depending on how derivatives and coefficients are arranged algebraically [13,14]. Likewise, alternative coordinate representations of dynamical systems may reveal different structural features even though the underlying trajectories remain unchanged [3,24].

Within this framework, the symbolic descriptors provide a reproducible means of detecting such representational differences at the level of the operator expression itself.

7.2. Sensitivity to Weight Schemes

The admissible weight scheme defined in Section 2.3 assigns symbolic costs to grammar atoms such as derivatives, coefficients, and algebraic operations. Because the descriptors depend on these weights, different admissible weight schemes may produce different quantitative values for $\tau$, $\kappa$, and $\mathsf{\Gamma }$. However, the robustness result established in Section 3 shows that the diagnostic classification induced by the screening relation $\mathsf{\Gamma }\le \sigma$ is locally stable under sufficiently small perturbations of the encoding specification, provided the operator lies away from the threshold surface $\mathsf{\Gamma }=\sigma$. This property shows that modest variations in symbolic weight assignments preserve the classification of an operator representation across a local admissible neighborhood.

Similar robustness considerations appear in perturbation theory and spectral analysis, where small changes in operator structure lead to continuous changes in eigenstructure or spectral quantities [7,8]. The robustness result obtained here serves an analogous purpose in the symbolic setting.

The formal perturbation analysis underlying this robustness property is developed in Appendix D, which supplies the extended derivations for both descriptor perturbations and weight perturbations and therefore serves as the main theoretical support for the robustness claims stated in Section 3.

7.3. Relationship to Classical Analytical Frameworks

The symbolic descriptors introduced in this paper operate alongside classical analytical methods for studying differential equations. Stability theory, Lyapunov analysis, spectral theory, and semigroup methods remain the primary tools for understanding the dynamical behavior of solutions [1,2,4,10,11].

The symbolic framework addresses an earlier, more sharply delimited question about the representation of the operator itself. Before trajectory analysis, spectral computation, or discretization is performed, differential operators are typically written in explicit symbolic form. In many analytical and computational workflows, these expressions undergo symbolic manipulations such as normalization, regrouping of nonlinear terms, or coefficient scaling.

The descriptors introduced here provide a deterministic method for comparing such symbolic encodings within a declared representation. The framework operates upstream of classical stability analysis, serving as a structural diagnostic that complements rather than replaces dynamical analysis.

7.4. Computational Reproducibility

Another motivation for the proposed framework is the need for reproducible symbolic analyses. Modern symbolic and scientific computing environments routinely manipulate differential operators as expression trees [16,19]. By defining explicit grammar rules, weight schemes, and compression procedures, the descriptor computation becomes fully deterministic.

The workflow ensures that independent implementations operating under the same symbolic specification produce identical descriptor values. This reproducibility property aligns with broader trends in computational mathematics and scientific computing that emphasize transparent algorithmic pipelines and reproducible symbolic workflows [20,21].

7.5. Limitations

Several limitations of the framework should be noted. The descriptors characterize symbolic representations rather than solution dynamics and therefore do not provide direct information about stability, attractors, or long-term behavior of the underlying differential equation. The framework also relies on a declared symbolic grammar, so if the grammar is expanded—for example, by introducing new operator atoms or alternative compression rules—the descriptor values may change accordingly. In addition, the descriptors are currently defined for differential operators expressed in explicit symbolic form; extensions to broader classes of operators, such as nonlocal or integral operators, would require modifications to the grammar and weight scheme. These limitations reflect the paper’s chosen scope: the goal is to introduce a reproducible symbolic diagnostic within a clearly defined representational setting.

Appendix E complements these remarks with explicit instances of vanishing strain, non-admissible reformulation, degenerate compression, and other cases in which the framework becomes undefined or diagnostically weak.

7.6. Concluding Perspective

A symbolic framework is formulated in this study to compare algebraically equivalent differential operator encodings using deterministic structural descriptors. Once the grammar, admissible weight scheme, and canonical compression rules are fixed, the resulting quantities can be evaluated reproducibly and interpreted directly in terms of operator-expression structure.

Examples from both ordinary and partial differential equations—including diffusion, advection–diffusion, and Burgers-type operators—show that equivalent formulations may still differ in systematic ways at the symbolic level. The resulting descriptor space therefore captures meaningful variation in representation even when the underlying equation itself is unchanged.

What emerges is a structural diagnostic that operates at the representation level, upstream of trajectory-based, spectral, and numerical analysis. Its value lies in providing a precise and reproducible language for comparing operator formulations before more familiar analytical tools are brought to bear. Together, the main development and Appendix A, Appendix B, Appendix C, Appendix D and Appendix E provide a complete treatment of the framework’s definitions, robustness results, computational evaluation, perturbation behavior, and limits of applicability.

This perspective provides a clear path toward broader operator classes, richer symbolic grammars, and automated workflows for representation-level analysis in modern symbolic computing environments.