Generalized Research--Expression Model (GREM)

1. Classical Forms of Academic Expression

1.1. The Possibility of Arbitrary Expression

Written expression has nearly infinite possibilities. One may express a concept, a question, a viewpoint, evidence, a counterexample, a process, a relation, or a complex structure. Such content may further be organized according to different orders and relations. It is impossible here to exhaust this open space of expression. Instead, following the idea of the generalized analytical coordinate system discussed previously, we abstract some basic structures that repeatedly appear in expression.

For this purpose, we introduce the expression coordinate:

$$\mathbf{e}=\{{\mathbf{e}}_1,{\mathbf{e}}_2,\dots \}$$

where ${\mathbf{e}}_1$ denotes a content point selected in the current expression. It may be a question, a viewpoint, evidence, a counterexample, a definition object, a process node, or any other content that needs to be expressed. Here, the expression coordinate is not yet required to correspond to the previously discussed generalized analytical coordinate. It first merely indicates “what is to be expressed at present.”

• m-point expression modes.

According to the number m of expression coordinates selected in one expression, one may form one-point expression, two-point expression, and multi-point expression. We denote such m-point expression as

$$\mathrm{EM}\left(\mathbf{e}\right)={\mathrm{EM}}_m({\mathbf{e}}_1,{\mathbf{e}}_2,\dots ,{\mathbf{e}}_m).$$

One-point expression mode:${\mathrm{EM}}_1\left({\mathbf{e}}_1\right)$.

1.

One-point expression unfolds around one central expression coordinate ${\mathbf{e}}_1$. For example, one may define a concept, describe an object, give an example, present a number or a figure, or merely raise a question or give a conclusion.

Taking ${\mathbf{e}}_1=$ “cloud” as an example, one may explain what a cloud is, what phenomena it has, or give a figure related to clouds.

One-point expression is one of the most basic forms of expression. From very early on, people have been able to express themselves around a person, an object, an event, or a question without necessarily establishing relations between it and other content.

Two-point expression mode:${\mathrm{EM}}_2({\mathbf{e}}_1,{\mathbf{e}}_2)$.

1.

Two-point expression brings two expression coordinates ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ simultaneously into the current expression. In the simplest case, the two contents may merely appear successively or in parallel, without necessarily establishing an explicit relation.

As the expressive task deepens, a clearer relation may be formed between the two expression coordinates. For example:

$${\mathbf{e}}_1\approx {\mathbf{e}}_2\left(\mathrm{similarity}\right),{\mathbf{e}}_1\ne {\mathbf{e}}_2\left(\mathrm{difference}\right),{\mathbf{e}}_1\to {\mathbf{e}}_2\left(\mathrm{direction}\right).$$

There may also be other relations such as dependence, inclusion, correspondence, and succession.

For example, taking “bird wing” and “aircraft wing” as two expression coordinates, one may express their similarity; taking two sets of experimental results as two expression coordinates, one may express their difference; taking a cause and a result as two expression coordinates, one may form a directional expression.

Therefore, the important change in two-point expression is not merely that “there is one more content,” but that the two expression coordinates begin to make it possible to establish a relation.

Multi-point expression mode:${\mathrm{EM}}_m({\mathbf{e}}_1,{\mathbf{e}}_2,\dots ,{\mathbf{e}}_m),m\ge 3$.

1.

Multi-point expression involves a number of coordinates $m\ge 3$. Multiple expression coordinates may simply be juxtaposed, or may form order, hierarchy, network, or other complex relations.

For example, in a simplified argumentative expression, evidence, reasoning, and claim may be taken as three expression coordinates:

$$\mathrm{Evidence}\to \mathrm{Reasoning}\to \mathrm{Claim}.$$

Here, the three expression coordinates form an argumentative structure according to a certain relation.

Another example is to take problem, method, result, and conclusion as four expression coordinates:

$$\mathrm{Problem}\to \mathrm{Method}\to \mathrm{Result}\to \mathrm{Conclusion},$$

forming an expression of a paper structure.

Actual texts are usually far more complex than these simple forms. However, no matter how content increases, its basic structure can first be observed from the perspective of “how many expression coordinates are currently present, and how these coordinates are organized.”

One-point, two-point, and multi-point expression modes may be further combined. For example, a passage may first give a one-point description around ${\mathbf{e}}_1$, then establish a comparative relation between ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$, and then introduce ${\mathbf{e}}_3$ to explain the cause. A complex text may contain a large number of such local expression units.

Such a combined relation may be generally represented as

$$\mathbf{EM}={\mathbf{EM}}^{\left(1\right)}\oplus {\mathbf{EM}}^{\left(2\right)}\oplus \dots \oplus {\mathbf{EM}}^{\left(n\right)},$$

where “⊕” denotes the combination of different expression modes, without pre-specifying the concrete manner of combination.

Therefore, a complex text does not need to run from beginning to end according to one single mode. Instead, it may be formed by continuous, nested, or superimposed expressive structures of different scales and relations. Papers, monographs, and textbooks usually have such combinatory characteristics.

1.2. Expression Based on Discourse Modes

In classical writing research and practice, people have also used discourse modes to confront nearly infinite content and organizational possibilities.

Discourse modes are also called patterns of development. Specifically, long-term writing practice has formed a set of discourse modes, also called modes of discourse, that can be repeatedly used across objects and disciplines. Discourse modes include basic rhetorical modes, such as description, narration, definition, classification, decomposition, comparison and contrast, analogy, process analysis, cause and effect, exemplification, evaluation and argumentation, and question and answer. These rhetorical modes have long existed in rhetoric, college writing, and academic writing traditions (Connors, 1981, 1997; Kirszner & Mandell, 1986; Corbett & Connors, 1999; Langan, 2013; Nadell, Langan & Coxwell-Teague, 2019; Hacker & Sommers, 2020; Crowther et al., 2022).

Academic papers, monographs, and textbooks may adopt different structures, and the same structure may contain nearly infinite concrete content. Nevertheless, complex expression can usually still be decomposed into smaller discourse tasks. For example, a study may need to explain what the research object is like, define a concept, divide several categories, explain an experimental process, compare different results, identify differences, analyze possible causes, provide evidence, and argue for a conclusion. The objects, knowledge, and concrete language involved in these tasks may continually change, but basic operations such as description, definition, classification, process analysis, comparison, contrast, causation, evidence provision, and argumentation can repeatedly appear.

In this paper, we denote rhetorical modes as

$$\mathbf{M}\equiv \{{\mathbf{M}}_i\mid i=1,2,\dots ,I\}.$$

To reduce the complexity caused by the correspondence between symbols and terminology, we do not form a correspondence between specific rhetorical modes and ${\mathbf{M}}_i$.

The discourse mode invoked in one expressive process is uniformly denoted by $\mathrm{Md}$. $\mathrm{Md}$ may denote a single rhetorical mode, namely $\mathrm{Md}={\mathbf{M}}_i$, or a mixed mode formed by a combination of multiple rhetorical modes, namely $\mathrm{Md}=\{{\mathbf{M}}_i,{\mathbf{M}}_j,\dots \}$,.

Existing rhetorical modes may appear relatively independently, or may appear in combination, superposition, or nesting. In actual texts, even one sentence or one paragraph may present one rhetorical mode, and sometimes it is a combination or superposition of multiple rhetorical modes.

Such combination and superposition is a mixed mode (see, for instance, Darling 2004). Sometimes a chapter may center on one mode, for example describing quantitative research results, while in other cases multiple rhetorical modes may be superimposed. In addition, some modes may be composed of other more basic rhetorical modes. For example, exposition may use multiple modes such as description, definition, comparison, contrast, classification, decomposition, and analogy. Mixed modes may appear in parallel or nested forms (Wu 2026b).

According to the organizational relation among different modes, mixed modes may be divided into parallel structures and embedded structures.

Parallel structure. A parallel structure indicates that multiple rhetorical modes participate successively in the same expressive process, while the modes remain relatively independent. Two adjacent modes may be denoted by the symbol “//”:

$$\mathrm{Md}={\mathbf{M}}_i//{\mathbf{M}}_j.$$

Embedded structure. An embedded structure indicates that one rhetorical mode internally contains one or more auxiliary rhetorical modes, and may be represented as

$$\mathrm{Md}={\mathbf{M}}_i\left({\mathbf{M}}_j\right),$$

where ${\mathbf{M}}_j$ denotes an auxiliary mode embedded within the main mode ${\mathbf{M}}_i$.

For example, $\mathrm{Md}=$ description (comparison) indicates that description is completed through comparison of different objects or different results. Similarly, $\mathrm{Md}=$ argumentation (evidence provision) indicates that concrete cases are invoked as support in the process of argumentation.

These forms of existence, together with the direct appearance of mode terms or other information that can indicate mode type, make some rhetorical modes highly manifest; other rhetorical modes may not directly appear. Similar to harmonics in music, after multiple components are superimposed, it is sometimes difficult to directly judge whether a certain frequency component is contained.

Therefore, from the perspective of the degree of mode recognizability, the following situations may appear in a text: mode manifest, where it can clearly be identified as definition, comparison, contrast, analogy, causation, etc.; mode superimposed, where multiple ${\mathbf{M}}_i$ can be identified simultaneously; mode latent, where a certain mode exists but its surface form is not obvious; mode undetermined/ambiguous, where, according to the current granularity of analysis, it cannot be reliably determined which ${\mathbf{M}}_i$ it belongs to, or even whether it should be assigned to some ${\mathbf{M}}_i$.

Mode undetermined or ambiguity actually touches a more basic question: can any text necessarily be composed independently, in parallel, or by superposition of various rhetorical modes? Are there text units that do not belong to the organization of any rhetorical mode? This is difficult to define precisely. On the surface, a sentence or paragraph may not appear to belong to any specific rhetorical mode. However, if the scope of application of certain modes is broadened, it may again be classified into an existing mode. For example, suppose a passage organizes a classification mode concerning aircraft, and a later passage is a decomposition mode concerning the composition of aircraft. Then the transitional discourse inserted between the two passages—“The above introduced the classification of aircraft; the following introduces the system composition of aircraft”—seems not to belong to any rhetorical mode. Yet under a broader understanding of description, it may also be regarded as a description of the transitional relation between the preceding and following passages.

Therefore, this paper does not equate “cannot be identified as a certain mode” with “does not belong to any mode,” nor does it assume that the above modes have exhausted all forms of textual organization.

Discourse modes provide a reduction mechanism relatively separate from concrete content: what continually changes is the concrete content entering the task, while what remains relatively stable are the basic cognitive and expressive operations that researchers can repeatedly invoke.

Although different rhetorical modes play different roles, they share several common features: the mode names and their basic functions are usually highly recognizable; different objects and disciplines can repeatedly invoke the same mode; a limited number of modes can reduce complex cognitive and expressive tasks; and combinations among modes can further expand the expressive space.

The names of various rhetorical modes usually have a certain semantic self-evidence. The names of various rhetorical modes usually have a certain semantic self-evidence. For example, names such as “description,” “definition,” “comparison,” “causation,” and “argumentation” already suggest the general direction of the corresponding discourse operation. As for how exactly to organize and unfold it, there is still a certain flexibility and room for choice.

Universality. Rhetorical modes are universal because the same description can describe the external contour of an aircraft, the morphology of a cell, a data distribution, or the basic content of a theory; the same definition can define a physical concept, a mathematical object, a social phenomenon, or a research term; comparison and contrast can handle experimental results, theoretical predictions, different literature, or different research plans; analogy, causation, evidence provision, and argumentation can likewise be repeatedly used across different disciplines and objects.

Strong reduction capacity. The reduction capacity of rhetorical modes is strong because the basic operations involved are limited, decomposing complex and diverse overall research and expression tasks into a relatively small number of basic tasks that can be identified and repeatedly invoked. The continuous increase of concrete objects and knowledge does not require a synchronous increase in basic discourse modes. Such reduction does not reduce the content that academic research and expression may involve; rather, it reduces the basic operations that researchers need to recreate when facing different tasks. A researcher does not need to invent a new “comparison” for every object, nor establish a new “definition” or “argumentation” for every theory. Relatively stable basic modes can enter continually changing concrete content.

This relation may be summarized as: complex and open cognitive and expressive tasks → limited basic Rhetorical modes. Limited modes therefore provide a set of basic operational resources that can be repeatedly used across tasks.

Large expansion space. Finally, the space for expanding expression is large. Rhetorical modes do not restrict complex cognition and expression to a few single modes. On the contrary, complex academic activities are rarely completed by only one mode; different modes may be successively, parallely, nestedly, or repeatedly invoked according to task needs. For example, a study may describe phenomena, compare different results, analyze causes, and form an argument. It may also involve defining concepts, classifying, giving examples, and providing evidence. Different modes therefore are not only used separately from one another, but can form combinations of different scales and levels.

Wu (2025) used relations such as split-unit, forward-backward, expansion-reduction, and orthogonal duality to discuss the generation and combination of rhetorical modes, producing new academic functions. Therefore, discourse modes actually form two processes that run in opposite directions but connect with each other:

$$\mathrm{complex}\mathrm{task}\to \mathrm{decomposition}\mathrm{into}\mathrm{limited}\mathrm{basic}\mathrm{modes}\to \mathrm{recombination}\mathrm{of}\mathrm{modes}\mathrm{into}\mathrm{complex}\mathrm{tasks},$$

or simply:

$$\mathrm{open}\mathrm{cognitive}\mathrm{and}\mathrm{expressive}\mathrm{tasks}\to \mathrm{limited}\mathrm{discourse}\mathrm{modes}\to \mathrm{repeated}\mathrm{invocation}\mathrm{and}\mathrm{combination}.$$

In short, limited basic modes do not close the expressive space; rather, they provide relatively stable basic units for open combination.

2. Generalized Research–Expression Model

2.1. Brief Introduction to Generalized Analytical Coordinates

Wu (2026b) proposed a generalized coordinate system composed of ten dimensions. This paper no longer considers ten dimensions, but divides generalized coordinates into two major categories:

Analytical coordinates:

$$\mathrm{AC}\equiv (O,F),$$

which are the collective name of analytical object O and analytical feature F. O mainly determines “what is analyzed,” while F mainly determines “from which direction and what concrete content is analyzed.”

Analytical-result coordinates:

$$\mathrm{RC}\equiv (S,Q,X)$$

or

$$\mathrm{Res}\equiv (S,Q,X),$$

which are the collective name of qualitative feature S, quantitative feature Q, and knowledge formulation X. In brief, S, Q, and X carry the qualitative result, quantitative result, and knowledge formulation formed or invoked at the corresponding analytical position. If the coordinate aspect is emphasized, $\mathrm{RC}$ is used; if the result aspect is emphasized, $\mathrm{Res}$ is used.

The generalized coordinate system is denoted as

$$\mathrm{GCS}\equiv (O,F,S,Q,X),$$

which is composed of five types of coordinates: O, F, S, Q, and X. The aforementioned $\mathrm{AC}$ and $\mathrm{RC}$ both belong to $\mathrm{GCS}$. The generalized coordinate C refers generally to one or more coordinates selected from O, F, S, Q, and X.

2.2. Brief Introduction to Generalized Analysis

Generalized analysis is analysis conducted with respect to analytical features. Detailed discussion will be discussed in subsequent manuscripts; here the main points are introduced.

Analytical features are divided into eight categories: morphology, composition, state, dynamics, function, relation, origin, and history. The following symbolization is adopted:

$$F=\{F_1,F_2,F_3,F_4,F_5,F_6,F_7,F_8\},$$

where $F_1\to F_8$ successively denote morphology, composition, state, dynamics, function, relation, origin, and history.

Each category has its concrete analytical content $F_{ij}$, where i denotes the category of the first-level analytical feature, and j denotes the concrete analytical content within that category.

The Generalized Analytical Method (GAM) means selecting one or more analytical features $F_i$ from $F_1\to F_8$, and selecting one or more concrete analytical contents $F_{ij}$ within the corresponding analytical feature to carry out analysis. Typical $F_{ij}$ for each $F_i$ will be given in subsequent manuscripts.

According to the number of analytical features selected in one analysis, generalized analytical methods may be divided into local analytical method (LA) and cross analytical method (CA), namely

$$\mathrm{GAM}\equiv \{\mathrm{LA},\mathrm{CA}\}.$$

The local analytical method (LA) selects one analytical feature from $F_1$–$F_8$,

$$F^s\equiv F_i,$$

and further selects one or more concrete analytical contents $F_{ij}$ within that analytical feature to carry out analysis.

When directed at one concrete analytical content, local analysis is represented as

$$\mathrm{LA}\left(F^s\right)\equiv F^s\to F_{ij}.$$

When directed at multiple concrete analytical contents within the same analytical feature, it is represented as

$$\mathrm{LA}\left(F^s\right)\equiv F^s\to \left\{F_{ij}\right\},$$

where $\left\{F_{ij}\right\}$ denotes the multiple concrete analytical contents actually selected from the same analytical feature $F_i$.

The cross analytical method (CA) selects two or more analytical features from $F_1$–$F_8$. Let its index set be

$$I^m\equiv \{i_1,i_2,\dots ,i_m\},m\ge 2.$$

The corresponding multiple analytical features are denoted as

$$F^m\equiv \{F_i\mid i\in I^m\}.$$

When two different analytical features $F_i$ and $F_j$ are selected, the cross analytical method may be represented as

$$\mathrm{CA}\left(F^m\right)\equiv F_i\leftrightarrow F_j,i\ne j.$$

Here “↔” indicates that the current investigation concerns what relation exists between two analytical features, for example, “morphology $F_1$ and function $F_5$ are related.”

Actual analysis usually also needs to enter concrete analytical content. A connection between two analytical features such as “morphology and function are related” is still relatively broad. Therefore, establishing relations between different analytical features usually also requires entering the concrete analytical contents of the two analytical features respectively. For $F_i$ and $F_j$, one may respectively select $F_{ik}$ and $F_{jl}$, and further investigate the relation between them.

After entering concrete analytical content, the cross analytical method may be represented as

$$\mathrm{CA}\left(F^m\right)\equiv F_{ik}\leftrightarrow F_{jl},i\ne j.$$

Here “↔” indicates that the current investigation concerns what relation exists between the respective analytical contents of two different analytical features.

2.3. Brief Introduction to Generalized Analytical Pathways

The Generalized Analytical Pathway (GAP) connects the analytical object with the analytical result $\mathrm{Res}$ through the main-node path $\mathrm{MAP}\left(C\right)$, and realizes pathway transformation through re-anchoring $\mathrm{Ra}$. The generalized analytical pathway includes three levels: the main-node path $\mathrm{MAP}$, internal paths within $\mathrm{GAM}\equiv \{\mathrm{LA},\mathrm{CA}\}$, and the re-anchoring cyclic path. Detailed introduction to the generalized analytical pathway will be given in subsequent manuscripts; some key points are introduced here.

1. Main-node path MAP.

This level establishes the main-node path from the analytical object O to the generalized analytical method $\mathrm{GAM}\equiv \{\mathrm{LA},\mathrm{CA}\}$ to the analytical result $\mathrm{Res}$, namely

$$\mathrm{MAP}\left(C\right)\equiv O\Rightarrow \mathrm{GAM}\Rightarrow \mathrm{Res},$$

where $\mathrm{GAM}\equiv \{\mathrm{LA}\left(F^s\right),\mathrm{CA}\left(F^m\right)\}$. Here “⇒” denotes the main path, O denotes the analytical object, $\mathrm{LA}$ or $\mathrm{CA}$ denotes local analysis or cross analysis of the analytical object, and $\mathrm{Res}$ denotes the result formed by analysis, namely one or more coordinate contents among S, Q, and X. This level describes where a study starts, what type of analysis it undergoes, and what kind of result it finally forms.

The main-node path is highly general. Different disciplines, different research objects, and different concrete research methods may have completely different implementation processes, but in the main-node analytical path, they can all first be compressed into the basic structure “analytical object—local or cross analysis—analytical result.” Therefore, the first level mainly reflects the overall direction and basic node relations of the generalized analytical pathway.

2. Internal paths within GAM.

Local analysis unfolds within one analytical feature, while cross analysis establishes relations between two or more analytical features.

• Local analysis $\mathrm{LA}\left(F^s\right)\equiv F^s\to F_{ij}$ establishes a connection between a single analytical feature $F^s$ and a single analytical content $F_{ij}$; here “→” denotes an internal path.
• Local analysis $\mathrm{LA}\left(F^s\right)\equiv F^s\to \left\{F_{ij}\right\}$ establishes a connection between a single analytical feature $F^s$ and multiple analytical contents $\left\{F_{ij}\right\}$; here “→” denotes an internal path.
• Cross analysis $\mathrm{CA}\left(F^m\right)\equiv F_i\leftrightarrow F_j$, $i\ne j$, establishes a connection between different analytical features $F_i$ and $F_j$; here “↔” denotes an internal path.
• Cross analysis $\mathrm{CA}\left(F^m\right)\equiv F_{ik}\leftrightarrow F_{jl}$, $i\ne j$, establishes a connection between different analytical contents $F_{ik}$ and $F_{jl}$; here “↔” denotes an internal path.

3. Re-anchoring cyclic path.

After one generalized analysis forms $\mathrm{Res}$, if the F, S, Q, or X content therein becomes a new research center, it may be re-determined as a new analytical object $O^{\prime }$ through re-anchoring $\mathrm{Ra}\left(C\right)$, and the same set of main-node analytical pathways may be invoked again, namely

$$\mathrm{Ra}\left(C\right)\equiv C\mapsto O^{\prime }\Rightarrow \mathrm{GAP},C\in \{F,S,Q,X\}.$$

Here “↦” denotes role transformation, and “⇒” denotes that the transformed analytical object re-enters the generalized analytical pathway.

2.4. Generalized Research–Expression Model GREM

Let the set of rhetorical modes be

$$\mathbf{M}\equiv \{{\mathbf{M}}_i\mid i=1,2,\dots ,I\}.$$

The discourse mode may select one rhetorical mode, or a combination of some. In general, a discourse mode may act on any existing text or content:

$$E\equiv \mathrm{Md}\left(\mathrm{text}\right),$$

where E denotes the expression output, namely the text, sentence, figure-table combination, or other communicable result formed after content is organized by some discourse mode.

If the generalized analytical coordinate C is directly taken as input, then

$$E\equiv \mathrm{Md}\left(C\right).$$

At this time, the discourse mode acts on analytical content that has already been positioned, rather than arbitrarily selected expressive content.

If the generalized analytical pathway GAP is introduced into the expressive process, then

$$E^G\equiv \mathrm{Md}\left[\mathrm{GAP}\left(C\right)\right].$$

Here $E^G$ denotes the generalized expression output, namely the expressive result formed after the discourse mode takes generalized analytical coordinates and their analytical pathways as input.

Here, C determines what analytical content the discourse mode acts on, $\mathrm{GAP}\left(C\right)$ determines along what analytical pathway these contents unfold, and $\mathrm{Md}$ determines what discourse mode is used to organize and express these contents. Thus, generalized analytical coordinates, generalized analytical pathways, and discourse modes enter the same operational model.

Actual academic expression is also affected by representational forms, auxiliary resources, expression principles, and other conditions. These factors are uniformly denoted by operational factors $\mathrm{\kappa }$.

Therefore, the general form of the generalized research–expression model (GREM) proposed in this paper is written as

$$\mathrm{GREM}:E^G\equiv \mathrm{Md}[\mathrm{GAP}\left(C\right),\mathrm{\kappa }].$$

GREM may therefore be understood through the following four progressive levels:

$$\begin{array}{c}E=\mathrm{Md}\left(\mathrm{text}\right)\\ \downarrow \\ E=\mathrm{Md}\left(C\right)\\ \downarrow \\ E^G=\mathrm{Md}\left[\mathrm{GAP}\left(C\right)\right]\\ \downarrow \\ \mathrm{GREM}:E^G=\mathrm{Md}[\mathrm{GAP}\left(C\right),\mathrm{\kappa }].\end{array}$$

The first level allows discourse modes to act on general text; the second level uses generalized analytical coordinates to delimit the content entering expression; the third level further provides these contents with an analytical pathway through GAP; the fourth level adds the operational conditions required by actual academic expression and forms the complete GREM.

Therefore, the core of GREM may be summarized as follows: use generalized analytical coordinates C to determine expressive content, use generalized analytical pathways GAP to pull the unfolding of content, use discourse modes $\mathrm{Md}$ to organize expression, and form generalized expression output $E^G$ under the constraints of operational factors $\mathrm{\kappa }$.

2.5. Generalized Research–Expression Dual Reduction

Generalized research based on the generalized analytical pathway $\mathrm{GAP}\left(C\right)$ and expression based on discourse mode $\mathrm{Md}$ each have a certain reduction function. Here we further discuss their joint action, forming a mechanism of dual reduction.

The research side reduces open research content into limited analytical coordinates and their pathways:

$$\mathrm{open}\mathrm{research}\mathrm{content}\to \mathrm{GAP}\left(C\right).$$

The expression side reduces open cognitive and expressive tasks into limited basic discourse modes:

$$\mathrm{open}\mathrm{cognitive}\mathrm{and}\mathrm{expressive}\mathrm{tasks}\to \mathrm{Md}.$$

The reduction here does not shrink the research and expression space: the generalized analytical pathway $\mathrm{GAP}\left(C\right)$ can form a large number of different pathways through changes in objects, analytical features, concrete analytical contents, and their relations; discourse modes $\mathrm{Md}$ can also form an open discourse space through selection, ordering, combination, nesting, and repeated invocation.

The most basic operational relation of GREM is that $\mathrm{GAP}\left(C\right)$ provides analytical content and pathways to $\mathrm{Md}$. However, in some tasks, the operation of discourse modes may in turn drive new analysis. For example, comparison may expose previously unnoticed differences; analogy may lead to new analytical features; causal analysis may require finding new evidence; argumentation may also discover that existing results are insufficient, thereby causing research to re-enter GAP.

This bidirectional coordination may be represented as

$$\mathrm{GAP}\left(C\right)\leftrightarrow \mathrm{Md}.$$

This paper calls this operational form in GREM, in which research pathways and discourse modes mutually drive each other, leading to generalized research–expression dual reduction (GREDR).

Therefore,

$$\mathrm{GREDR}\subset \mathrm{GREM}.$$

Here GREDR is not another expression model independent of GREM, but an operational form in which research reduction and expression reduction interact bidirectionally within GREM.

When comparing two experimental results, $\mathrm{GAP}\left(C\right)$ may first determine the two objects and their common analytical features and concrete analytical contents, so that differences have clear analytical positions. If the comparison only presents already obtained differences, the research pathway mainly provides content to the discourse mode; if the comparison exposes a previously unnoticed anomaly between experimental results and theoretical predictions, and thereby turns to a new $F_{ij}$ for experiments, computations, or theoretical analysis, then the discourse mode in turn drives GAP operation.

Analogy may also produce such an effect. $\mathrm{GAP}\left(C\right)$ first delimits between which objects and analytical contents a correspondence is established, while an unexpectedly formed analogy may cause researchers to enter analytical features that were not originally selected.

Causal modes may drive researchers to continue searching for cause terms, effect terms, action conditions, and evidence; argumentation may discover insufficient evidence while organizing existing S, Q, and X, thereby re-entering GAP to supplement analysis.

These situations jointly manifest as

$$\mathrm{GAP}\left(C\right)\leftrightarrow \mathrm{Md},$$

namely research pathways and discourse modes jointly advance the current task.

In more cases, $\mathrm{GAP}\left(C\right)$ mainly provides already determined analytical positions, analytical contents, or analytical results to $\mathrm{Md}$, while $\mathrm{Md}$ is mainly responsible for organizing and expressing these contents. At this time it may be represented as

$$\mathrm{GAP}\left(C\right)\to \mathrm{Md}.$$

For example, a study has already obtained the qualitative feature S and quantitative feature Q of a certain object, and then uses description mode to organize these results into text.

Compared with pure discourse modes, after introducing GAP, it can be further clarified:

$$\mathrm{for}\mathrm{which}O;\mathrm{from}\mathrm{which}F\mathrm{to}\mathrm{enter};\mathrm{which}{F}_{ij}\mathrm{are}\mathrm{used}\mathrm{to}\mathrm{process}\mathrm{which}\mathrm{already}\mathrm{formed}S,Q,\mathrm{or}X.$$

For example, classification mode itself does not specify what classification criterion is adopted; after entering GAP, one may clearly choose function $F_5$, a certain concrete analytical content $F_{ij}$, or classify according to existing S, Q, or X.

This operation may still expose differences, anomalies, or gaps during expression, except that these discoveries have not yet obviously changed GAP itself.

If the current expression does not invoke generalized analytical coordinates and generalized analytical pathways, one may set

$$\mathrm{GAP}\left(C\right)=⌀.$$

At this time GREM degenerates into pure discourse mode operation:

$$\mathrm{GREM}\to \mathrm{Md}.$$

Traditional discourse modes themselves have independent cognitive and expressive value. Therefore, not all expression is required to pass through generalized analytical coordinates and GAP. The subsequent chapters of this paper mainly discuss the operation of discourse modes after they enter GREM, rather than repeating the general theory of pure discourse modes.

2.6. Closure of the Generalized Research–Expression Model

The Generalized Research - Expression Model GREM establishes a relation between generalized expression output $E^G$ and discourse mode $\mathrm{Md}$, in the form

$$\mathrm{GREM}:E^G\equiv \mathrm{Md}[\mathrm{GAP}\left(C\right),\mathrm{\kappa }],$$

where

• $\mathrm{Md}$ denotes one discourse mode actually invoked in one expression, or a mixed mode formed by multiple discourse modes;
• C denotes generalized analytical coordinates;
• $\mathrm{GAP}\left(C\right)$ denotes generalized analytical pathways;
• $\mathrm{\kappa }$ denotes operational factors.

1. Discourse mode.

Discourse mode $\mathrm{Md}$ is selected from the rhetorical modes set

$$\mathbf{M}\equiv \{{\mathbf{M}}_i\mid i=1,2,\dots ,I\},$$

including description, narration, definition, classification, decomposition, comparison and contrast, analogy, process analysis, causation, exemplification, evaluation and argumentation, question and answer, etc. It may also denote a mixed mode formed by a combination of multiple discourse modes, namely $\mathrm{Md}=\{{\mathbf{M}}_i,{\mathbf{M}}_j,\dots \}$. Combination may be through parallel structure, for example $\mathrm{Md}={\mathbf{M}}_i//{\mathbf{M}}_j$, or through embedded structure, for example $\mathrm{Md}={\mathbf{M}}_i\left({\mathbf{M}}_j\right)$.

2. Generalized analytical coordinates.

Generalized analytical coordinates C include analytical object O, analytical feature $F_i$, concrete analytical content $F_{ij}$, and analytical result $\mathrm{Res}\equiv \{S,Q,X\}$.

3. Generalized analytical pathways.

The generalized analytical pathway $\mathrm{GAP}\left(C\right)$ is composed of the main-node analytical pathway, internal paths of generalized analytical methods, and the re-anchoring cyclic path:

• Main-node analytical pathway:

  $$\mathrm{MAP}\left(C\right)\equiv O\Rightarrow \mathrm{GAM}\Rightarrow \mathrm{Res}.$$
• Internal paths of GAM:

  $$\mathrm{LA}\left(F^s\right)\equiv F^s\to F_{ij},\mathrm{CA}\left(F^m\right)\equiv F_i\leftrightarrow F_j,F_{ik}\leftrightarrow F_{jl}i\ne j.$$
• Re-anchoring cyclic path:

  $$\mathrm{Ra}\left(C\right)\equiv C\mapsto O^{\prime }\Rightarrow \mathrm{GAP}.$$

Here

$$\mathrm{GAM}\equiv \{\mathrm{LA}\left(F^s\right),\mathrm{CA}\left(F^m\right)\}$$

is the generalized analytical method composed of local analytical method LA and cross analytical method CA. The main-node analytical pathway MAP and the re-anchoring cyclic path will be discussed in subsequent manuscripts.

4. Closure requirement: operational factors.

To make GREM satisfy the closure requirement, operational factors $\mathrm{\kappa }$ also need to be clarified. We consider five operational factors; their specific forms will be introduced in the next section.

In summary, GREM is not simply applying a certain discourse mode to arbitrary content, but enabling discourse modes to invoke research content that has already been positioned and organized through analytical pathways.

Compared with general discourse modes, the core change of GREM is that expressive content is no longer merely arbitrarily selected text or concepts, but research content that has already entered generalized analytical coordinates and generalized analytical pathways. In this model,

• $\mathrm{Md}$ answers “what discourse mode is adopted for unfolding”;
• C answers “what analytical content is expressed”;
• $\mathrm{GAP}\left(C\right)$ answers “how these contents are organized through analysis”;
• $\mathrm{\kappa }$ provides further operational constraints.

3. The Set of Operational Factors $\mathrm{\kappa }$ in GREM

This section introduces the operational factors $\mathrm{\kappa }$ in GREM:

$$\mathrm{GREM}:E^G\equiv \mathrm{Md}[\mathrm{GAP}\left(C\right),\mathrm{\kappa }].$$

This paper mainly considers the following five types of operational factors:

$$\mathrm{\kappa }\equiv \{{\mathrm{\kappa }}_{\Phi },{\mathrm{\kappa }}_{\rho },{\mathrm{\kappa }}_C,{\mathrm{\kappa }}_M,{\mathrm{\kappa }}_P\},$$

where:

• ${\mathrm{\kappa }}_{\Phi }\equiv \Phi ,$academic representation factor;
• ${\mathrm{\kappa }}_{\rho }\equiv \rho ,$expression pathway factor;
• ${\mathrm{\kappa }}_C\equiv \mathrm{Aux}\left(C\right),$auxiliary coordinate factor;
• ${\mathrm{\kappa }}_M\equiv \mathrm{Aux}\left(M\right),$auxiliary discourse mode factor;
• ${\mathrm{\kappa }}_P\equiv P,$expression principle factor.

Different research and expression tasks need not invoke all operational factors; one or more may be selected according to actual needs. Specific tasks may also be affected by research purpose, disciplinary norms, evidence requirements, text type, length, and other conditions.

3.1. Academic Representation ${\Phi }_r$

Academic research and expression need appropriate representational forms. This paper collectively calls such representational forms academic representations $\Phi$, and denotes the one or more representational forms actually adopted as ${\Phi }_r$.

Unlike general expression based on pure text or natural language, academic research and academic expression often need to use multiple representational forms. The common set of academic representations is:

$$\Phi =\{{\Phi }_1,{\Phi }_2,\dots ,{\Phi }_n\}.$$

Here, ${\Phi }_1$ denotes natural language; ${\Phi }_2$ denotes numbers, where numbers may directly present quantitative features; ${\Phi }_3$ denotes symbols, including unit symbols, various operational and logical symbols, variable symbols, independent and dependent variable symbols, and symbols summarizing invariant quantitative features (e.g., pi, gravitational constant, fine-structure constant); ${\Phi }_4$ denotes figures and tables, where figures include concrete images, analytical schematic diagrams, and data visualizations, and tables include tables presenting quantitative numbers and tables of qualitative features; ${\Phi }_5$ denotes mathematical expressions, including formulas for presenting laws, equations for finding unknowns, and mathematical definitions; ${\Phi }_6$ denotes multimedia representations, such as audio, animation, and video; ${\Phi }_7$ denotes citation representations, namely pointing to relevant content through references, appendices, supplementary materials, databases, the remaining chapters of this paper, etc.; ${\Phi }_8$ denotes coordinate-based representation, namely representation through coordinate systems, coordinate values, and relations among coordinates.

These representational forms are not mutually exclusive categories. In actual use, these representations are selectable and combinable.

The same coordinate content, operational pathway, discourse operation, or operational constraint may be presented through one or more $\Phi$; the same $\Phi$ may also simultaneously carry multiple coordinate contents, operational relations, or other academic content.

Taking a concrete number in a quantitative feature as an example, besides being represented in Arabic numerals, it may also exist in other representational forms: it may be expressed in natural language, listed in a table, displayed in a data visualization figure, fitted by a mathematical expression, presented in multimedia form, recorded in cited resources such as databases.

Day (1998) pointed out that when there is only one or a small number of measurement results, they may be presented directly in the main text; for repeated or more numerous data, tables or figures may be used. If readers need to obtain accurate numerical values, tables are usually more suitable; if the main purpose is to show trends or morphology, figures are more effective. Rowland (n.d.) also pointed out that tables have advantages in precisely presenting numerical values, while figures and charts usually more easily help readers identify overall patterns. The concrete form should be determined by expressive purpose and reader needs.

For example, the relation among force, mass, and acceleration expressed by Newton’s second law may be simultaneously represented by natural language ${\Phi }_1$ (there is a definite relation among force, mass, and acceleration), mathematical expression ${\Phi }_5$ ($F=ma$), and schematic diagram ${\Phi }_4$ (marking the object, direction of force, and direction of acceleration).

Graphical representation ${\Phi }_4$ is also a common academic representation. Christiansen (2018) pointed out that various figures are distributed between concrete and abstract.

Concrete images are realistic images, such as various camera photographs, microscope images, telescope images, etc. They can present physical details at a glance. Analytical schematic diagrams are mainly used to help understand complex principles and processes, such as principle diagrams, circuit diagrams, and flowcharts. Data visualization figures are generally used to present quantitative research results, such as curve plots, scatter plots, pie charts, cloud charts, contour lines, etc. Various professional fields have professional tools for presenting data visualization, such as Tecplot. Scientific Workplace, etc., can also output data visualization figures.

Analytical schematic diagrams are sometimes also used to present research results, such as atomic structure diagrams.

The figure in Burbidge (1957), a paper on the synthesis of elements in stars, is both an analytical schematic diagram and a figure representing final results.

For ${\Phi }_8$, coordinate-based representation, a point may be located by $(x,y,z)$, a curve may be represented by $y=f\left(x\right)$, a surface may be represented by $z=f(x,y)$, and complex morphology may also be represented by discrete coordinate points, grids, or other coordinate data. Coordinate-based representation is especially convenient for converting objects that humans can observe or understand into forms that can be computed, addressed, and further operated.

One representational form may also represent multiple coordinates. For example, a figure ${\Phi }_4$ may simultaneously carry object O, morphology F (e.g., spatial contour), qualitative feature S (e.g., rough surface), quantitative feature Q (e.g., scale), and even X (e.g., structural relations).

It should be pointed out that mathematical relations are not necessarily expressed by symbolization based on mathematical formulas or equations. For example, the symbolic expression of Hubble’s law is $v=H_{0}D$, where v represents the recession velocity of a galaxy, D represents the distance between the galaxy and Earth, and $H_0$ is the Hubble constant. However, in Hubble’s original paper (Hubble 1929), he did not present the conclusion in the form of this formula, but expressed his discovery through verbal description and a linear relation drawn on a velocity–distance diagram.

Natural language representation ${\Phi }_1$ itself is not always isolated. When other representational forms appear, natural language can organize and command other representational forms.

For example, a paper may write: “As can be seen from Figure 3...”; “According to Eq. (5)...”; “Table 2 further shows...”; “Existing studies [18] point out...”; “For a right triangle, the two legs a, b and the hypotenuse c satisfy the Pythagorean theorem $a^2+b^2=c^2$.”

Here: figure is ${\Phi }_4$, mathematical expression is ${\Phi }_5$, literature is ${\Phi }_7$.

It can be seen that natural language can not only independently represent knowledge, but also organize numbers, symbols, formulas, figures, tables, and cited resources into the same academic discourse. Because of this commanding role of natural language, it needs to operate in a stable manner.

Then, according to what relatively stable manner does natural language organize these contents?

Although natural language can undertake the role of organizing multiple representational forms, complex academic discourse is not organized arbitrarily. Long-term cognitive, rhetorical, and writing practice has formed some basic organizational methods that can be repeatedly used, namely discourse modes $\mathbf{M}$.

3.2. Expression Operational Pathways

The expression pathway factor is used to describe the organizational directions that generalized coordinate contents may adopt during the operation of GREM. To vividly express organizational directions, a planar virtual expression coordinate system $\xi$–$\eta$ may be introduced. Here, $\xi$ and $\eta$ denote two mutually independent virtual expression directions. This virtual coordinate system is not required to correspond to actual physical space; it is only used to describe the organizational directions and connective relations of generalized coordinate contents in the expressive process.

This planar virtual expression coordinate system $\xi$–$\eta$ may be compared with a longitude–latitude map: virtual coordinate directions are similar to longitude and latitude directions on a map, expression pathways are similar to routes on a map, and generalized coordinate contents are similar to nodes on the routes.

For convenience of explanation, we use the expression coordinate $\mathbf{e}=\{{\mathbf{e}}_1,{\mathbf{e}}_2,\dots \}$ introduced in Section 1 to represent the coordinate nodes of expression operational pathways. These nodes may be any coordinate among O, $F_i$, $F_{ij}$, S, Q, or X.

We denote possible expression operational pathways as

$$\rho \in \{p,h,v,s,c,d,r\},$$

where p denotes point operation; h denotes horizontal operation; v denotes vertical operation; s denotes diagonal operation; c denotes curved operation; d denotes discrete operation; r denotes random operation.

1.

Point operation $\rho =p$.

Point operation indicates that expression mainly unfolds around one determined generalized coordinate content node:

$$\rho =p:{\mathbf{e}}_1.$$

In the planar virtual expression coordinate system, the position of point operation may be represented as

$$\xi =\mathrm{const},\eta =\mathrm{const}.$$

2.

Horizontal operation $\rho =h$.

Horizontal operation indicates that expression unfolds along the virtual horizontal direction:

$$\rho =h:{\mathbf{e}}_1\to {\mathbf{e}}_2\to \dots \to {\mathbf{e}}_m.$$

At this time, different generalized coordinate contents usually have a relatively parallel relation.

In the planar virtual expression coordinate system, the horizontal operation expression pathway may be represented as

$$\eta =\mathrm{const}.$$

3.

Vertical operation $\rho =v$.

Vertical operation indicates that expression unfolds along the virtual vertical direction:

$$\rho =v:{\mathbf{e}}_1\perp {\mathbf{e}}_2\perp \dots \perp {\mathbf{e}}_m.$$

At this time, different generalized coordinate contents usually have hierarchical, progressive, or superordinate-subordinate relations.

In the planar virtual expression coordinate system, the vertical operation expression pathway may be represented as

$$\xi =\mathrm{const}.$$

4.

Diagonal operation $\rho =s$.

Diagonal operation indicates that expression simultaneously crosses two virtual directions:

$$\rho =s:{\mathbf{e}}_1\setminus {\mathbf{e}}_2\setminus \dots \setminus {\mathbf{e}}_m.$$

This operation usually involves connections among different objects, different analytical features, or different levels.

In the planar virtual expression coordinate system, the diagonal operation expression pathway may be represented as

$$\eta =a\xi +b,a\ne 0.$$

5.

Curved operation $\rho =c$.

Curved operation indicates that the expression direction is not a fixed straight line, but changes during operation:

$$\rho =c:{\mathbf{e}}_1⇝{\mathbf{e}}_2⇝\dots ⇝{\mathbf{e}}_m.$$

It may represent complex operational processes such as continuous change of direction, turning, cycling, feedback, or stage change.

In the planar virtual expression coordinate system, the expression pathway running along a curve may be represented as

$$\eta =f\left(\xi \right),$$

where $f\left(\xi \right)$ is a nonlinear function of $\xi$.

6.

Discrete operation $\rho =d$.

Discrete operation indicates that expression selects and organizes information from multiple mutually separated generalized coordinate content nodes:

$$\rho =d:\{{\mathbf{e}}_1,{\mathbf{e}}_2,\dots ,{\mathbf{e}}_m\}.$$

These content nodes may further form connections, networks, or combinatory relations.

In the planar virtual expression coordinate system, discrete operation may be represented as a set of discrete coordinate points:

$$\rho =d:\{({\xi }_1,{\eta }_1),({\xi }_2,{\eta }_2),\dots ,({\xi }_m,{\eta }_m)\}.$$

Here, the points are not required to form a continuous path, nor to satisfy a unified straight-line or curved relation.

7.

Random operation $\rho =r$.

Random operation indicates that the selection, invocation order, or connective relation of coordinate contents in the expressive process has a certain uncertainty:

$$\rho =r:\mathrm{Rand}\left(\mathbf{e}\right).$$

Here it does not mean that a strict probability model has been established, but only that the current operational pathway cannot be completely predetermined.

In the planar virtual expression coordinate system, random operation may be represented as a set of coordinate points selected or connected randomly:

$$\rho =r:\mathrm{Rand}\left(\left\{({\xi }_i,{\eta }_i)\right\}\right).$$

3.3. Auxiliary Coordinate Factor ${\mathrm{\kappa }}_C$

In concrete analysis and expression, the coordinate content that currently needs to be clarified or handled often cannot unfold completely by itself, and may need to invoke other generalized coordinate contents as auxiliary. This paper uses the auxiliary coordinate factor ${\mathrm{\kappa }}_C$ to represent such operation:

$${\mathrm{\kappa }}_C\equiv \mathrm{Aux}\left(C\right).$$

For the current coordinate $C_i$, its auxiliary coordinate $\mathrm{Aux}\left(C_i\right)$ denotes one or more other generalized coordinate contents that may be invoked when $C_i$ is the current main content. Auxiliary coordinates are used to supplement, delimit, explain, compare, or relate to the current coordinate, but do not change the main coordinate role of $C_i$ in the current task.

Auxiliary coordinates may have two sources.

The first comes from generalized coordinate contents already formed within the current research. For example, when clarifying analytical object O, related analytical feature F may be invoked; when analyzing a certain function F, the quantitative feature Q already obtained in the current research may be invoked to explain the degree to which the function is exerted; when describing a certain Q, the corresponding S may also be invoked to help explain the actual state corresponding to the numerical value.

The second comes from generalized coordinate contents already formed in other studies. For example, when the current research analyzes a certain object O, F, S, Q, or X concerning similar objects in existing literature may be invoked as auxiliary; when the current research obtains a certain Q, existing experimental results, database data, or Q in literature may also be invoked as reference. At this time, although these auxiliary coordinates originate outside the current $\mathrm{GAP}\left(C\right)$, after entering the current task they may still participate in operation as $\mathrm{Aux}\left(C\right)$.

When a specific auxiliary direction needs to be represented, it may be written as

$$\mathrm{Aux}(C_i\to C_j),$$

indicating that coordinate $C_i$ is used to help clarify or handle the current coordinate $C_j$. For example:

$$\mathrm{Aux}(Q\to F)$$

indicates that quantitative feature Q is used to assist the unfolding of the current function, state, or other analytical feature.

One current coordinate may also simultaneously invoke multiple auxiliary coordinates, and these auxiliary coordinates may respectively come from the current research and existing research. For example, when evaluating the current experimental result Q, the state S in this study and the reference result Q in existing literature may be invoked simultaneously.

Therefore, the auxiliary coordinate factor ${\mathrm{\kappa }}_C$ is not only used to describe the cooperative invocation among generalized coordinates within the current research, but also provides an interface for introducing external existing generalized coordinate content into the current research and expression process.

3.4. Auxiliary Discourse Mode Factor ${\mathrm{\kappa }}_M$

A main discourse mode in actual operation also does not necessarily complete all expressive tasks independently; it may locally invoke other discourse modes as auxiliary. This paper uses the auxiliary discourse mode factor ${\mathrm{\kappa }}_M$ to represent such operation:

$${\mathrm{\kappa }}_M\equiv \mathrm{Aux}\left(M\right).$$

Here the current main discourse mode undertakes the main expressive task, and $\mathrm{Aux}\left(M\right)$ denotes one or more auxiliary discourse modes invoked to complete this task. Auxiliary modes may appear at local positions of the main mode operation, or may be invoked multiple times at different stages, but they do not thereby change the basic task of the current main discourse mode.

For example, in the process of argumentation, it may be necessary first to define key concepts, then invoke exemplification to provide evidence, use causal mode to explain action relations, and form a judgment through evaluation; when explaining a complex concept, definition, description, exemplification, or analogy may also be locally invoked. Here, definition, exemplification, causation, or analogy do not undertake the entire expressive task in parallel with the main mode, but help the main mode complete a certain local academic function.

Therefore, auxiliary discourse modes are not completely the same as simple combinations of discourse modes. A combination of discourse modes may form a mixed mode in which multiple modes jointly undertake the main task, whereas $\mathrm{Aux}\left(M\right)$ emphasizes the primary–auxiliary relation between the main mode and auxiliary modes. In one concrete operation, the main mode may remain unchanged, while auxiliary modes are added, replaced, or invoked multiple times according to local needs.

Wu (2025) proposed “main–auxiliary rhetorical modes” to explain how a main rhetorical mode invokes auxiliary modes to clarify and realize the academic function of the main rhetorical mode.

3.5. Expression Principle Factor ${\mathrm{\kappa }}_P$

The symbol P denotes principles affecting the organization of research expression. Wu (2026a) introduced the four-step enhancement method for academic expression, including primary enhancement, intermediate enhancement, advanced enhancement, and top-level enhancement. We may take the principles involved in the four-step enhancement method as expression principle factors.

1. Primary enhancement.

Primary enhancement is mainly used to avoid obvious non-academic expression. Expression should remain as neutral, verifiable, relevant, evidence-based, cautious, and objective as possible, reducing subjective, emotional, absolute, or unsupported statements, and especially avoiding overgeneralization.

Aull, Bandarage, and Miller (2017) compared generality in the writing of first-year college students, upper-level students, and published academic texts through corpus analysis. The study found that generality markers appeared more frequently in first-year college students’ writing, and generalizations often involved larger groups and longer time ranges; published academic writing had the fewest generality markers, and its expression was generally more cautious. The study shows that mature academic expression does not simply avoid generalization, but pays more attention to limiting the scope of application of claims, so that the degree of generalization matches existing evidence.

2. Intermediate enhancement.

Intermediate enhancement mainly focuses on conciseness, readability, and organizational quality of expression. Ruben (2012) emphasized the use of minimalist language to reduce the burden of understanding.

Borja (2015) suggested using the breath rule, namely that a sentence should not exceed the time of one breath, to limit sentence length. Borja (2015) also mentioned the single-meaning principle, namely that a sentence or paragraph should contain only one topic and one meaning.

When sentences and paragraphs mix text, numbers, symbols, formulas, and cross-references, the principle of smooth reading, speaking, and listening needs to be satisfied. For example, grammar should be correct when read aloud, punctuation should be appropriate (Rowland, n.d.), sentence meaning should not become grammatically erroneous due to mixing, and sentences should not become fragmented when heard (Lee, n.d.).

Important viewpoints, concepts, and terms should be placed at the core position of the corresponding expression unit; singular and plural, reference, scope of qualification, and logical relations should be as explicit as possible to avoid ambiguity.

3. Advanced enhancement.

McIntyre (1997; 2023a, Chapter 2) discussed the requirement of lucidity in scientific expression from the perspective of pattern perception. McIntyre (2023b) further proposed the principles of organic change, explicitness, and coherent ordering for improving lucidity.

The principle of organic change emphasizes that change must proceed step by step and maintain continuity, and must not be abrupt or disorderly. Terms and symbols should not change easily, grammatical patterns and sentence structures should not change easily, and style should not change easily.

The principle of explicitness requires that expression must be direct and transparent, leaving no room for readers to guess. In academic writing, obscure expressions, ambiguous pronouns, unexplained abbreviations or symbols all force readers to reason repeatedly and increase the extra burden of understanding. It requires explaining terms and symbols, making semantics explicit (try to avoid using pronouns), re-explaining complex concepts or results, and clearly marking sources.

The principle of coherent ordering requires that the order of different contents satisfy certain coherence or continuity requirements. Coherent ordering involves both the ordering of different contents and the organic connection, transition, or transition between different contents, thereby establishing effective contextual relations. When content involves a turn, transition signals should be used to make contextual relations easier for readers to identify. University of Technology Sydney (n.d.-a) summarized transition signals as words or phrases that connect ideas and enhance textual cohesion, and summarized their functions as introducing examples, supplementing content, indicating order and time, comparison and contrast, indicating causation, and summarizing or drawing conclusions.

The most common form of transition markers is words and phrases, such as “in addition,” “subsequently,” “by contrast,” “therefore,” and “for example.” When connecting larger content units, they may also be expanded into complete transition sentences and other discourse navigation expressions. Monash University (n.d.) divided such signposting into main navigation markers, transition sentences, connectives, and retrospective reminders, and pointed out that transition sentences can directly explain why and how to move from one content to another.

4. Top-level enhancement.

Top-level enhancement emphasizes that expression units should undertake clear academic functions. Wu (2025) proposed the concept of academic mapping, namely linking concrete expression with corresponding academic functions, so that text not only completes linguistic expression, but also undertakes academic tasks such as presenting information, describing features, analyzing relations, organizing processes, providing evidence, and supporting argumentation.

The expression principle factor ${\mathrm{\kappa }}_P$ runs through the entire academic expression process. Primary, intermediate, advanced, and top-level enhancement respectively propose requirements from basic academicity, expressive organization, scientific lucidity, and academic function. In concrete expression, these should be considered simultaneously as far as possible. Different principles may have different emphases in different tasks, but none of these requirements should therefore be neglected.

4. Summary of GREM

GREM takes discourse mode as expressive operation, generalized analytical coordinate as the basis of expressive content, generalized analytical pathway as the organization of the unfolding of generalized coordinate content, and different operational factors as constraints on the concrete expression process. Its unified form is:

$$\mathrm{GREM}:E^G\equiv \mathrm{Md}[\mathrm{GAP}\left(C\right),\mathrm{\kappa }],$$

where $E^G$ denotes generalized expression output; $\mathrm{Md}$ denotes one discourse mode actually invoked in one expression; C denotes generalized analytical coordinates; $\mathrm{GAP}\left(C\right)$ denotes generalized analytical pathways; $\mathrm{\kappa }$ denotes limiting conditions affecting expression operation.

Discourse mode $\mathrm{Md}$ includes rhetorical modes (such as description, narration, definition, classification, decomposition, comparison and contrast, analogy, process analysis, causation, exemplification, evaluation, argumentation, question and answer), as well as combinations of rhetorical modes.

The generalized analytical pathway $\mathrm{GAP}\left(C\right)$ includes:

$$\mathrm{MAP}\left(C\right)\equiv O\Rightarrow \mathrm{GAM}\Rightarrow \mathrm{Res},$$

$$\mathrm{GAM}\equiv \{\mathrm{LA}\left({\mathrm{F}}^{\mathrm{s}}\right),\mathrm{CA}\left({\mathrm{F}}^{\mathrm{m}}\right)\}:\mathrm{LA}\left(F^s\right)\equiv F^s\to F_{ij},\mathrm{CA}\left(F^m\right)\equiv F_i\leftrightarrow F_j,i\ne j,$$

$$\mathrm{Ra}\left(C\right)\equiv C\mapsto O^{\prime }\Rightarrow \mathrm{GAP}.$$

The operational factors that GREM may invoke are:

$$\mathrm{\kappa }\equiv \{{\mathrm{\kappa }}_{\Phi },{\mathrm{\kappa }}_{\rho },{\mathrm{\kappa }}_C,{\mathrm{\kappa }}_M,{\mathrm{\kappa }}_P\},$$

where: ${\mathrm{\kappa }}_{\Phi }\equiv \Phi ,$academic representation factor;

• ${\mathrm{\kappa }}_{\rho }\equiv \rho ,$expression pathway factor;
• ${\mathrm{\kappa }}_C\equiv \mathrm{Aux}\left(C\right),$auxiliary coordinate factor;
• ${\mathrm{\kappa }}_M\equiv \mathrm{Aux}\left(M\right),$auxiliary discourse mode factor;
• ${\mathrm{\kappa }}_P\equiv P,$expression principle factor.

The four components C, $\mathrm{GAP}\left(C\right)$, $\mathrm{Md}$, and $\mathrm{\kappa }$ in GREM may form a large number of different combinations. In actual research and academic writing, researchers may select corresponding analytical coordinates, analytical pathways, discourse modes, and operational conditions according to concrete tasks, thereby forming research - expressive structures of different levels and purposes.

Usually, the generalized analytical pathway in GREM appears as continuous linking operation among coordinate contents. However, in some cases, pathway pause also needs to be considered, namely temporarily stopping the current operation.

Let the symbol “→” denote continuous operation, and let the symbol “↛” denote paused operation. Suppose the analytical process is

$$C_{i-1}\to C_i\to C_{i+1}.$$

Then

$$C_{i-1}\nrightarrow C_i\nrightarrow C_{i+1}$$

indicates that the analytical process pauses at coordinate $C_i$. At this time, the coordinate $C_i$ in the paused state is denoted as

$$\mathrm{Stop}\left(C_i\right).$$

At the pause position, the generalized analytical pathway changes from continuous operation to

$$\mathrm{GAP}\left(C\right)\to \mathrm{Stop}\left(C_i\right).$$

Forming $\mathrm{Stop}\left(C_i\right)$ may have different purposes, for example: clarifying the coordinate; conducting local in-depth analysis; temporarily suspending the advancement of the current analysis.

Acknowledgement. This paper is prepared in Chinese, and then translated into English using large language models. Ma Xin-Yu offered text checking during this work.