The Generalized Coordinate System for Rhetorical Modes

1. Introduction

Rhetorical modes have a long and rich history of development, dating back to the foundational works of Aristotle (c. 335 BCE/2007), Bain (1866), Hill (1895), and Connors (1997). Modern comprehensive composition textbooks continue to highlight an overlapping cluster of rhetorical modes (e.g., Corbett & Connors, 1999; Hacker & Sommers, 2022; Kirszner & Mandell, 1986; Lunsford, 2021; Lunsford, Ruszkiewicz & Walters, 2021; Nadell, Langan & Coxwell-Teague, 2019; Oshima & Hogue, 2007; Smalley, Ruetten & Kozyrev, 2011). These modes are primarily known through their application in writing practice. For example, rhetorical modes serve as the second layer in the progressive four-level writing method that progresses from the basic expression layer, through the academic mapping layer, to the functional unit layer, and finally, to the thesis integral layer (Wu, 2026).

In contemporary times, despite the ongoing evolution of academic discourse, a core group of rhetorical modes remains central in numerous authoritative writing and rhetoric textbooks (e.g., Corbett & Connors, 1999; Lunsford, 2021). This enduring presence underscores the theoretical vitality and practical utility of rhetorical modes, both as analytical tools and pedagogical methods. While the spectrum of rhetorical modes continues to evolve, recent proposals, such as Wu’s (2025) duality mode operation method, offer new ways to generate and expand these modes.

However, a review of this tradition reveals that much of the research and pedagogical focus has centered on identifying and imitating rhetorical modes, often treating them as isolated textual techniques. Despite the growing diversity of modes, there remains a lack of an integrated theoretical framework that bridges linguistic form, cognitive processes, and real-world applications. Specifically, existing systems have yet to systematically address several interconnected core questions that can be pointed to multi-modes in the real world (Figure 1):

• What specific objects (entities) do these modes operate upon?
• What multimodal semiotic resources (media) do they rely on?
• How do they align with fundamental human cognitive functions (mind)?
• How do they ultimately serve real-world social practices (purpose)?
• More importantly, how does an individual’s competence in utilizing these modes progressively develop and differentiate (agent development)?

To address these questions, this paper proposes a theoretical framework - the General Coordinate System (GCS) of Rhetorical Modes, following ontologies which explicitly represent domains by defining the entities, properties, and relationships that constitute the real world and serve as semantic models, capturing and representing various aspects of reality (Obrst et al., 2007). The GCS consists of ten dimensions as shown in Table 1, each corresponding to a generalized coordinate axis, with specific expressions represented as generalized coordinate points within this system.

These generalized coordinate axes fall into three types: low-dimensional axes, mediating and ontological axes, and high-dimensional axes. Low-dimensional refers to the foundational, more basic axes and includes the Thing Axis (Object Category), the Feature Axis (Analytic Feature), the Quantitative Attribute Axis, the Qualitative Attribute Axis, the Formal Attribute Axis. The mediating axes makes bridge between low-dimensional axes and high-dimensional axes, and include Basic Expressive-Representational Elements Axis (Multimodal Symbolic Resources) and the Ontological or Rhetorical Mode axis (Various Rhetorical Modes). High-dimensional refers to higher-level axes, including the Cognitive Function axis, the Epistemic Purposes axis, and the Five-Level Expression Staircase Model Axis (User Competency Development Ladder).

The generalized coordinate axes are designed to encompass all elements relevant to the application of rhetorical modes. Each of the ten generalized coordinate axes listed in Table 1 is divided into several representative (though not necessarily exhaustive) subcategories, which we dub major ticks (MT). The representativeness of these major ticks can be further specified by minor ticks beneath them. Section 2-4 describe the low-dimensional axes, the mediating axes and the high-dimensional axes.

The GCS allows for the construction of a pyramid-like mapping model that spans from "what acts upon" (things, features, and attributes), "what to express with" (basic elements), "how to express" (mode types), "what is the purpose" (cognition and application), to "who expresses" (competency development). This model converges infinite expressive possibilities within the finite intersections of the ten axes. Section 5 delves into the transition from static "axes" to a dynamic "system," exploring how various mapping methods can generate and operationalize meaning. The nesting functions mentioned in Section 3 will be defined in Appendix A.

Based on the exposition of the coordinate system’s static composition and dynamic mapping in the preceding sections, Section 6 subjects the descriptive power of this framework to quantitative verification. By calculating the "Generalized Modes Number" (GMn) under mappings from low-dimensional to high-dimensional and full-dimensional configurations, this section visually reveals the near-infinite combinatorial possibilities inherent within the system, thereby providing mathematical confirmation of its richness and expansive potential as a powerful engine for meaning generation.

Section 7 provides a short summary of the work and limitation.

2. Low-Dimensional Coordinate Axes

Low-dimensional axes include the Thing Axis, Analytic Feature Axis, Quantitative Attribute Axis, Qualitative Attribute Axis, and Formal Attribute Axis.

2.1. Introduction of Low-Dimensional Axes

2.1.1. Thing Type Axis (Th)

The first generalized coordinate axis is the Thing Type Axis. We denote the set of major ticks on the Thing Axis (Things) as $\mathrm{Th}=\{{\mathrm{th}}^{\left(1\right)},{\mathrm{th}}^{\left(2\right)},\dots ,{\mathrm{th}}^{\left(K_1\right)}\}$, where $K_1$ denotes the total number of major ticks on this axis, and ${\mathrm{th}}^{\left(k\right)}$ (for $k=1,2,\dots ,K_1$) represents the name of each major tick, i.e., the thing category. Here, we introduce five basic thing categories, setting $K_1=5$. As illustrated in Figure 2 for the thing axis, the major ticks are:

1. Physical Entity (k=1)

2. Abstract Thing (k=2)

3. Event and Process (k=3)

4. Relation and System (k=4)

5. Mind and Experience (k=5)

Each major tick can be further divided into minor ticks, with varying numbers, potentially inexhaustible. See Table 2 for examples.

2.1.2. Analytic Feature Axis (Ft)

The second generalized coordinate axis is the Analytic Feature Axis. We denote the set of major ticks on the Feature Axis (Features) as $\mathrm{Ft}=\{{\mathrm{ft}}^{\left(1\right)},{\mathrm{ft}}^{\left(2\right)},\dots ,{\mathrm{ft}}^{\left(K_2\right)}\}$, where $K_2$ denotes the total number of major ticks on this axis, and ${\mathrm{ft}}^{\left(k\right)}$ (for $k=1,2,\dots ,K_2$) is the name of each major tick, i.e., the feature category. This paper classifies seven basic feature types, setting $K_2=7$. As illustrated in Figure 3 for the feature axis, there are seven major ticks:

1. Morphology and Composition (k=1)

2. State (k=2)

3. Dynamic (k=3)

4. Function (k=4)

5. Relation (k=5)

6. Cognition and Representation (k=6)

7. Origin and History (k=7)

Each major tick can be further divided into minor ticks. For example, minor ticks under "Morphology and Composition" include: - External form - Internal composition - Organizational structure, etc. More details are given in Table 3. For instance, Cognition & Representative focuses on how a phenomenon is received, understood, and expressed by a cognitive agent (such as an observer, user, or theorist). It covers the entire chain from direct sensory experience to the formation of abstract concepts, aiming to analyze how we know a phenomenon and how we analyze it.

2.1.3. Quantitative Attribute Axis (Qt)

The third generalized coordinate axis is Quantitative Attribute. We denote the set of major ticks on the Quantitative Attribute Axis (Quantitative Attributes) as $\mathrm{Qt}=\{{\mathrm{qt}}^{\left(1\right)},{\mathrm{qt}}^{\left(2\right)},\dots ,{\mathrm{qt}}^{\left(K_3\right)}\}$, where $K_3$ denotes the total number of major ticks on this axis, and ${\mathrm{qt}}^{\left(k\right)}$ (for $k=1,2,\dots ,K_3$) is the name of each major tick, i.e., the quantitative attribute category. This paper classifies three basic quantitative attribute types, setting $K_3=3$. As illustrated in Figure 4 for this axis, the major ticks are:

1. Basic Measurement (k=1)

2. Quantity and Frequency (k=2)

3. Ratio and Intensity (k=3)

Each major tick can be further divided into minor ticks. For example, minor ticks under "Basic Measurement" include: - Size - Length - Area - Volume - Mass - Temperature - Time point - Coordinates, etc. See Table 4 for more details.

2.1.4. Qualitative Attribute Axis (Ql)

The fourth generalized coordinate axis is Qualitative Attribute. We denote the set of major ticks on the Qualitative Attribute Axis (Qualitative Attributes) as $\mathrm{Ql}=\{{\mathrm{ql}}^{\left(1\right)},{\mathrm{ql}}^{\left(2\right)},\dots ,{\mathrm{ql}}^{\left(K_4\right)}\}$, where $K_4$ denotes the total number of major ticks on this axis, and ${\mathrm{ql}}^{\left(k\right)}$ (for $k=1,2,\dots ,K_4$) is the name of each major tick, i.e., the qualitative attribute category. This paper classifies five basic qualitative attribute types, setting $K_4=5$. As illustrated in Figure 5 for this axis, the major ticks are:

1. Shape and Configuration (k=1)

2. Color and Pattern (k=2)

3. Texture and Perception (k=3)

4. Position and Orientation (k=4)

5. Material and Composition (k=5)

Each major tick can be further divided into minor ticks. Each qualitative attribute may have both concrete and abstract ones, see Table 5 for more details.

2.1.5. Formal Attribute Axis (Fm)

The fifth generalized coordinate axis is Formal Attribute. We denote the set of major ticks on the Formal Attribute Axis (Formal Attributes) as $\mathrm{Fm}=\{{\mathrm{fm}}^{\left(1\right)},{\mathrm{fm}}^{\left(2\right)},\dots ,{\mathrm{fm}}^{\left(K_5\right)}\}$, where $K_5$ denotes the total number of major ticks on this axis, and ${\mathrm{fm}}^{\left(k\right)}$ (for $k=1,2,\dots ,K_5$) is the name of each major tick, i.e., the formal attribute category. This paper classifies four basic formal attribute types, setting $K_5=4$. As illustrated in Figure 6 for this axis, the major ticks are:

1. Logical Relation (k=1)

2. Functional Structure (k=2)

3. Rule and Constraint (k=3)

4. Symbol and Formula (k=4)

Each major tick can be further divided into minor ticks. For example, minor ticks under "Logical Relation" include: - Inclusion - Equivalence - Causality - Implication, etc. See Table 6 for details.

2.2. Perspectival Low-dimensional Axis Multiplicity

Low-dimensional axes form the foundational bases. Figure 7 shows the relations between various low-dimensional axes, showing that analytical features may be used for each thing, quantitative attribute can be used for things, features, qualitative attribute can be used for things, features, quantative attribute, and formal attribute can be used for things, features, quantative attribute and qualitative attribute.

Perspectival low-dimensional axis multiplicity means that the same referent (thing, feature, or attribute) can be assigned to different major ticks or elements of another low-dimensional axis.

2.2.1. Thing Axis: Perspectival Categorical Multiplicity

For the thing axis, the same referent can be assigned to different major ticks on the Thing Axis, depending on the perspective or the aspect under consideration. We term this property "perspectival categorical multiplicity" or simply "multi-aspectual belonging". For example, consider the entity "student". Depending on which thing category we adopt, the way the student belongs to that category shifts:

• When a student is treated as a Physical Entity, the belonging concerns objective existence: e.g., whether the student exists, is enrolled, or belongs to a certain grade or class.
• When treated as an Abstract Thing, the belonging concerns the attribution of identity or role: e.g., whether the person is a student, a registered student, a domestic or international student, a current or former student.
• When treated as an Event and Process, the belonging concerns the "studenthood" period: its start, end, duration, or stages (e.g., whether the person has received higher education).
• When treated as a Relation and System, the belonging concerns the relational network: e.g., whether the person has a teacher, classmate, or alumni relationship with others.
• When treated as Mind and Experience, the belonging concerns internal perception and experiential facts: e.g., whether the person has a student-like temperament or has experienced a particular learning event.

This example demonstrates that the Thing Axis is not a rigid taxonomy but a flexible framework where the same referent can be re-categorized along different perspectives, each yielding a distinct mode of belonging.

2.2.2. Feature Axis: Perspectival Feature Multiplicity

The same thing can be analyzed along different feature categories depending on which aspect of its nature we focus on. We term this property "perspectival feature multiplicity" or "multi-aspectual feature attribution". For example, consider a "river". Depending on the feature category adopted, the way we characterize the river shifts:

• Under Morphology and Composition: shape, length, width, depth, sediment, water quality.
• Under State: current flow velocity, temperature, turbidity, flooding or drying condition.
• Under Dynamic: erosion, meandering, seasonal variation, long-term course shifts.
• Under Function: water supply, ecosystem support, transportation, hydroelectric power.
• Under Relation: tributaries, watershed boundaries, role within a drainage system.
• Under Cognition and Representation: map naming, cultural significance, scientific measurements.
• Under Origin and History: geological age, glacial source, historical course changes.

This example shows that the Feature Axis, like the Thing Axis, is a flexible framework where the same target can be characterized under multiple feature categories, each offering a distinct analytical lens.

In general, features are considered as features of things. But features can also be considered as things. For instance, each feature can be considered as an abstract thing, and the Relation feature can further be considered as a Relation and System.

2.2.3. Quantitative Attribute Axis: Perspectival Quantitative Multiplicity

Quantitative attributes can be applied to things and features. The same thing or feature can be described using different types of quantitative measures, depending on what aspect of "how much" or "how often" we emphasize. We term this “perspectival quantitative multiplicity”. For example, consider the "growth of a plant", a dynamic feature:

• As Basic Measurement: height 15 cm, leaf area 25 cm², biomass 30 g.
• As Count and Frequency: number of leaves = 8, watering frequency = 3 times per week.
• As Ratio and Intensity: growth rate = 1.2 cm/day, photosynthetic efficiency = 42%, nutrient concentration = 150 ppm.

Thus, the same plant growth process can be quantified from purely dimensional, frequency-based, or derived intensity perspectives, each revealing a different layer of information.

2.2.4. Qualitative Attribute Axis: Perspectival Qualitative Multiplicity

Thing, Features and Quantitative attributes can be characterized using different qualitative categories, depending on the sensory or metaphorical lens we adopt. We term this "perspectival qualitative multiplicity". For example, consider a "piece of music", an abstract thing:

• Under Shape and Configuration: the melody has an arch-shaped contour, repetitive circular phrases.
• Under Color and Pattern: the timbre is bright and warm, with shimmering texture.
• Under Texture and Perception: the rhythm feels smooth and flowing, the harmony is rich and dense.
• Under Position and Orientation: the main theme appears at the beginning and recurs at the end.
• Under Material and Composition: the composition contains elements of jazz and classical idioms.

Hence, the same musical piece can be qualitatively described from geometric, chromatic, tactile, spatial, and material perspectives, each providing a distinct experiential framing.

2.2.5. Formal Attribute Axis: Perspectival Formal Multiplicity

For formal attributes, the same phenomenon (thing, feature, quantitaive attribute and qualitative attribute) can be expressed through different formal systems, depending on the level of abstraction or the modeling language we choose. We term this "perspectival formal multiplicity". For example, consider the "causal relationship between study time and exam performance", a relation feature:

• As Logical Relation: if study time increases, then exam performance improves (implication).
• As Action Structure: a flow of information from input (study hours) through a processing unit (learning) to output (score).
• As Rule and Constraint: the relationship must satisfy that total study time does not exceed 24 hours per day.
• As Symbol and Formula: $P=a·ln\left(T\right)+b$, where P is performance and T is study time.

Thus, the same underlying dependency can be formalized logically, structurally, normatively, or mathematically, each offering a different formal grip on the phenomenon.

In summary, the low-dimensional axes-thing, Feature, Quantitative Attribute, Qualitative Attribute, Formal Attribute-each exhibit perspectival multiplicity. This means that any given entity, property, or relation can be relocated across different major ticks within the same axis, or even across axes, depending on the analytical purpose. Such multiplicity is not a defect but a design feature of the coordinate system, enabling flexible, multi-perspective description and reasoning.

3. Mediating/Ontological Axes

The mediating axes serve to transform raw materials represented by lower dimensional axes into a communicable package.

3.1. Basic Expressive-Representational Elements Axis or Basic Element Axis (Be)

Any presentation needs to rely on specific symbolic materials, just as an architect needs bricks, stones, wood, and glass. These materials together form a Basic Expressive-Representational Elements Axis. Language is a natural element. However, language is just one type of basic element. We denote the set of major ticks on the Basic Expressive-Representational Elements Axis as $\mathrm{Be}=\{{\mathrm{be}}^{\left(1\right)},{\mathrm{be}}^{\left(2\right)},\dots ,{\mathrm{be}}^{\left(K_6\right)}\}$, where $K_6$ denotes the total number of major ticks on this axis, and ${\mathrm{be}}^{\left(k\right)}$ (for $k=1,2,\dots ,K_6$) is the name of each major tick, i.e., the basic element category. In practical applications, the types of basic elements are diverse. This paper summarizes them into seven major types of expressive resources, setting $K_6=7$. The major ticks are:

1. Language (k=1)

2. Numbers and Symbols (k=2)

3. Mathematical Formulas and Equations (k=3)

4. Images and Diagrams (k=4)

5. Data and Relation Visualization (k=5)

6. Sound, Gesture, and Dynamic Demonstration (k=6)

7. Notation/Markup (k=7)

Each major tick can be further divided into numerous and diverse minor ticks. For example, minor ticks under "Language" include: - Terms and directional words - Topic sentences of paragraphs - Transitional phrases - Cross-references - Outline and heading hierarchy, etc.

Table 7 gives more details on the seven major ticks of the Basic Expressive-Representational Elements Axis.

Gesture, though difficult to appear in text, may have importance in oral communication (Rizzo, Berger, & Zhou, 2025). Note that rhetorical modes are not only used in written text but also in oral communication. The use of gestures can play a significant role in enhancing meaning and engagement in face-to-face communication, helping to express nuances that are difficult to convey through words alone.

3.2. Ontological Axis: Rhetorical Modes

The ontological axis, placed at the seventh, refers to rhetorical modes (Rm), and is a mediating layer between low- and high-dimensional coordinate axes.

We denote the set of major ticks on the Rhetorical Mode Axis as $\mathbf{Rm}=\{r{m}^{\left(1\right)},r{m}^{\left(2\right)},\dots ,r{m}^{\left(K_7\right)}\}$, where $K_7$ denotes the total number of major ticks on this axis, and $r{m}^{\left(k\right)}$ ($k=1,2,\dots ,K_7$) is the name of the major tick, i.e., the specific rhetorical mode.

This paper considers 18 rhetorical modes ($K_7=18$), the major ticks are The major ticks are:

1. Description (k=1)

2. Comparison (k=2)

3. Contrast (k=3)

4. Analogy (k=4)

5. Cause and Effect (k=5)

6. Exemplification (k=6)

7. Evidence (k=7)

8. Classification (k=8)

9. Division (k=9)

10. Process Analysis (k=10)

11. Narration (k=11)

12. Definition (k=12)

13. Evaluation (k=13)

14. Argumentation (k=14)

15. Persuasion (k=15)

16. Exposition (k=16)

17. Question (k=17)

18. Answer (k=18)

The simple meaning are presented in Table 8. In Table 8, rhetorical modes are presented and divided into four major categories: foundational rhetorical modes, relational rhetorical modes, organizational rhetorical modes, and comprehensive rhetorical modes.

3.3. Beyond Language: The six non-linguistic basic elements as Rhetorical Enhancers

The basic expressive-representational elements include the language and the six non-linguistic basic elements (Numbers and Symbols; Mathematical Formulas and Equations; Images and Diagrams; Data and Relation Visualization; Sound, Gesture, and Dynamic Demonstration; Notation/Markup). Traditionally, we build rhetorical modes only with language. But if we also use numbers, symbols, formulas, images, diagrams, charts, sound, gestures, animations, and markup as additional expressive-representational elements, we can make our communication clearer, more complete, more useful, and more logically consistent. This matters most in high-level tasks, which we will describe later with high-dimensional axes.

3.3.1. Co-Cognitive Prosthetics: They Change How We Think

The six non-linguistic basic elements are not just extra channels for sending out ready-made thoughts. They work like external thinking aids – prosthetics for your mind. They reshape how meaning comes into being. Language alone pushes us toward linear, step-by-step reasoning. But when we combine symbols, images, motion, and sound at the same time, we can process meaning in parallel, much like a neural network where each sense sharpens and limits the others.

3.3.2. Collapsing Medium and Message: A Formula Does the Work Itself

With six non-linguistic basic elements, the old line between the medium (how you say it) and the message (what you say) disappears. A math equation is not just a translation of words. It is a rhetorical move that packs cause-and-effect logic into a form you can flip, combine, and reuse. From that, you get new insights that no single sentence could give you. Likewise, a gesture or a live data chart does not just illustrate a point. It acts out the point through time, space, and body feeling. The audience becomes part of the thinking process, not just listeners.

3.3.3. Trans-Biological Rhetoric: Talking to Machines and Math Spaces

Beyond human-to-human talk, these six non-linguistic basic elements let us do trans-biological rhetoric. That means we can communicate with AI, lab instruments, or even abstract math spaces. A well-built formula, run inside a computer, becomes a rhetorical act that "persuades" a simulator to give you a certain result. A piece of markup, placed inside a neural network’s attention layer, can steer hidden meaning paths without ever turning into natural language.

3.3.4. Rhetorical Superposition: Holding Multiple Meanings at Once through Quantum Rhetoric

Here is an idea that goes beyond current cognitive science: rhetorical superposition. This is when all seven elements are active at the same time, but none is fully fixed. A single act of communication carries many – even contradictory – logical paths. Only when the context collapses do we force one interpretation. This is like quantum decoherence, but applied to meaning. We call it "quantum rhetoric". The same set of symbols, gestures, and visuals can be true in one way of knowing and purely beautiful in another. The listener’s own mental state decides which side becomes real.

3.3.5. Cross-Linguistic Rhetoric: Posssibly No Translation Needed

These six non-linguistic basic elements also help us jump over language barriers. A picture, a formula, or a well-made diagram can carry the same core meaning to a Chinese speaker, an English speaker, and an Arabic speaker – without going through translation. This is not just about universal symbols. It is about anchoring meaning in spatial, logical, or sensory basics that all humans share. In high-stakes situations, like international science, airplane safety rules, or global health warnings, using only words risks confusion. But diagrams, number codes, color maps, or standard gestures can give immediate understanding across languages. So the seven elements act as a universal base for rhetoric that brings people together, rather than splitting them apart.

3.3.6. Meta-Rhetorical Systems: What Clarity, Completeness, Utility, and Consistency Really Mean

So six non-linguistic basic elements are not just a better toolbox for explaining things. They are the building blocks of meta-rhetorical systems. These systems change what we mean by "clear", "complete", "useful", and "consistent". In such a system: - Clarity is not the absence of ambiguity. It is the ability to move through controlled ambiguity. - Completeness is not listing everything. It is being closed under rules that let you transform one expression into another. - Utility is not just getting a job done. It is resonating across many cognitive dimensions at once. - Consistency is not avoiding contradiction. It is staying coherent when you shift your point of view. These properties are hard to imagine if you only use language. But they become normal working conditions for high-dimensional rhetorical acts.

3.3.7. Conclusion: Stretching the Horizons of Rhetorical Modes

In short, using six non-linguistic basic elements challenges old views by introducing new concepts: parallel semantic processing, trans-biological rhetoric, rhetorical superposition, cross-linguistic rhetoric, and meta-rhetorical systems. Our goal is to stretch what you think rhetorical modes can become.

4. High-Dimensional Coordinate Axes

High-dimensional axes include Cognitive Function Axis, the Epistemic Purpose Axis, and the Five-Level Expression Staircase Model Axis. Within the Generalized Coordinate System of Rhetorical Modes, elements on high-dimensional axes primarily serve as dependent variables, although some can also act as independent variables.

4.1. Cognitive Function Axis (Cf)

Cognitive functions are the internal mental operations that rhetorical modes directly serve. Each rhetorical act, whether describing, comparing, arguing, or questioning, ultimately supports one or more underlying cognitive functions. We denote the set of major ticks on the Cognitive Function Axis (Cognitive functions) as $\mathrm{Cf}=\{{\mathrm{cf}}^{\left(1\right)},{\mathrm{cf}}^{\left(2\right)},\dots ,{\mathrm{cf}}^{\left(K_8\right)}\}$, where $K_8$ denotes the total number of major ticks on this axis, and ${\mathrm{cf}}^{\left(k\right)}$ (for $k=1,2,\dots ,K_8$) represents the name of each major tick, i.e., the specific cognitive function. Wu (2025) introduced 14 cognitive functions ($K_8=14$). In the present framework, we replace three of them, Compare, Classify, and Evaluate, with Alignment, Partition, and Judgement, respectively, while keeping the remaining eleven unchanged. The resulting major ticks are as follows:

1. Observation (k=1)

2. Identification (k=2)

3. Alignment (k=3)

4. Partition (k=4)

5. Abstraction (k=5)

6. Hypothesis (k=6)

7. Modeling (k=7)

8. Inference (k=8)

9. Testing/Verification (k=9)

10. Explanation (k=10)

11. Judgment (k=11)

12. Prediction (k=12)

13. Integration/Synthesis (k=13)

14. Reflection/Metacognitive Evaluation (k=14)

The meanings of each cognitive functions are briefly described in Table 9, where possible correspondence to rhetorical modes are outlined.

4.2. Epistemic Purpose Axis (Ep)

Epistemic purposes represent the ultimate real-world applications that cognitive functions serve. In other words, while cognitive functions describe internal mental operations, epistemic purposes locate those operations in concrete social and professional practices. Each rhetorical act, when examined at this level, aims to fulfill one or more epistemic goals, such as forming knowledge, making a discovery, solving a problem, or implementing a policy.

We denote the set of major ticks on the Epistemic Purpose Axis (Epistemic functions) as $\mathrm{Ep}=\{{\mathrm{ep}}^{\left(1\right)},{\mathrm{ep}}^{\left(2\right)},\dots ,{\mathrm{ep}}^{\left(K_9\right)}\}$, where $K_9$ denotes the total number of major ticks on this axis, and ${\mathrm{ep}}^{\left(k\right)}$ (for $k=1,2,\dots ,K_9$) represents the name of each major tick, i.e., the specific epistemic purpose category.

We consider the 8 categories introduced by Wu (2025), thus setting $K_9=8$. The major ticks are:

1. Knowledge Formation (k=1)

2. Scientific Discovery (k=2)

3. Writing and Communication (k=3)

4. Teaching/Learning (k=4)

5. Problem-Solving (k=5)

6. Innovation/Design (k=6)

7. Evaluation/Decision-Making (k=7)

8. Policy/Action Implementation (k=8)

See Table 10 for more details.

4.3. Five-Level Expression Staircase (Depth) Axis (Es) and Level of Competence

The Expression Staircase Axis, the depth axis, captures the developmental progression of rhetorical competence. It answers the question: At what level of mastery does a person use rhetorical modes? From basic sensory description to advanced knowledge design, each level reflects a higher order of cognitive integration and communicative purpose. This axis thus directly corresponds to the notion of level of competence, the increasing ability to select, combine, and transform rhetorical modes for increasingly complex tasks.

We denote the set of major ticks on the Expression Staircase Axis (Expression staircase) as $\mathrm{Es}=\{{\mathrm{es}}^{\left(1\right)},{\mathrm{es}}^{\left(2\right)},\dots ,{\mathrm{es}}^{\left(K_{10}\right)}\}$, where $K_{10}$ denotes the total number of major ticks on this axis, and ${\mathrm{es}}^{\left(k\right)}$ (for $k=1,2,\dots ,K_{10}$) represents the name of each major tick, i.e., the specific expression level.

This paper adopts a five-layer developmental model from basic to high-order, thus setting $K_{10}=5$. The major ticks are:

1. Sensory Level (k=1)

2. Autonomous Expression Level (k=2)

3. Academic Standard Level (k=3)

4. Methodological Level (k=4)

5. Knowledge Enlightenment Level (k=5)

See Table 11 for explanation of each level.

4.4. How the Three High-Dimensional Axes Fit Together

We can think of the three high-dimensional axes as a simple chain:

1. Cognitive functions are what your mind does internally, observing, comparing, inferring, judging, etc.

2. Epistemic purposes are what you want to achieve in the real world with those mental operations, forming new knowledge, solving a problem, making a decision, teaching someone.

3. The five-level staircase describes how well you can perform those operations to achieve those purposes. It is about your level of competence.

Here is the logical flow: You cannot serve an epistemic purpose (say, scientific discovery) without using certain cognitive functions (hypothesizing, testing, explaining). And you cannot use those cognitive functions effectively unless you have reached a certain level on the staircase.

For example, a student at the Sensory Level can observe and describe a falling apple, but cannot yet form and test a hypothesis about gravity. At the Methodological Level, the same person can design an experiment, analyze data, and explain the results, because the necessary cognitive functions have matured through practice and training.

Thus, the three axes are not separate lists. They form a nested progression:

$$\mathrm{Cognitive}\mathrm{functions}\to \mathrm{Epistemic}\mathrm{purposes}\to \mathrm{Level}\mathrm{of}\mathrm{competence}$$

Lower levels of the staircase rely on basic cognitive functions (observation, identification) and serve simple epistemic goals (knowledge formation for oneself). Higher levels integrate complex functions (synthesis, reflection, prediction) to serve advanced purposes (innovation, policy implementation, knowledge enlightenment for others).

In short: what you can do (cognitive) determines what you can aim for (epistemic), and how well you can do it is your level on the staircase (competence).

5. Navigation and Mapping Across Generalized Coordinate Axes

The various axes of the Generalized Coordinate System of Rhetorical Modes are not isolated but interwoven, forming a multidimensional, layered network for mode production. Elements on low-dimensional axes (e.g., Things, Features, Attributes) are systematically combined and elevated to ultimately construct meaning units on the ontological axis (Rhetorical Modes) using the seven expressive-representational elements, to achieve purpose described by high-dimensional axes (e.g., Cognitive Functions, Epistemic Purposes). There are three levels of generative frameworks for meaning production in the Generalized Coordinate System of Rhetorical Modes: low-dimensional mapping, high-dimensional mapping, and full-dimensional mapping. These frameworks correspond to the Pyramid Multilayer Mapping (Wu, 2025). The Sensory Level and Autonomous Expression Level primarily involve low-dimensional mapping, the Academic Standard Level and Methodological Level can employ high-dimensional mapping, and the Knowledge Enlightenment Level may engage with full-dimensional mapping. When introducing these mappings, symbolic expressions will be used for illustrative purposes. It should be noted that these symbolic expressions are intended only as illustrations and are not strict mathematical formulations. If research in relevant fields requires translating these mappings into computational and programmable algorithms, necessary transformations will be needed.

Figure 8 is a schematic display of low- and high- dimensional mappings.

5.1. How Axes Work Together: The Logic of Mapping

Before we dive into the three kinds of mappings (low-dimensional, high-dimensional, and full-dimensional), it helps to see how the different axes relate to each other in practice.

Low-dimensional axes (Things, Features, Attributes) provide the raw material that you talk about, which aspect you focus on, what measurements or qualities you use. The basic expressive-repretational axis provides what physical signs (words, numbers, images, gestures) carry the meaning. The Rhetorical Mode axis is where you assemble those materials into a recognizable pattern of expression (description, comparison, argument, etc.).

High-dimensional axes (Cognitive Functions, Epistemic Purposes, Expression Staircase) describe why and how well you use those patterns, what mental operation you are performing, what real-world goal you are aiming for, and at what level of competence.

So a typical flow looks like this:

$$\mathrm{Low}-\mathrm{dimensional}\mathrm{axes}⟶\mathrm{Rhetorical}\mathrm{Mode}⟶\mathrm{High}-\mathrm{dimensional}\mathrm{axes}$$

But the relationship is not one-way. A skilled writer or speaker can also start with a high-dimensional intention (e.g., "I want to solve a problem?" and then choose low-dimensional materials and a suitable rhetorical mode. That is why the full-dimensional mapping (explained later) treats all axes as potentially influencing each other.

5.2. Low-Dimensional Mapping

Low-dimensional mapping refers to the process of treating rhetorical modes as dependent variables and elements on low-dimensional axes (e.g., Things, Features, Attributes) as independent variables. It constructs different rhetorical modes by nesting various elements on the Basic Expressive-Representational Elements Axis. Low-dimensional mapping corresponds to the base mapping in the Pyramid Multilayer Mapping. See Appendix A for three classes of nesting functions.

Specifically, low-dimensional mapping starts with a Thing (Th), selects its Features (Ft) and Attributes (Qt/Ql/Fm), and mobilizes Basic Elements (Be) for organization, thereby instantiating a Rhetorical Mode (Rm). This is essentially a generative process, transforming multiple elements into a single mode.

Symbolic representation:

$${\mathrm{Rm}}_i=\mathsf{\Phi }(\left\{{\mathrm{th}}_a\right\},\left\{{\mathrm{ft}}_b\right\},\{{\mathrm{qt}}_c,{\mathrm{ql}}_d,{\mathrm{fm}}_e\},\left\{{\mathrm{be}}_f\right\})$$

Explanation: $\mathsf{\Phi }$ represents a nesting function. It accepts non-empty sets of elements (denoted by $\left\{\right\}$) from various foundational axes and generates a specific instance of a rhetorical mode ${\mathrm{Rm}}_i$ through selection and integration.

For example, generating a mode of "describing the morphology of a mountain peak" can be represented as:

$${\mathrm{Rm}}_{\mathrm{Description}}=\mathsf{\Phi }(\left\{{\mathrm{Th}}_{\mathrm{Natural}\mathrm{Object}}\right\},\left\{{\mathrm{Ft}}_{\mathrm{Morphology}}\right\},\left\{\right\},\left\{{\mathrm{Be}}_{\mathrm{Language}}\right\})$$

Example: "K2 (Karakoram mountain range, located on the border between Pakistan and China) presents a steep pyramidal massif, with a perilous north face covered by glaciers, a straight southeastern ridge forming a sharp outline, its overall morphology reflecting intense erosion and complex geological structure."

Depending on the number of selected low-dimensional coordinates, the detail level and even expressive depth of a rhetorical mode can be flexibly adjusted.

For example, when defining a concept, we can start with the most direct definition, enrich the expression by adding its features, and further supplement different types of attributes, such as quantitative, qualitative, or formal attributes, making the definition increasingly complete and precise. Suppose we want to define a "smartphone" using a rhetorical mode:

- Basic definition: A smartphone is a mobile phone that can be carried around.

- Adding features: A smartphone is a mobile phone equipped with a touch screen, capable of installing applications, and connecting to the internet.

- Supplementing attributes:

• Quantitative attributes: Typically equipped with a 6-inch screen, 8GB RAM, and 4000mAh battery capacity.
• Qualitative attributes: Provides a smooth user experience and a highly personalized interface.
• Formal attributes: Uses an operating system based on Android or iOS, supporting multi-layer encryption protocols.

As the level of detail increases, the description of a "smartphone" evolves from a simple functional statement to encompass its technical characteristics and user experience, significantly enhancing the information content and expressive depth of the rhetoric.

5.3. High-Dimensional Mapping

High-dimensional mapping extends low-dimensional mapping by adding layers of complexity. It begins with mapping from rhetorical modes to cognitive functions (the middle mapping in the Pyramid Multilayer Mapping), and then further maps cognitive functions to epistemic purposes (the apex mapping in the Pyramid Multilayer Mapping).

5.3.1. Rhetorical Mode → Cognitive Function

High-dimensional mapping starts by treating cognitive function as the dependent variable and rhetorical modes as independent variables. A sequence of rhetorical modes is combined in an ordered manner to achieve a specific cognitive function. This can be symbolically represented as:

$$C{f}_j=\mathsf{\Psi }\left([R{m}_1,R{m}_2,\dots ,R{m}_n]\right)$$

where $\mathsf{\Psi }$ represents a nesting function that accepts an ordered list of rhetorical modes, which collectively embody a specific cognitive operation.

For example, achieving the "Integration/Synthesis" cognitive function may require a sequential application of rhetorical modes, such as:

• Definition (to establish key terms),
• Classification, Comparison, and Contrast (to categorize studies by dimensions and highlight consistencies and divergences),
• Cause and Effect (to explain mechanisms producing differences),
• Evidencing and Argumentation (to link evidence and build a cohesive argument).

5.3.2. Cognitive Function → Epistemic Purpose

The next step in high-dimensional mapping is to map cognitive functions to epistemic purposes. Taking cognitive functions as independent variables and epistemic purposes as dependent variables, this process can be symbolically represented as:

$$E{p}_j=\Omega \left([C{f}_1,C{f}_2,\dots ,C{f}_m]\right)$$

where $\Omega$ represents a nesting function that accepts an ordered list of cognitive functions, forming a sequence that collectively embodies a specific epistemic purpose.

For example, the "Teaching/Learning" epistemic purpose may involve the following cognitive function sequence:

• Observation, Identification, and Modeling (to activate prior knowledge and construct theoretical frameworks),
• Explanation, Juadgement, and Reflection (to deepen understanding and solidify learning outcomes).

In a classroom, this sequence helps guide the process of teaching and learning, facilitating the transition from passive observation to active, reflective learning.

5.3.3. Self-Mapping: Rhetorical Mode → Rhetorical Mode

Rhetorical modes can also undergo self-mapping, where higher-level modes embed lower-level modes to enhance expressiveness. For instance, embedding the Description mode within Narration enriches the narrative, allowing it to not only tell a story but also provide context and analysis.

An example could be a documentary where Narration serves as the framework, with embedded Description to reveal event backgrounds, Contrast to highlight differences, and Process Analysis to demonstrate event development. This transforms the narrative from mere storytelling to a tool for cognitive construction.

5.4. Full-Dimensional Mapping

Full-dimensional mapping involves constructing an element within a dependent variable axis by taking any of the high-dimensional axes (e.g., Rhetorical Mode, Cognitive Function, Epistemic Purpose) as the dependent variable, while considering elements from the other coordinate axes as independent variables.

For example, if the Rhetorical Mode axis is the dependent variable, elements from low-dimensional axes (e.g., Thing, Feature, Attribute) automatically become independent variables, while high-dimensional axes (e.g., Cognitive Function, Epistemic Purpose) also act as independent variables. This comprehensive mapping pushes rhetorical modes to a full-dimensional height, where rhetorical modes are not just goals but must also consider cognitive functions and epistemic purposes as constraints.

The symbolic expression for full-dimensional mapping is:

$$G{a}_j=\mathsf{\Pi }(t{h}_a,f{t}_b,q{t}_c,b{e}_f,R{m}_g,C{f}_h,E{s}_i)$$

where $\mathsf{\Pi }$ represents a nesting function that integrates elements from various axes to construct a fully integrated rhetorical mode.

Example: Full-Dimensional "Definition"

To illustrate full-dimensional mapping, let us revisit the definition of a "smartphone" from the previous low-dimensional example. By adding layers of cognitive functions (e.g., Explanation) and epistemic purposes (e.g., Problem-Solving), we can elevate the definition to a higher level.

Full-Dimensional Definition: "A smartphone is a mobile phone that can be carried around. It features a touch screen, supports applications, and connects to the internet. It typically includes a 6-inch screen, 8GB of RAM, and a 4000mAh battery capacity. Users experience smooth functionality and personalized interfaces. The device operates on Android or iOS and supports multi-layer encryption protocols. Functionally, it enables multitasking through dynamic scheduling, allowing users to seamlessly switch between tasks. Additionally, with GPS and real-time traffic data, it helps users avoid traffic and provides turn-by-turn navigation, solving everyday travel challenges."

In this full-dimensional example, the definition is enriched by cognitive functions such as Explanation (explaining multitasking) and epistemic purposes such as Problem-Solving (solving navigation issues). This elevates the original definition from a basic functional description to a multi-dimensional expression that incorporates not just the features of the object but also its use cases and cognitive implications.

6. The Potential of the Generalized Coordinate System

This section explores the number of coordinate points in the generalized coordinate system when mapping from low-dimensional to high-dimensional to full-dimensional, aiming to reveal the near-infinite modal possibilities generated by various combinations within the system. This quantification will help intuitively demonstrate the descriptive potential and expressive richness contained in the framework.

6.1. Generalized Modes Number (GMn)

The number of coordinate points corresponds to the total number of combinatorial possibilities, called the Generalized Modes Number (GMn), which varies depending on the chosen "dependent variable axis."

Definition: For a chosen dependent variable axis, its Generalized Modes Number (GMn) equals the product of the number of major ticks on all axes considered as independent variables.

According to the above definition, we can calculate the GMn for four core application layers, from basic to advanced, step by step.

Recall the number of major ticks for each axis given in Section 2: $K_1=5$, $K_2=7$, $K_3=3$, $K_4=5$, $K_5=4$, $K_6=7$, $K_7=18$, $K_8=14$, $K_9=8$, $K_{10}=5$.

Let

$$GMn\left(k_{max}\right)={\displaystyle \prod _{i=1}^{k_{max}}}{K}_i$$

The quantity $GMn\left(k_{max}\right)$ is shown in Table 12.

Table 12 quantifies the explosive growth of descriptive possibilities within the framework. As the mapping scope expands, progressively incorporating the Rhetorical Mode axis (Rm), the Cognitive Function axis (Cf), the Epistemic Purpose axis (Ep), and finally the Five-Level Expression Staircase (Es) as independent variables, the Generalized Modes Number escalates from 14,700 to nearly 148 million.

This exponential increase, governed by the product formula $GMn\left(k_{max}\right)$, means nearly infinite modal possibilities. It visually substantiates the role of the Ontological Axis (Rm) as a pivotal mediator: by combining low-dimensional elements (axes 1–6), it generates a vast, structured space of options that serve high-dimensional cognitive and epistemic purposes. The global GMn value (approximately 1.5$\times {10}^8$) defines the comprehensive combinatorial space available for computational text analysis under this model.

6.2. From Infinite Possibilities to Finite Purposes

The data of Generalized Modes reveals the fact that the modal number is extremely large. This fact, on one hand, illustrates the infiniteness of rhetorical mode expression; on the other hand, it indicates the need to consider finite goals in specific practice.

The combinatorial calculations based on major ticks have revealed a structurally hierarchical modal landscape of staggering scale: without considering minor tick refinements, the Generalized Modes Number for Rhetorical Modes has reached 14,700; the Cognitive Function modal number leaps to 264,600; the Epistemic Purpose modal number soars to 3,704,400; and the modal number for the Five-Level Expression Staircase reaches an astronomical 29,635,200. This is merely the combination of major ticks. If minor ticks under each major coordinate axis, with quantities not less than an order of magnitude of 10, are taken into account, the number of unique modes that can be generated for rhetorical modes alone will surpass ${10}^{11}$, while the modal numbers for the rest will move towards profound mathematical spaces difficult for human intuition to grasp.

This nearly infinite modal number is not a theoretical fiction but a faithful mapping of the infinite complexity of the world and cognition itself. It symbolizes four aspects:

1.

Infinite Details of the Objective World

All things exhibit infinite details due to their wide spectrum of kinds, multi-dimensional features, and triple attributes.

2.

Infinite Paths of Cognitive Processes

Human understanding and interpretation of things inherently have countless possible entry points and combinations of thought.

3.

Infinite Possibilities of Meaning Generation

Language and expression are not closed boxes but open, creative systems that can be infinitely combined and regenerated.

4.

Infinite Levels of Competency Advancement

From sensory capture to wisdom enlightenment, the ascent of expressive ability is a never-ending, finely describable process, not merely divisible into Sensory Level, Autonomous Expression Level, Academic Standard Level, Methodological Level, and Knowledge Enlightenment Level.

Therefore, the Generalized Modes Number is not a fixed list to be memorized but a coordinate system capable of navigating infinite possibilities, a powerful meaning generation engine.

A mechanical application of the Generalized Coordinate System of Rhetorical Modes is to create static classification tables, mechanically applying low-dimensional, high-dimensional, and full-dimensional mapping methods. Such mechanical application can be achieved with the help of artificial intelligence tools like large language models. The Generalized Coordinate System of Rhetorical Modes we propose provides a viable path for how future large language models might approach rhetorical modes.

However, in human practice, we need to jump out from static classification table mode and adopt a dynamic, dialectical method to construct rhetorical modes and cognitive functions. For this purpose, there are three directions to choose from.

1.

Forward Navigation

Forward navigation is the most direct application logic. We start from insights into things, features, and attributes, select and combine multimodal resources from the basic element library to form rhetorical modes, ultimately aiming to achieve specific cognitive functions and application purposes. This is the goal of low-dimensional and high-dimensional mapping.

2.

Reverse Navigation

However, this process can also operate in reverse: when we have clear intentions for cognitive functions and epistemic purposes, the coordinate system will guide us in reverse to discover or design the rhetorical modes and basic elements that best achieve these purposes. This is a method that can be considered for full-dimensional mapping.

3.

Omni-directional Navigation

This is precisely the essence of the "map and compass" metaphor: the map shows the whole picture, the compass guides the direction—only by combining the two can one chart their own course amidst infinite possibilities. We are no longer confined to irreversible thinking; reversible thinking allows us to navigate in various directions.

Regardless of the direction of navigation, achieving cognitive functions and purposes, and advancing along the Five-Level Expression Staircase, will be the yardstick by which we measure the height and depth of navigation, marking the competency progression from sensory depiction to knowledge creation.

7. Summary, limitation and Future Directions

The core contribution of this work lies in the construction of the Generalized Coordinate System of Rhetorical Modes, an integrative theoretical framework. This system unifies the operational logic of rhetorical modes across five dimensions: Acting Upon What (objects, features, attributes), Expressed With What (multimodal basic elements), What They Are (four categories of 18 core modes), How to Express (cognitive functions and epistemic purposes), and Who Expresses (the Five-Level Expression Staircase). By doing so, it elevates rhetorical modes from a static typology to a dynamic, navigable cognitive and practical operating system.

This coordinate system reveals the tripartite nature of rhetorical modes: they serve as formal discourse structures, functional cognitive tools, and strategic communicative acts. Its fundamental value lies in providing a set of "grammars of thought" that converge infinite expressive possibilities into a finite, structured, and analyzable space, guiding practitioners from unconscious use to methodological awareness.

This paper presents a complete theoretical coordinate system and paradigm, rather than an exhaustive enumeration of its near-infinite combinations. Its "finiteness" serves as the starting point for practical application. The primary future direction is to develop a formal, derivable, and programmable theory based on this Generalized Coordinate System. Specific pathways for future development include:

• Formal Modeling: Transforming the axes and their relationships into computable formal rules and constraint logics.
• Algorithmic Implementation: Exploring the integration of this formal system into computational architectures, such as Large Language Models (LLMs), enabling them to understand, generate, and evaluate complex rhetorical structures aligned with cognitive purposes.
• Tool Development: Building intelligent assistants for analysis, writing, and pedagogy based on the above model.

Ultimately, by empowering various applications with this structured "grammar of thought," this research aims to provide a solid theoretical foundation and a programmable pathway for enhancing the level of deep cognitive collaboration and creative expression between humans and machines.

Acknowledgements

This paper was initially drafted in Chinese. Large language models, including ChatGPT and DeepSeek, were utilized interchangeably to assist in translating the text into English, and to improve some text. HAN Min-en (Master student) and MA Xin-Yu (an intern) helped to design most of the figures.

Appendix A.1. Combinatory Structure

Definition: Combinatory Structure refers to the direct combination of basic elements (from Th, Ft, Qt, Ql, Fm, Be) across different independent variable dimensions at the lexical, phrasal, or simple sentence level to form a meaningful expression unit. This is the atomic operation for constructing all complex expressions.

Manifestation: Often manifested as multiple modifiers, compound noun phrases, or closely related short sentences.

Example: "The fitness trainer’s waist-to-hip ratio is 0.80." This sentence combines elements from three dimensions: Thing, Feature, and Quantitative Attribute.

• Thing (Th): Person among living organisms (fitness trainer)
• Feature (Ft): Morphology and composition (waist-to-hip ratio)
• Quantitative Attribute (Qt): Ratio and intensity (0.80)

Symbolic representation: When describing the combination of basic elements, use the multiplication symbol "|" to connect elements from different dimensions, emphasizing their cross-composite relationship. The general expression is $R{m}_k=\mathsf{\Phi }$, where $\mathsf{\Phi }=Th\left|Ft\right|Qt\left|Ql\right|Fm$.

For example: $R{m}_{Description}=\mathsf{\Phi }$, where $\mathsf{\Phi }=T{h}_{Human\&LivingBeing}\left|F{t}_{Morphology\&Composition}\right|Q{t}_{Ratio\&Intensity}$.

Function: Achieving precise reference and basic propositional statements. It is the most basic and frequently invoked operation in the base mapping function $\mathsf{\Phi }$, responsible for “gluing” discrete coordinate ticks into expression blocks with specific meanings. We focus on explaining core combinatory patterns such as "$Ft|Qt$" (Feature–Quantitative), "$Ft|Ql$" (Feature–Qualitative), "$Ft|Fm$" (Feature–Formal), etc.

Appendix A.2. Parallel Structure

Definition: Parallel Structure refers to multiple elements (Rm or Cf) of the same dimension being deployed sequentially in text or thought flow, with equal status and relatively independent semantics. They are related by jointly serving the same higher-level goal, akin to parallel circuits.

Symbolic representation: Use "$\left|\right|$" to connect elements in a sequence, emphasizing their parallel nature and cumulative relationship. For example:

$$C{f}_{Integration}=\mathsf{\Psi }\left(\right[R{m}_{Definition}\left|\right|R{m}_{Classification}\left|\right|R{m}_{Comparison}\left|\right|R{m}_{Argumentation}\left]\right)$$

$$E{p}_{Decision-Making}=\Omega \left(\right[C{f}_{Analysis}\left|\right|C{f}_{Evaluation}\left|\right|C{f}_{Prediction}\left]\right)$$

Function: Expanding breadth and coverage. Suitable for situations requiring multi-angle, multi-evidence, modular coverage of a complex topic.

Appendix A.3. Embedded Structure

Definition: Embedded Structure refers to the organic integration of one or more other elements (often from the same dimension) within the unfolding process of an element (Rm or Cf), forming a hierarchical and encapsulating relationship, like Russian nesting dolls.

Symbolic representation: Use "→" to denote embedding, with the main element on the left and the embedded element on the right. For example, "$R{m}_{Description}\to R{m}_{Comparison}$" means "comparing within description." Another example: "$C{f}_{Observation}\to C{f}_{Identification}$" means "identifying while observing." Sometimes, multiple nested structures can be formed, e.g.,

$$C{f}_{Integration}=\mathsf{\Psi }\left([R{m}_{Description}\to R{m}_{Contrast}\to R{m}_{Classification}]\right)$$

$$E{p}_{Decision-Making}=\Omega \left([C{f}_{Modeling}\to C{f}_{Evaluation}\to C{f}_{Prediction}]\right)$$

Function: Increasing depth and complexity. Suitable for situations requiring instant completion of multi-layer cognitive processing within a core operation, or making expression more precise and thinking more economical.

Appendix A.4. Comprehensive Example

A complex “Scientific Discovery” purpose ($E{p}_{ScientificDiscovery}$) realization can clearly demonstrate the coordination of the three structures:

$$\begin{array}{c}E{p}_{ScientificDiscovery}=\Omega \left(\right[(C{f}_{Observation}\to C{f}_{Identification})\hfill \\ \left|\right|C{f}_{Hypothesis}\\ \left|\right|\mathsf{\Psi }\left(\right[R{m}_{ExperimentalProcess}\\ \left|\right|(R{m}_{QuantitativeDescription}\left(Th\right|Ft\left|Qt\right)\to R{m}_{Comparison}\left)\right])\\ \left|\right|C{f}_{Inference}\\ \hfill \left|\right|C{f}_{Evaluation}\left]\right)\end{array}$$

This sequence indicates that scientific discovery begins with an observation activity embedded with identification, followed by the parallel proposal of hypotheses. It then achieves the “testing” function through a combination of two parallel modes: experimental procedures and data analysis (which is embedded with comparison). Finally, inference and evaluation are conducted in parallel. Moreover, the generation of each rhetorical mode itself relies on the combination of fundamental dimensional elements through a large number of combinatory structures at the underlying level.

By formalizing nesting relationships ($\mathsf{\Phi },\mathsf{\Psi },\Omega$) and the three logical structures ($|,||,\to$), pyramid mapping transforms from a theoretical diagram into an operational blueprint that can be parsed, designed, and replicated. This provides a key methodological bridge for readers to move from understanding the framework to actively constructing with it.