Generalized Analytical Pathway (GAP) and Generalized Research (GR)

1. Introduction

A paper usually consists of elements such as problem/purpose–method–result–conclusion (cf. Cargill & O’Connor, 2013). In a research paper adopting the IMRAD structure (Introduction, Methods, Results, and Discussion; see Sollaci & Pereira, 2004), the introduction generally contains the two elements of problem and purpose; the two major sections of Methods and Results respectively correspond to the two elements of method and result; and the discussion often gives the element of conclusion. The research process and paper organization also broadly unfold along the path of problem/purpose–method–result–conclusion.

More specifically, research usually starts from a problem worth solving, obtains results through the clarification of research objectives and ideas, the selection and implementation of research methods, and forms conclusions through the analysis, interpretation, and testing of results. The path of research is often analogized by the “hourglass model,” just as sand flows down through an hourglass.

In the hourglass model, researchers gradually narrow from a broader research background to a specific research problem, which corresponds to the wider upper part of the hourglass. In the Methods and Results sections, they enter the most specific research process and findings, which enters the narrowest part of the hourglass, indicating that research needs to focus on one point. Then, the Discussion and Conclusion expand again to the significance, scope of application, and further research directions of the results. This corresponds to the lower part of the hourglass, which becomes wider again.

This model can be traced back to the classic exposition of the structure of experimental research papers by Hill et al. (1982), who compared the overall structure of a research paper to an hourglass and pointed out that the introduction proceeds from general to specific, while the discussion proceeds in the opposite direction; Swales (1990), in his monograph on genre analysis, further introduced and promoted this metaphor, making it a common framework for describing the macro-structure of research papers. Subsequently, scholars such as Schulte (2003) and Derntl (2014) demonstrated the applicability of the hourglass model in scientific writing from different disciplinary perspectives. Fetters and Freshwater (2015) systematically applied and expounded this model in mixed methods research writing, explicitly distinguishing the “top of the hourglass” from the “middle of the hourglass,” and pointed out that the introduction of a mixed methods paper needs to present both content objectives and methodological objectives. Fetters (2020) further operationalized the hourglass design model in a dedicated chapter, making it a systematic guiding framework for mixed methods paper writing. From this structure, problem, motivation, method, result, and conclusion are not isolated components of a paper, but are connected through a series of research transformations and integrations.

Standard research methods and academic writing traditions usually further explain how this process unfolds: entering a research problem from interest and phenomena, clarifying research objectives and argumentation ideas, selecting and justifying research methods, collecting and analyzing materials, forming results, and then drawing conclusions through discussion, interpretation, and testing (Booth et al., 2024; Creswell & Creswell, 2023; O’Leary, 2021). Scientific methods and specific research traditions are diverse, and specific research methods are selectable (Hepburn & Andersen, 2021; McCombes & George, 2025). Critical thinking and questioning research further point out that questions determine the organization of subsequent evidence, reasoning, and conclusions, and that questioning and follow-up questioning are the basic driving forces for deepening research (Browne & Keeley, 2017; Elder & Paul, 2019). These studies explain from different angles how research raises questions, selects methods, and forms conclusions, but when facing a specific research object, one may further ask: from which aspects can analysis enter, which analytical contents can be selected, and what relations can be established among different analytical contents. The systematic unfolding of these possibilities helps to expand research horizons and enlarge the research space.

To expand research horizons and enlarge the research space, Wu (2026c) proposed the generalized analytical method based on the generalized coordinate system. The generalized coordinate system (GCS) is a coordinate system composed of analytical object O, analytical feature F, and analytical result $\mathrm{Res}\equiv (S,Q,X)$, where S, Q, and X respectively denote qualitative characteristic, quantitative characteristic, and knowledge formulation. Wu (2026c) expanded analytical feature F as

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

where $F_1$–$F_8$ successively denote morphology, composition, state, dynamics, function, relation, origin, and history. Each analytical feature $F_i$ has several concrete analytical contents $F_{ij}$, which constitute analytical positions.

The generalized analytical method GAM includes local analytical method $\mathrm{LA}\left(F^s\right)$ and cross analytical method $\mathrm{CA}\left(F^m\right)$, namely

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

In the local analytical method, one analytical feature $F^s\equiv F_i$ is selected for analysis, including two cases:

• analysis directed at one concrete analytical content $F_{ij}$, namely $\mathrm{LA}\left(F^s\right)\equiv F^s\to F_{ij}$;
• analysis directed at multiple concrete analytical contents $\left\{F_{ij}\right\}$ within the same analytical feature, namely $\mathrm{LA}\left(F^s\right)\equiv F^s\to \left\{F_{ij}\right\}$.

In the cross analytical method, multiple analytical features $F^m\equiv \{F_i\mid i\in I^m\}$ are selected for analysis, where $I^m\equiv \{i_1,i_2,\dots ,i_m\}$, $m\ge 2$.

The cross analytical method also has two cases:

• selecting two different analytical features $F_i$ and $F_j$ for analysis, namely $\mathrm{CA}\left(F^m\right)\equiv F_i\leftrightarrow F_j$, $i\ne j$;
• selecting two concrete analytical contents $F_{ik}$ and $F_{jl}$ for analysis, namely $\mathrm{CA}\left(F^m\right)\equiv F_{ik}\leftrightarrow F_{jl}$, $i\ne j$.

In brief, the generalized analytical method (GAM) takes analytical feature F as the analytical entry, concrete analytical content $F_{ij}$ as the analytical position, and local analytical method LA and cross analytical method CA as the core analytical operations. LA unfolds within a single analytical feature, while CA establishes relations between different analytical features. Compared with traditional research methods, GAM is superordinate; traditional research methods provide selectable implementation pathways for completing concrete analytical contents.

However, the generalized analytical method based on the generalized coordinate system mainly answers questions such as “from which analytical feature to enter,” “which concrete analytical contents to select,” and “how to establish relations among different analytical positions,” but has not further indicated along what pathway analysis or research can be carried out, nor has it provided a suitable framework for judging when research can terminate or move to the next round of analysis.

In other words, determining analytical positions and their relations does not equal forming a complete analytical process; how different analytical positions are connected in sequence, how they are advanced, and how to enter new analysis from existing analytical results still need to be explained.

For this purpose, this paper proposes the Generalized Analytical Pathway (GAP) based on the generalized coordinate system and the generalized analytical method. It takes GAM as a core link, organizes analytical objects, analytical features, concrete analytical contents, analytical operations, and analytical results into a traceable, testable, and re-enterable pathway, and discusses its advancement, convergence, termination, and transformation conditions, thereby forming the generalized research method in the Generalized Research–Expression Model proposed by Wu (2026b).

The remainder of this paper is organized as follows. Section 2 introduces the generalized adjustment method from initial analytical object to formal analytical object. Section 3 establishes the main-node analytical path MAP, gives four basic analytical paths, and introduces local analysis LA and cross analysis CA. Section 4 discusses re-anchoring, namely the method of setting analytical features and analytical results again as analytical objects. Section 5 comprehensively introduces the generalized analytical pathway GAP and generalized research method GR.

2. From Initial Object to Formal Analytical Object: Generalized Adjustment

In the generalized analytical method based on the generalized coordinate system, analytical objects can be divided into three types: existing analytical object $O^e$, composite analytical object $O^c$, and prospective analytical object $O^p$. Regardless of which type, at the beginning of the current research, it can first serve as an initial analytical object $O_0$.

To make the initial analytical object suitable for the current research, it also needs to be adjusted according to the research task. Common adjustment modes include:

• delimitation (object delimitation): clarify the scope of the current research object and the content temporarily not included;
• simplification: temporarily ignore factors weakly related to the current problem and retain the main content;
• modeling: use a substitute object that can retain key features for research;
• decomposition: split a relatively complex whole into several smaller local objects;
• combination: establish associations among several existing objects according to the current research task, so that they participate in subsequent analysis as a whole;
• granularity adjustment: change the analytical level at which the object is located, so that it is neither too broad nor too fine.

The initial analytical object $O_0$, after adjustment, formally enters subsequent research as analytical object O. Finding an initial analytical object with research value is itself an important part of research problem formation, while object adjustment further makes the initial analytical object a formal analytical object suitable for the current research task.

Here we introduce generalized adjustment, denoted as ADJ:

$$\mathrm{ADJ}=\{{\mathrm{ADJ}}_1,{\mathrm{ADJ}}_2,\dots ,{\mathrm{ADJ}}_6\},$$

where:

• ${\mathrm{ADJ}}_1$: delimitation;
• ${\mathrm{ADJ}}_2$: simplification;
• ${\mathrm{ADJ}}_3$: modeling;
• ${\mathrm{ADJ}}_4$: decomposition;
• ${\mathrm{ADJ}}_5$: combination;
• ${\mathrm{ADJ}}_6$: granularity adjustment.

Generalized adjustment is not completed entirely by intuition, but invokes the analytical features $F_i$ and concrete analytical contents $F_{ij}$ provided by the generalized analytical method GAM (Wu, 2026c) as the basis for object adjustment. The generalized analytical method consists of local analytical method and cross analytical method:

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

Therefore, it can be represented as:

$$O={\mathrm{ADJ}}_h(O_0;\mathrm{GAM}),h=1,\dots ,6.$$

Here, GAM does not directly analyze $O_0$ into O, but provides ADJ with invocable analytical features and concrete analytical contents. In other words, generalized adjustment is responsible for completing:

$$O_0\to O,$$

while after formally entering the subsequent generalized analytical pathway, GAM undertakes:

$$O\to \mathrm{GAM}\to \mathrm{Res}.$$

It should be particularly noted that the $F_i/F_{ij}$ invoked in generalized adjustment are not necessarily the same as the $F_i/F_{ij}$ actually adopted in the subsequent generalized analytical pathway. The former serves the determination and adjustment of the object, while the latter serves further analysis of the formal analytical object.

2.1. Delimitation (ADJ ₁)

Delimitation means using relevant analytical features $F_i$ and their concrete analytical contents $F_{ij}$ to delimit the research scope of the initial analytical object.

For example, one may start from the morphology $F_1$ of the analytical object, whose typical analytical contents include:

• $F_{11}=$ shape;
• $F_{12}=$ external contour;
• $F_{13}=$ boundary morphology;
• $F_{14}=$ surface topography;
• $F_{15}=$ spatial distribution.

Example based on external contour $F_{12}$: suppose the initial analytical object is an aircraft wing. One may first delimit its external contour, and if necessary further delimit its boundary morphology. One may further invoke existing qualitative or quantitative characteristics for delimitation, for example delimiting the external contour as streamlined, the planform as swept wing, or further delimiting the sweep angle as ${75}^{\circ }$.

Therefore, the pathway of generalized delimitation is:

$$O_0\to \mathrm{select}\mathrm{relevant}F/F_{ij}\to \mathrm{delimit}\mathrm{scope}\to O.$$

2.2. Simplification (ADJ ₂)

Simplification means using relevant analytical features $F_i$ and their concrete analytical contents $F_{ij}$ to simplify the initial analytical object, retaining the main content directly related to the current research task.

For example, one may start from the composition $F_2$ of the analytical object, whose typical analytical contents include:

• $F_{21}=$ basic unit;
• $F_{22}=$ constituent category;
• $F_{23}=$ hierarchical composition;
• $F_{24}=$ functional module;
• $F_{25}=$ material composition;
• $F_{26}=$ phase composition;
• $F_{27}=$ sample composition;
• $F_{28}=$ stage composition;
• $F_{29}=$ amount of matter or scale of composition.

Example based on material composition $F_{25}$: suppose the initial analytical object is air. Dry air, by volume fraction, mainly consists of about $78.08\%$ nitrogen, about $20.95\%$ oxygen, about $0.93\%$ argon, and about $0.04\%$ carbon dioxide, and also contains trace gases such as neon, methane, helium, hydrogen, and argon; actual atmosphere also contains water vapor whose content varies with the environment.

When studying aircraft aerodynamics, one may, according to research needs, ignore trace components and approximate air as a mixed gas mainly composed of nitrogen and oxygen; further, it may be equivalent to a single ideal gas with an average molar mass of about $28.97\mathrm{g}/\mathrm{mol}$.

Therefore, simplification is not arbitrarily deleting object content, but retaining the main content according to the analytical features and concrete analytical contents of current research concern, and forming a formal analytical object suitable for the current research.

The pathway of generalized simplification is:

$$O_0\to \mathrm{select}\mathrm{relevant}F/F_{ij}\to \mathrm{retain}\mathrm{main}\mathrm{content}\to \mathrm{simplify}\to O.$$

2.3. Modeling (ADJ ₃)

Modeling means using relevant analytical features $F_i$ and their concrete analytical contents $F_{ij}$ to transform the initial analytical object into a substitute object that can retain the key content required by the current research. This modeling is what is commonly called modeling.

For example, one may start from the morphology $F_1$ of the analytical object, whose typical analytical contents include:

• $F_{11}=$ shape;
• $F_{12}=$ external contour;
• $F_{13}=$ boundary morphology;
• $F_{14}=$ surface topography;
• $F_{15}=$ spatial distribution.

Example based on spatial distribution $F_{15}$: Suppose the initial analytical object is a celestial body such as the sun, a planet, or a satellite. When studying the motion of a celestial system, these celestial bodies with actual size, shape, and internal structure may be modeled as point masses, retaining only their spatial positions and information such as mass required by the current research, while no longer considering their specific shape, external contour, boundary morphology, and surface topography. Thus, real celestial bodies are replaced by a point-mass model suitable for the current research task.

Therefore, modeling is not simply deleting part of the initial object, but re-determining the formal analytical object with a substitute object according to the analytical features and concrete analytical contents of current research concern.

The pathway of generalized modeling is:

$$O_0\to \mathrm{select}\mathrm{relevant}F/F_{ij}\to \mathrm{determine}\mathrm{model}-\mathrm{retained}\mathrm{content}\to \mathrm{model}\to O.$$

2.4. Decomposition (ADJ ₄)

Decomposition means using relevant analytical features $F_i$ and their concrete analytical contents $F_{ij}$ to split a relatively complex initial analytical object into several smaller objects.

For example:

• according to composition $F_2$, it may be decomposed by components, constituent units, or hierarchical composition;
• according to function $F_5$, it may be decomposed by functional modules;
• according to dynamics $F_4$, it may be decomposed by different stages or processes.

Suppose the initial analytical object is an aircraft. According to composition, it may be decomposed into objects such as wing, fuselage, tail, and engine; according to function, it may be decomposed into functional units such as flight control, propulsion, and load bearing.

The pathway of generalized decomposition is:

$$O_0\to \mathrm{select}\mathrm{relevant}F/F_{ij}\to \mathrm{decompose}\to \{O_1,\dots ,O_m\}.$$

If subsequent research selects only one of the decomposed objects as the formal analytical object, then further:

$$\{O_1,\dots ,O_m\}\to O_k=O.$$

Therefore, decomposition can transform one initial analytical object into multiple objects that can respectively enter subsequent analysis.

2.5. Combination (ADJ ₅)

Combination means, for multiple initial analytical objects $\{O_1,\dots ,O_n\}$, using relevant analytical features $F_i$ and their concrete analytical contents $F_{ij}$ to reorganize several of them into one whole object.

Combination may be carried out according to different analytical features such as relation, function, composition, and dynamics.

For example, the heart, blood vessels, and local circulation may originally be taken separately as analytical objects; if the current research concerns the blood circulation function they jointly undertake, these objects may be combined into a composite analytical object $O^c$, namely the human circulatory system.

Therefore, combination is not simply increasing the number of objects, but re-determining multiple objects as one unified analytical object according to the current research task.

The pathway of generalized combination is:

$$\{O_1,\dots ,O_n\}\to \mathrm{select}\mathrm{relevant}F/F_{ij}\to \mathrm{combine}\to O^c\to O.$$

2.6. Granularity Adjustment (ADJ ₆)

Granularity adjustment means using relevant analytical features $F_i$ and their concrete analytical contents $F_{ij}$ to adjust the analytical level at which the initial analytical object is located, so that it is neither too broad nor too fine.

For example, if studying the overall function of an aircraft, “aircraft” may be taken as the formal analytical object; if studying the aerodynamic characteristics of the wing, the object needs to be adjusted to “wing”; if studying the local pressure distribution of an airfoil, it may further be adjusted to a certain local region.

Therefore, the core of granularity adjustment is not simply changing the physical size of the object, but making the level at which the object is located match the analytical content of current research concern.

The pathway of generalized granularity adjustment is:

$$O_0\to \mathrm{select}\mathrm{relevant}F/F_{ij}\to \mathrm{adjust}\mathrm{object}\mathrm{level}\to \mathrm{granularity}\mathrm{adjustment}\to O.$$

In summary, generalized adjustment, through the six basic operations of delimitation, simplification, modeling, decomposition, combination, and granularity adjustment, transforms the initial analytical object $O_0$ into a formal analytical object O suitable for the current research task. Its core lies not in adding new analytical results to the object, but in re-determining the object’s scope, composition form, substitute form, and analytical level with the help of the analytical features and concrete analytical contents provided by GAM. After the formal analytical object O is determined, it enters the subsequent generalized analytical pathway.

3. Main-Node Analytical Path MAP

3.1. Basic Analytical Paths

The basic analytical path of the generalized coordinate system can simultaneously face single or multiple analytical objects and single or multiple analytical features. For convenience of distinction, we introduce the following notation.

Single analytical object: $O^s\equiv O$, where O denotes an analytical object, which may be an existing analytical object $O^e$ (namely an analytical object already clarified before or at the beginning of research), a composite analytical object $O^c$ (namely an analytical object formed by combining multiple analytical objects), or a prospective analytical object $O^p$ (an achievement-type or work-type object that the current research attempts to design, form, improve, or realize; it is not treated as an analytical result but as an analytical object).

Multiple analytical objects: $O^m\equiv \{O_1,\dots ,O_p\}$, $p\ge 2$, where p is the number of analytical objects, and the most common case is $p=2$. Here, each analytical object may be an existing analytical object $O^e$, a composite analytical object $O^c$, or a prospective analytical object $O^p$.

Single analytical feature: $F^s\equiv F_i$, where $F_i$ is one feature selected from the eight analytical features $F_1$–$F_8$: Morphology, composition, state, dynamics, function, relation, origin, and history.

Multiple analytical features: $F^m\equiv \{F_i\mid i\in I^q\}$, where $I^q\equiv \{i_1,i_2,\dots ,i_q\}$, $q\ge 2$, and $F^m$ is two or more analytical features selected from $F_1$–$F_8$.

The basic analytical path is used to prescribe the basic skeleton by which an analytical object reaches an analytical result via analytical features. Specifically, the basic analytical path prescribes the path by which a single analytical object $O^s$ or multiple analytical objects $O^m$ pass through a single analytical feature $F^s$ or multiple analytical features $F^m$ and enter the analytical result $\mathrm{Res}\equiv (S,Q,X)$. Here, Res denotes the type of result that may be formed or invoked, and it is not required that all three types of results be obtained simultaneously each time.

According to the number of analytical objects and analytical features selected, the basic analytical path covers the following four cases:

• Single-object single-feature analysis ${\mathrm{AP}}_{1,1}$. For a single analytical object $O^s$, one analytical feature $F^s$ is selected from $F_1$–$F_8$ for analysis, obtaining the corresponding analytical result Res. For example, for a certain bird wing, analyze its morphological feature and obtain the qualitative result that it has a “streamlined” shape, forming the analytical path

  $${\mathrm{AP}}_{1,1}:O^s\to F_1\to S.$$
  Here: $O^s=$ bird wing; $F^s=F_1$ (morphology); $S=$ “streamlined.”
• Single-object multi-feature analysis ${\mathrm{AP}}_{1,q}$. For a single analytical object $O^s$, multiple analytical features $F^m$ are selected from $F_1$–$F_8$ for analysis, obtaining the corresponding analytical result Res. For example, for a certain bird wing, simultaneously analyze its morphology and function, obtaining two qualitative results: “streamlined” and “able to generate lift,” forming the analytical path

  $${\mathrm{AP}}_{1,2}:O^s\to \{F_1,F_5\}\to \{S_1,S_2\}.$$
  Here: $O^s=$ bird wing; $F^m=\{F_1,F_5\}$; $S_1=$ “streamlined”; $S_2=$ “able to generate lift.”
• Multi-object single-feature analysis ${\mathrm{AP}}_{p,1}$. For multiple analytical objects $O^m$, one analytical feature $F^s$ is selected from $F_1$–$F_8$ for analysis, obtaining the corresponding analytical result Res. For example, simultaneously analyze the morphological features of a bird wing and an aircraft wing, and find that both have streamlined external contours, forming the analytical path:

  $${\mathrm{AP}}_{2,1}:\{O_1,O_2\}\to F_1\to \{S_1,S_2\}.$$
  Here: $O^m=\left\{\mathrm{bird}\mathrm{wing},\mathrm{aircraft}\mathrm{wing}\right\}$; $F^s=F_1$ (morphology); $S_1=$ “bird wing is streamlined”; $S_2=$ “aircraft wing is streamlined.”
• Multi-object multi-feature analysis ${\mathrm{AP}}_{p,q}$. For multiple analytical objects $O^m$, multiple analytical features $F^m$ are selected from $F_1$–$F_8$ for analysis, obtaining the analytical result Res. For example, simultaneously analyze the morphology and function of a bird wing and an aircraft wing. For the bird wing, one may obtain “streamlined” and “able to generate lift”; for the aircraft wing, one may also obtain “streamlined” and “able to generate lift,” forming the analytical path:

  $${\mathrm{AP}}_{2,2}:\{O_1,O_2\}\to \{F_1,F_5\}\to \{S_1,S_2,S_3,S_4\}.$$
  Here: $O^m=\left\{\mathrm{bird}\mathrm{wing},\mathrm{aircraft}\mathrm{wing}\right\}$; $F^m=\{F_1,F_5\}$; $S_1=$ “bird wing is streamlined”; $S_2=$ “bird wing is able to generate lift”; $S_3=$ “aircraft wing is streamlined”; $S_4=$ “aircraft wing is able to generate lift.”

Here, ${\mathrm{AP}}_{2,2}$ actually forms four basic “object–analytical feature” positions:

$$O_1-F_1,O_1-F_5,O_2-F_1,O_2-F_5.$$

Generally, p analytical objects and q analytical features can form at most $p\times q$ basic analytical positions.

3.2. Formation of the Main-Node Analytical Path

The four basic analytical paths prescribe the basic skeleton by which analytical object O passes through a single analytical feature $F^s$ or multiple analytical features $F^m$ and enters qualitative characteristic S, quantitative characteristic Q, and knowledge formulation X. These four basic analytical paths may be represented as:

$${\mathrm{AP}}_{1,1}\equiv O^s\to F^s\to \mathrm{Res},$$

$${\mathrm{AP}}_{1,q}\equiv O^s\to F^m\to \mathrm{Res},$$

$${\mathrm{AP}}_{p,1}\equiv O^m\to F^s\to \mathrm{Res},$$

$${\mathrm{AP}}_{p,q}\equiv O^m\to F^m\to \mathrm{Res}.$$

To construct the main-node analytical path, we introduce the generalized analytical method GAM presented by Wu (2026c). The generalized analytical method includes local analytical method LA and cross analytical method CA.

In the generalized analytical method, in addition to determining analytical feature $F_i$, it is also necessary to determine the concrete analytical contents $F_{ij}$ under each analytical feature $F_i$. For example, for morphological feature $F_1$, the concrete analytical contents are:

$$F_{11}=\mathrm{shape},F_{12}=\mathrm{external}\mathrm{contour},F_{13}=\mathrm{boundary}\mathrm{morphology},F_{14}=\mathrm{surface}\mathrm{topography},F_{15}=\mathrm{spatial}\mathrm{distribution}.$$

The local analytical method LA enters $F_{ij}$ or $\left\{F_{ij}\right\}$ from $F^s\equiv F_i$, namely

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

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

The cross analytical method CA relates different analytical features or analytical contents of different analytical features, namely

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

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

The Main-node Analytical Path (MAP): Introducing LA and CA into the four analytical paths ${\mathrm{AP}}_{1,1}$, ${\mathrm{AP}}_{1,m}$, ${\mathrm{AP}}_{n,1}$, and ${\mathrm{AP}}_{n,m}$ forms the Main-node Analytical Path.

Thus, the Main-node Analytical Path $\mathrm{MAP}\left(C\right)$ may be represented as

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

where O generally refers to the single or multiple analytical objects currently entering analysis; $\mathrm{GAM}\equiv \{\mathrm{LA}\left(F^s\right),\mathrm{CA}\left(F^m\right)\}$; $\mathrm{Res}\equiv (S,Q,X)$.

Corresponding to the four basic analytical paths ${\mathrm{AP}}_{1,1}$, ${\mathrm{AP}}_{1,m}$, ${\mathrm{AP}}_{n,1}$, and ${\mathrm{AP}}_{n,m}$, the Main-node Analytical Path has the following four cases:

$$\mathrm{MAP}-1\left(C\right)\equiv O^s\to \mathrm{LA}\left(F^s\right)\to \mathrm{Res},$$

$$\mathrm{MAP}-2\left(C\right)\equiv O^s\to \mathrm{CA}\left(F^m\right)\to \mathrm{Res},$$

$$\mathrm{MAP}-3\left(C\right)\equiv O^m\to \mathrm{LA}\left(F^s\right)\to \mathrm{Res},$$

$$\mathrm{MAP}-4\left(C\right)\equiv O^m\to \mathrm{CA}\left(F^m\right)\to \mathrm{Res}.$$

Here, MAP-1, MAP-2, MAP-3, and MAP-4 respectively correspond to the four basic cases of single-object single-feature, single-object multi-feature, multi-object single-feature, and multi-object multi-feature.

4. Re-Anchoring and Recursive Re-Anchoring

In a study or a paper, whether it is an analytical feature or analytical content currently positioned at the intermediate level, or an analytical result, it may need further analysis and discussion. Re-anchoring converts these roles requiring further analysis into new analytical objects and invokes the Main-node Analytical Path for analysis.

4.1. Re-Anchoring and Analytical Pathway Transformation

Re-anchoring means that in a new analytical task, the F, S, Q, or X coordinate content in the current analysis is re-determined as analytical object $O^{\prime }$, and the Main-node Analytical Path MAP is invoked with $O^{\prime }$ as the new starting point.

Let the re-anchoring operator be denoted by Ra:

$$\mathrm{Ra}\left(C\right)\equiv C\to O^{\prime }\to \mathrm{MAP},$$

where $C\in \{F,S,Q,X\}$ denotes the coordinate content currently being re-anchored, and $O^{\prime }$ denotes the content that obtains the role of analytical object after re-anchoring.

The following are several cases of re-anchoring:

• $\mathrm{Ra}\left(F\right)\equiv F\to O^{\prime }\to \mathrm{MAP}$, namely re-anchoring an analytical feature as an analytical object, for example studying “morphology.”
• $\mathrm{Ra}\left(F_{ij}\right)\equiv F_{ij}\to O^{\prime }\to \mathrm{MAP}$, namely re-anchoring an analytical content as an analytical object, for example studying “shape.”
• $\mathrm{Ra}\left(S_m\right)\equiv S_m\to O^{\prime }\to \mathrm{MAP}$, namely re-anchoring a qualitative characteristic as an analytical object, for example studying “streamlined.”
• $\mathrm{Ra}\left(Q\right)\equiv Q\to O^{\prime }\to \mathrm{MAP}$, namely re-anchoring a certain quantitative characteristic or a set of quantitative results as an analytical object, for example studying “a set of anomalous pressure data.”
• $\mathrm{Ra}\left(X_n\right)\equiv X_n\to O^{\prime }\to \mathrm{MAP}$, namely re-anchoring a knowledge formulation as an analytical object, for example studying “the Pythagorean theorem.”

In the introduction of a paper, what is determined is often an existing analytical object, a composite analytical object, or a prospective analytical object.

A research method itself may also become an analytical object. If an existing method is originally a research object, it may serve as an existing analytical object; if the research goal is to form a new method, it may serve as a prospective analytical object; if a certain method is originally only a pathway for realizing the current analytical task, but its accuracy, robustness, or scope of application needs to be evaluated during use, it may also be temporarily re-anchored as $O^{\prime }$. This role transformation does not change its original nature as a research method.

After a paper gives research results, when discussing the results, they may be regarded as new analytical objects and the generalized analytical method may be invoked for analysis.

Re-anchoring changes the role of a certain coordinate content in the current research, not its original knowledge nature. After a qualitative characteristic is re-anchored as $O^{\prime }$, it does not thereby lose its original meaning as a qualitative characteristic, but merely becomes the center around which a new round of analysis revolves. The same applies to other coordinates.

For example, “approximately spherical” may initially be a qualitative characteristic discovered when studying the Earth. When the research question turns to examining what manifestations “approximately spherical” itself has, what differences exist at different positions, and how the degree of deviation from a sphere is characterized, this qualitative characteristic may again become a new analytical object $O^{\prime }$.

An existing analytical object may also be re-determined in a new research task in terms of its scope, boundary, or analytical role and enter analysis again, but this case does not involve transformation between coordinate types; it belongs to re-anchoring of the analytical object itself, $O\to O^{\prime }$. For example, the first round of research:

$$\mathrm{Earth}O\to \mathrm{analyze}\mathrm{morphology}\to \mathrm{obtain}``\mathrm{approximately}\mathrm{spherical}’’S.$$

Later, when the research question changes, “Earth” is again taken as the object to study its internal composition or history:

$$\mathrm{Earth}O\to \mathrm{Earth}{O}^{\prime }\to \mathrm{new}\mathrm{round}\mathrm{of}\mathrm{analysis}.$$

Here O and $O^{\prime }$ still point to the same real object, the Earth; $O^{\prime }$ merely denotes a new analytical anchor.

For another example, if the analytical result is a mathematical relation, then this mathematical relation may be re-anchored as an analytical object and further analyzed according to “analytical object → local or cross analysis → analytical result.”

Re-anchoring is very common in actual research, although it is not expressed as “re-anchoring.” The analytical results of a research achievement do not have to be preserved only as the results of the current research; when new questions require further examination of these contents, they may again become $O^{\prime }$.

4.2. Reduction of Analytical Pathways

Re-anchoring enables objects from different sources to obtain a unified role. Regardless of whether it was originally O, F, S, Q, or X, as long as it needs to be examined as an object in subsequent analysis, it can be uniformly converted into $O^{\prime }$, without establishing new object categories and analytical rules for different types of content. This effect forms an obvious reduction in the generalized analytical pathway: The new $O^{\prime }$ can continue to use the unified Main-node Analytical Path $\mathrm{MAP}\left(C\right)$. This reduces the basic rules required when knowledge of different types and levels enters analysis.

The Main-node Analytical Path:

$$\mathrm{MAP}\left(C\right)\equiv O\to \mathrm{GAM}\to \mathrm{Res},\mathrm{GAM}\equiv \{\mathrm{LA}\left(F^s\right),\mathrm{CA}\left(F^m\right)\}.$$

Once a new research object $O^{\prime }$ is obtained through re-anchoring, it is invoked again:

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

namely $\mathrm{Ra}\left(C\right)\equiv C\to O^{\prime }\to \mathrm{MAP}$, $C\in \{F,S,Q,X\}$.

If, each time a new research center appears, another set of analytical systems and operating rules had to be established, then the basic operations would continuously increase with knowledge content and knowledge levels. Re-anchoring uniformly converts these research centers from different sources into new analytical objects $O^{\prime }$, so that they can subsequently continue to unfold according to the unified path MAP. Therefore, research content and knowledge levels can continuously increase, while the basic coordinate roles and analytical rules that need to be mastered do not have to increase synchronously.

Therefore, re-anchoring is both an advanced coordinate operation and an important mechanism for the generalized coordinate system to further achieve minimal reduction. It enables newly formed knowledge to re-enter the same analytical framework, repeatedly explore the open research space with limited basic pathways, and continuously transform existing results into possible new research starting points.

4.3. Recursive Re-Anchoring

If the new analytical result formed after one re-anchoring again becomes a research object, a second re-anchoring occurs. Such a process may be repeated, forming recursive re-anchoring.

For example, an experiment obtains an anomalous quantitative characteristic. Initially, this anomaly is only a quantitative result obtained by a certain object under specific conditions; if subsequent research turns to asking what distribution this anomaly has, under what conditions it appears, whether it has stable characteristics, and whether it can be explained, then “this anomaly” becomes a new analytical object, forming recursive re-anchoring: A set of $Q\to$ discover anomaly → anomaly becomes $O^{\prime }\to$ obtain $X\to X$ again becomes $O^{\prime }$.

In general, recursive re-anchoring may be represented as:

$${\mathrm{Ra}}^n:{\mathrm{Res}}^{(n-1)}\to O^{\left(n\right)}\to {\mathrm{MAP}}^{\left(n\right)},$$

where the superscript n denotes the iteration step.

Recursive re-anchoring does not mean that research needs to continue indefinitely. Whether to carry out the next round depends on whether the new research can produce meaningful new positions, new characteristics, new relations, or new knowledge formulations. If continued operation mainly repeats existing content and the newly added information is no longer sufficient to change the current understanding, then it may stop under the current research purpose and granularity. Re-anchoring provides the ability to continue entering the knowledge space, rather than prescribing that research must reach some fixed endpoint.

Recursive re-anchoring may continue indefinitely and be actively used to discover new knowledge and promote the continuous development of a research direction or discipline.

This process of continuing from existing knowledge into unknown positions may also be understood with the help of the distinction between “known and unknown.” Rumsfeld (2002) once proposed “known knowns,” “known unknowns,” and “unknown unknowns”: Content that has been mastered and whose relevance is recognized roughly corresponds to “known knowns”; questions that one is aware need answering but has not yet obtained answers to are close to “known unknowns”; and some questions have not even entered the current scope of cognition and may be regarded as “unknown unknowns.” Here, this distinction is not directly equated with the generalized coordinate system, but is borrowed to explain that the unknown faced by research is not limited to questions that have already been explicitly raised.

Logan (2009) further applied this framework to scientific research, pointing out that scientific exploration usually starts from known unknowns that one is aware of, while accidental discoveries may transform previously unrecognized unknowns into new, known unknowns that can continue to be studied. Similar knowledge boundary issues have also been used to discuss knowledge and decision-making under uncertainty (Daase & Kessler, 2007), and certain unknown unknowns that can be identified in advance in complex projects (Ramasesh & Browning, 2014). From the perspective of the generalized coordinate system, changing the analytical object, analytical feature, concrete analytical content, or coordinate combination, as well as re-anchoring newly appearing S, Q, or X, may gradually reveal content that originally did not enter the current research horizon and transform it into new research objects and questions.

5. Generalized Analytical Pathway and Generalized Research

5.1. Generalized Analytical Pathway

The Generalized Analytical Pathway (GAP) is an analytical pathway composed of three consecutive links: Generalized adjustment, Main-node Analytical Path, and re-anchoring. It may be represented as:

$$\mathrm{GAP}\left(C\right)\equiv \mathrm{ADJ};\mathrm{MAP}\left(C\right);\mathrm{Ra},$$

where ADJ is used to adjust the initial analytical object into a formal analytical object; $\mathrm{MAP}\left(C\right)$ is used to form analytical results from the formal analytical object via the generalized analytical method; Ra is used to re-anchor existing coordinate content as a new analytical object and cause research to enter the next round of GAP. Here,";" denotes the sequential connection of three different functional links in GAP.

$\mathrm{MAP}\left(C\right)$ internally contains the local analytical path and cross analytical path of GAM.

The starting point of the generalized analytical pathway is generalized adjustment of the initial analytical object:

$$O={\mathrm{ADJ}}_h(O_0;\mathrm{GAM}),h=1,\dots ,6.$$

ADJ includes delimitation, simplification, modeling, decomposition, combination, and granularity adjustment.

Therefore, the basic structure of GAP consists of three links: Generalized adjustment ADJ, Main-node Analytical Path MAP, and re-anchoring Ra, where MAP further contains the internal paths of the generalized analytical method GAM.

To distinguish different types of paths and relations, this paper introduces different arrow symbols:

• ⇒: Overall operational path between main nodes;
• → or ↔: Internal analytical relation;
• ↦: Role transformation, namely re-anchoring.

• Main-node Analytical Path MAP.

This level establishes the main-node path from analytical object O to generalized analytical method GAM and then to analytical result Res, namely:

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

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

Here, “⇒” denotes the main path, O denotes the analytical object, LA or CA denotes local analysis or cross analysis of the analytical object, and 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 Analytical 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 Main-node Analytical Path mainly reflects the overall direction and basic node relations of the generalized analytical pathway.

2.

Internal paths within GAM.

The generalized analytical method consists of local analytical method LA and cross analytical method CA. 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.

Therefore, the internal paths of GAM are a further unfolding of the generalized analytical method node in MAP.

3.

Re-anchoring cyclic path.

After one generalized analysis forms 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 enter the generalized analytical pathway again, namely:

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

Here “⇒” denotes role transformation, and “⇒” denotes that the transformed analytical object enters the Main-node Analytical Path again.

Further, recursive re-anchoring may also be invoked, namely:

$${\mathrm{Ra}}^n:{\mathrm{Res}}^{(n-1)}\mapsto O^{\left(n\right)}\Rightarrow {\mathrm{MAP}}^{\left(n\right)}.$$

Therefore, after one MAP is completed, Ra may be used to enter the next round of MAP, thereby forming a continuous analytical process.

5.2. Generalized Research Method

Generalized research method, or Generalized research (GR) for short, means research conducted using the Generalized Analytical Pathway GAP.

In brief, generalized research is research conducted using $\mathrm{GAP}\left(C\right)$, namely:

$$\mathrm{GR}\equiv \mathrm{GAP}\left(C\right).$$

The basic operational structure of $\mathrm{GAP}\left(C\right)$ is:

$$\mathrm{GAP}\left(C\right)\equiv \mathrm{ADJ};\mathrm{MAP}\left(C\right);\mathrm{Ra},$$

where the main path is:

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

Around this main path, generalized research has corresponding pathways at the three positions of analytical object, generalized analytical method, and analytical result.

• Generalized adjustment path at the object end.

Before formally entering the Main-node Analytical Path, the initial analytical object $O_0$ may form a formal analytical object O through generalized adjustment:

$$O={\mathrm{ADJ}}_h(O_0;\mathrm{GAM}),h=1,\dots ,6.$$

Therefore, ADJ acts on the object end of the main path to complete the transformation from initial analytical object to formal analytical object.

2.

Internal GAM path at the method end.

After the formal analytical object O enters the Main-node Analytical Path, analysis is carried out via the generalized analytical method GAM. GAM internally further includes local analytical path and cross analytical path, for example:

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

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

Therefore, the internal GAM path acts on the method end of the main path to determine from which analytical features and concrete analytical contents analysis enters, and how to establish relations among different analytical positions.

3.

Re-anchoring path at the result end.

When the F, S, Q, or X in the analytical result Res needs to become a new research center, it may be converted into a new analytical object through re-anchoring:

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

Therefore, Ra acts on the result end of the main path to convert existing analytical content or analytical results into new analytical objects, causing them to enter MAP again.

In addition, it is emphasized again that the $F_i/F_{ij}$ invoked in generalized adjustment are not necessarily the $F_i/F_{ij}$ invoked by the generalized analytical method in the subsequent Main-node Analytical Path.

In summary, in generalized research, the initial analytical object forms a formal analytical object through generalized adjustment; the formal analytical object enters MAP, forms analytical results via GAM, and may, when needed, re-enter a new GAP through re-anchoring. This continuous operation constitutes the basic process of generalized research.

Generalized research is called generalized not only because it is built on the generalized coordinate system and the generalized analytical method, but also because it can run through different stages of research and papers.

• Raising research questions.

The research question in the introduction may itself be the result of one round of generalized research. Researchers may take existing literature, existing knowledge, existing research results, or observed phenomena as the original analytical object, analyze them through the generalized analytical method, and thereby discover some qualitative characteristics, such as:

• deficiencies in existing research;
• contradictions;
• anomalies;
• gaps;
• relations not yet established.

Based on these qualitative characteristics, a new initial analytical object may be formed, generalized adjustment may again be invoked to form a formal analytical object, and a new GAP may be entered.

2.

Research on research methods.

A research method itself may also become an analytical object. When selecting, evaluating, improving, or establishing a research method, an existing method, method system, or method to be established may be taken as the analytical object, and GAP may be invoked to analyze its morphology, composition, state, dynamics, function, relation, origin, or history, forming corresponding qualitative characteristics, quantitative characteristics, or knowledge formulations, in order to judge the rationality of the method and provide guidance for its use.

Therefore, a concrete research method may serve both as an implementation means for completing a certain analytical task and as an analytical object converted into generalized research.

3.

Discussion of research results.

The S, Q, or X obtained by research does not necessarily mean that the research process has ended. When these analytical results need further discussion, they may be converted into new analytical objects through re-anchoring, and GAP may be invoked again, namely analyzing their morphology, composition, state, dynamics, function, relation, origin, history, etc., further obtaining new analytical results and forming conclusions.

Therefore, the discussion of research results may be regarded as another round of generalized research conducted with existing analytical results as new objects, and the further formed results may become deeper research results or conclusions.

4.

Further research on other coordinate contents..

At any stage of the research process, analytical feature F, concrete analytical content $F_{ij}$, and already formed S, Q, or X, as long as they need to become a new research center, may be converted into new analytical objects through re-anchoring and re-enter GAP.

Therefore, generalized research does not require research to always revolve around the initially determined object, but allows the research center to continuously transform as knowledge is formed. Limited basic coordinates and analytical pathways can thereby repeatedly enter new research content.

Generalized research solves what is studied, from where analysis enters, what is analyzed, how results are formed, and how research continues to advance. However, research also needs to be further transformed into academic expression that can be understood, communicated, and tested. The generalized analytical pathway itself does not directly prescribe how these research contents should be organized and expressed.

The Generalized Research–Expression Model GREM proposed by Wu (2026b) further unifies the research process and the expression process as:

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

where $\mathrm{GAP}\left(C\right)$ provides research content and its analytical pathway, Md is responsible for organizing and expressing these contents with discourse modes, and $\kappa$ denotes operational factors affecting the concrete expression process.

Therefore, $\mathrm{GAP}\left(C\right)$ constitutes the research-side core of GREM; on this basis, introducing discourse mode Md and operational factors $\kappa$, generalized research further enters the Generalized Research–Expression Model.