## 4 Materials and Methods
### Extenics-Based Identification Primitive Modeling of Coil Manufacturing Process Information
#### 4.4.1 All Elements Information Fusion Based on Extensible Petri Net Model
The extensible element model mentioned above serves as an example of an identifiable traceability unit. 2. Identifiable Traceability Scenarios (Site, abbreviated as \"\(s\)\" below)  
The manufacturing process of coil products often involves complex variations in location, time, and element information. Information traceability may involve multiple traceability items and traceability indicators. In an identification resolution traceability environment consisting of multiple traceability indicator items (such as within a specific space, within a specific time frame, and within a specific production process), several basic units may serve as input traceability units (\(u\)) and interact to generate several new traceability units, forming more complex traceability scenarios. Traceability in the manufacturing process of coil products must be based on the identification resolution code of each physical entity and traceability unit. This code must be unique, capable of accompanying the entire lifecycle of traceability entities, and able to distinguish any different traceability entities at any time, in any place, and in any scenario. Furthermore, even after the disappearance of traceability entities, their uniqueness representation must remain valid for a considerable period. By analyzing the transformation principles of traceability information, theoretically, traceability activities can be classified into three types of transformations: traceability scenario acquisition transformation, leading traceability unit acquisition transformation, and subsequent traceability unit acquisition transformation. 1. Traceability Scenario Acquisition Transformation \(\Gamma\)  
The transformation of obtaining traceability scenarios based on traceability units as conditions are achieved, where '\(h\)' has two options (0 or 1), indicating whether the transformation needs to obtain the pioneer traceability scenario or the follower traceability scenario of traceability unit 'u'. When \(h=0\), it indicates obtaining the pioneer traceability scenario of the trace unit. When \(h=1\), it refers to obtaining the follower traceability scenario of the trace unit. The mathematical expression is as follows:  
\[\Gamma(h,u)= \begin{cases}S^{prioneer},h=0\\ S^{follower},h=1\end{cases}  \tag{6}\]
2. Pioneer Traceability Unit Acquisition Transformation \(P\)  
Assume \(u_{k}\) is a trace unit. If it is a source trace unit (indicated as true), then it has no pioneer trace unit. If \(u_{k}\) is a non-source trace unit (indicated as false), then a pioneer trace unit exists. The mathematical expression is as follows:  
\[P(u_{k})= \begin{cases}\varnothing,true\\ S^{prioneer}(u_{k}),false\end{cases}  \tag{7}\]From the definition of transformation \(P\), the following relationship holds for non-source traceability unit \(u_{k}\), (where \(\Gamma^{-1}\) is the inverse transformation of \(\Gamma\)). \[P(u_{k})= S^{\text{prioneer}}(u_{k})= \Gamma^{-1}(0,u_{k})(u_{k}) \tag{8}\]  
(3) Follower Traceability Unit Acquisition Transformation \(F\)  
Similarly, if \(u_{k}\) is a source trace unit (indicated as true), then it has no follower trace unit. If \(u_{k}\) is a non-source trace unit (indicated as false), then a follower trace unit exists. The mathematical expression is as follows:  
\[F(u_{k})=\left\{ \begin{aligned} &\varnothing,true\\ & S^{follower}(u_{k}),false\end{aligned} \right. \tag{9}\]  
From the definition of transformation \(F\), the following relationship holds for the non-terminal traceability unit \(u_{k}\):  
\[F(u_{k})= S^{\text{follower}}(u_{k})= \Gamma(1,u_{k})(u_{k}) \tag{10}\]  
According to the basic configuration, the traceability scenario types of coil production process information can be categorized into three types: linear, disassembly, and merging. (1) Linear Type  
The fundamental type is represented by a traceability scenario \"\(s\)\" with a fan-in of 1 and a fan-out of 1, referred to as a linear traceability scenario, as shown in Figure 7.  
As illustrated in Figure 5, \((u_{1},1)\) and \((u_{2},1)\) are two traceability units.