PFUSRC-098 Three Topological Forms of 3D Projection—From Passive Slices to Containerized Expressions of Dimensional Hierarchy

All structures in three-dimensional space are, in essence, projection traces left by four-dimensional flow variables under different conditions. However, “projection” is not a single mode. Based on the PFUSRC system, this paper establishes a taxonomy of three topological forms of 3D projection, strictly distinguishing three hierarchical levels—Passive Slices, Slice-Expressions, and Containerized Expressions—and demonstrates that there exists a strict hierarchical progression among them.Passive Slices are passive traces left by four-dimensional flow variables after local cessation or solidification, such as apple scars, crystal defects, and fossils. They are records of flow-variable “cessation.” Slice-Expressions are three-dimensional structures in which the operational process of four-dimensional flow variables is actively encoded and solidified, such as Möbius strips, biconical geometry, and DNA double helices. They are records of flow-variable “operation.” Containerized Expressions are projections of the global container properties of four-dimensional flow variables onto three-dimensional space, such as Klein bottles, biconical waist-rings, and black hole event horizons. The Klein bottle itself is a 4D topological container; all 3D physical models are dimensional projections of it, uniformly classified under Containerized Expressions. They are records of flow-variable “container.”This paper further demonstrates that the three projection forms correspond precisely to the hierarchical progression of “Flow-Variable Cessation → Flow-Variable Operation → Flow-Variable Container” within the PFUSRC system, aligning respectively with “Passive Slices,” “Slice-Expressions,” and “Containerized Expressions.” This paper provides a complete taxonomic framework for 3D projection phenomena within the PFUSRC system and establishes the theoretical foundation for subsequent research on cross-dimensional information encoding.