PFUSRC-118 On Discrete Line Segments——A Conceptual Transition from Geometric Objects to Topological Records

This paper completes a foundational conceptual paradigm shift: the reconstruction of the mathematical primitive “line segment”—from a geometric object as traditionally defined, to a high-dimensional topological projection record within the PFUSRC framework. Classical mathematics defines a line segment as a widthless one-dimensional continuous set of points, presupposing continuity as its inherent ontological property. Within the PFUSRC 11-dimensional triple coaxial 45° biconical topological framework, however, a line segment is fundamentally the trajectory record left by four-dimensional flow variables at the 3D macroscopic projection layer. Its apparent continuity is determined solely by observational resolution, not by any intrinsic ontological attribute. This paper formally establishes the Discrete Line Segment as an independent topological concept: a finite ordered set of topological anchor points governed by a unified phase evolution rule within the biconical topology. The Discrete Line Segment is not a coarse approximation of the continuous line segment; it is the authentic manifestation of the line-type topological record under finite observational resolution. The continuous line segment is merely the limiting appearance of the Discrete Line Segment as the number of observational sampling points tends to infinity. The paper further clarifies the hierarchical correspondence between discrete anchoring and continuous projection within the PFUSRC system, demonstrating that both the number “1” and the geometric line segment are merely symbolic markers of topological structures at the projection layer. As a direct conceptual extension of the Kakeya conjecture topological repositioning work in PFUSRC-117, this paper completes the ontological recalibration of the fundamental geometric unit, establishing a rigorous conceptual foundation for subsequent research in discrete topological dynamics and anchor-point combinatorial topology.