PFUSRC-116: On the Integration and Demarcation of Modern Mathematics——The PFUSRC Biconical Topology as the Boundary-Setting Framework for Mathematical Tools

Mainstream definitions of mathematics only provide functional descriptions without physical ontological foundations. They merely answer the research objects of mathematics yet fail to interpret its intrinsic ontological origin. Based on the PFUSRC 11-dimensional triple coaxial 45° biconical topological axiomatic system and inheriting the three-tier mathematical classification framework proposed in PFUSRC-112, this paper puts forward a core ontological proposition: mathematics is neither a purely subjective artificial symbolic system nor an independently existing Platonic ideal entity, but a structural record generated by multi-layer projection of the 11D global topological skeleton. Modern mathematics is divided into two major categories: topologically anchorable instrumental mathematics and purely formal axiomatic systems without objective topological carriers. This manuscript establishes two standardized topological judgment criteria to rigidly divide their applicable boundaries: 1. Topological Integration Criterion: The core structures of a mathematical branch can realize one-to-one mapping with layered 45° bicone topology; its core equations can degenerate into PFUSRC topological rheology tensors and four-dimensional flow-variable evolution equations under designated steady-state limits; it can output quantitative, falsifiable physical observational predictions on the 3D projection layer. 2. Topological Demarcation Criterion: The core concepts of a mathematical branch have no corresponding topological structural layers; its deduction relies on man-made conventions such as the Axiom of Infinity and Axiom of Choice without topological anchors; it only holds internal formal self-consistency and cannot generate testable physical inferences. In accordance with the above two sets of standards, this paper completes classification for mainstream mathematical branches: calculus, Euclidean geometry, linear algebra, differential equations, complex analysis, classical probability, classical mechanics, General Relativity and Quantum Mechanics all fall under topological integration as low-dimensional approximations under specific steady-state limits of the bicone topology. ZFC set theory, category theory, pure number theory without topological correspondence, abstract algebra lacking symmetric isomorphism and abstract topology without physical carriers only possess formal self-consistency and no ontological correspondence, thus subject to topological demarcation. This paper further clarifies the limited effective scope of modern mathematical tools: all existing mathematical frameworks only intercept local steady-state slices on the 3D projection layer and lose validity once exceeding steady-state constraints. The PFUSRC system is not a subordinate branch of mathematics, but an ontological boundary framework that defines the physical applicable range of all mathematical systems. This manuscript acts as the closure document that drives the PFUSRC system from cosmic topological description to mathematical ontological reconstruction, and provides standardized starting coordinates for subsequent dimensional ascent research within the PFUSRC series.