PFUSRC-150 From Complex Analysis to Tensor Analysis: The Universal Unfolding Protocol of the Topological One at the Projection Layer ——Topological Reassignment of Calculus, Complex Analysis, Vector Analysis, Functional Analysis, and Tensor Analysis Document No.: PFUSRC-150

Abstract

PFUSRC-056, PFUSRC-086, PFUSRC-086B and PFUSRC-086C separately completed the topological reconstruction of complex analysis, calculus, the Topological One, and tensor analysis, forming a closed theoretical loop for the ontology of mathematical analysis. This paper serves as their unified concluding monograph, integrating scattered research conclusions into an integrated topological map of mathematical analysis without adding new derivations or proofs. It elaborates the triple classification of mathematics, the definition and intrinsic properties of the Topological One, the topological redefinition of five major branches of mathematical analysis, and the core conclusion that mathematical analysis is the unfolding record of the Topological One on the projection layer rather than independent objective truth. The paper clarifies that calculus, complex analysis, vector analysis, functional analysis and tensor analysis all take the Topological One as their unified convergence endpoint instead of zero. It further distinguishes ontological, instrumental and abstract mathematics, points out the physical boundary of tensor dimensionality, and links the mathematical ontology framework with the L4-L7 hierarchical dynamics proposed in PFUSRC-148 and PFUSRC-149.