PFUSRC-050 The Root of Mathematics: From Empty Set Dogma to Primordial Topological Ontology

This paper targets four foundational crises of mainstream mathematical formalism: the empty-set definition of zero, the circular symbolic proof of 1+1=2, irreducible residual terms in arithmetic operations like 1+2, and the logical trap of Russellian logicism. It exposes the core defect of modern formal mathematics: prioritizing artificial symbolic conventions over physical and topological ontological anchoring, reducing mathematical deduction to detached syntactic games. Based on the PFUSRC transcendental topological system and three primordial logical units (Primordial One, Coupling Zero, Background One), this paper reconstructs ontological definitions of core numerical symbols: 0 is redefined as coupling gap / 0-Interface, 1 as concrete existential Primordial One, and 2 as steady-state 45° bicone coupling structure. The work strictly deduces that 1+1=2 is not a man-made symbolic rule, but a necessary geometric outcome of equal radial expansion and axial convergence velocity within bicone topological manifolds. The residual term in arithmetic is re-interpreted as inherent information loss during the conversion from high-dimensional ontological topology to low-dimensional projection arithmetic, with a quantitative topological formula provided for residual magnitude. The paper systematically criticizes Russell’s logicism, which splits logic, mathematics and physics into disconnected self-consistent symbolic systems, creating a fundamental ontological schism. A comparative benchmark between Russellian set-theoretic formalism and PFUSRC topological ontology is established, proving that topological ontology rooted in cosmic steady-state structure is more primordial and fundamental than empty-set-based mathematical logic. This research completes the paradigm shift from conventional symbolic mathematics to ontological mathematics anchored in universal topological structure.