Axioms of Primordial Topology and Dynamic Emergence in New Number Theory — Universally Closed Proof Compatible with Modern Mathematics, Including the 1⁰ Background Element, 12 Generation Mechanism, Prime-Node Ontology, 4D Cosmic Expansion, and the Global Theorem of Prime-Driven Flowing Topological Spaces

This paper establishes a universally closed axiomatic foundation for the PFUSRC framework, fully compatible with modern mathematics and traditional prime number theory. It reveals the ontological roots of numbers by proving that the coupling of the numerical primitive element 1 and the background element 1⁰ generates the structure of 2. It redefines 0 not as absolute nothingness, but as a field, a coupling surface, and an interface. The ₁ drive enables the simultaneous expansion of 1D, 2D, 3D, and 4D space. The core contribution is the Global Theorem of Prime-Driven Flowing Topological Spaces, which proves that: (1) The primitive element 1 acts as a universal anchor, pairing with any prime number to form flowing, deformable, self-stable but non-globally closed topological spaces. (2) Prime 3 forms non-closed, self-stable flowing spaces. (3) Prime 5 is the minimal stable closed space in 4D topology. (4) Prime 11 is the globally optimal steady-state structure. (5) Prime 13 forms semi-flowing, long-lived structures incapable of sustaining stable galaxies. (6) 2D geometry is a projection, not a real physical space. Natural and cosmic observations, from octopuses and amoebas to irregular galaxies, are presented as direct evidence. The framework achieves a seven-fold logical closure, providing an unshakable foundation for the entire PFUSRC system.